\begin{theorem}\label{theorem: intro}
	Consistently from a measurable cardinal, there are two $\sigma$-complete ultrafilters $\mathcal{V}, \mathcal{W}$ such that $\mathcal{V}$ is Lipschitz below $\mathcal{W}$, but $\mathcal{V}$ and $\mathcal{W}$ are Ketonen-incomparable. Furthermore, the same is also consistent with the assumption that the Weak Ultrapower Axiom holds.
\end{theorem}

The study and analysis of the various connections between $\sigma$-complete ultrafilters plays a central role in the theory of large cardinals. Building on this perspective, Goldberg introduced and studied the Ultrapower Axiom (UA), which asserts
that for every pair of $\sigma$-complete ultrafilters, $U,W$, there are $\sigma$-complete ultrafilters $W^*\in M_U$ and $U^*\in M_W$ such that $M^{M_U}_{W^*} =M^{M_W}_{U^*}$ and $j^{M_U}_{W^*}\circ j_U = j^{M_{W}}_{U^*}\circ j_{W}$.\footnote{For a $\sigma$-complete ultrafilter $U$, $M_U$ denotes its ultrapower model, and $j_U\colon V\to M_U$ denotes the corresponding elementary embedding. If $U\in N$ for some inner model $N$ of $V$, we denote by $(M_U)^N$ and $j^{N}_U \colon N\to (M_U)^N$ the corresponding ultrapower and elementary embedding over $N$.}
Goldberg’s analysis of UA has led to a series of deep structural results about the set-theoretic universe. One key discovery is that, under UA, the class of $\sigma$-complete ultrafilters is well-ordered by a natural ordering called the Ketonen order (see Definition~\ref{Def: Ketonen order} below). Goldberg further showed that UA is equivalent to the linearity of the Ketonen order.

In his work, Goldberg identified that Ketonen comparability between ultrafilters implies determinacy of certain infinite games, reminiscent of the Lipschitz order used in descriptive set theory to compare subsets of the Cantor space (see Definition~\ref{Def: Lipschitz order on ultrafilters} below). He proved that, under UA, the Ketonen order and the Lipschitz order coincide when restricted to $\sigma$-complete ultrafilters, and raised the question whether it is consistent for the two orders to disagree (see \cite[Question 9.2.10]{GoldbergUABook}). Our main result gives a positive answer to this question:

\begin{definition}[The Ketonen order\footnote{The modern formulation of the Ketonen order is due to Goldberg, building on Ketonen’s earlier work, which treated only weakly normal ultrafilters \cite{Ketonen1972strongcompactnesscardinalsins}.}]\label{Def: Ketonen order}
	Let $U,W$ be $\sigma$-complete ultrafilters. We say that $U$ is Ketonen below $W$, and denote $U<_k W$, if and only if one of the following equivalent conditions hold:
	\begin{enumerate}
		\item There exists $I\in W$ and a sequence $\la U_\xi \colon \xi\in I \ra$ of $\sigma$-complete ultrafilters, such that each $U_\xi$ concentrates on $\xi$, and, for every $X\subseteq \kappa$ (where $\kappa$ is the underlying ordinal of $U$),
		$$X\in U \iff \{ \xi\in I \colon X\cap \xi \in U_\xi \}\in W.$$
		\item There exists a $\sigma$-complete ultrafilter $U^*\in M_W$ and an elementary embedding $k\colon M_U\to M^{M_W}_{U^*}$ such that $k\circ j_U = j^{M_W}_{U^*}\circ j_W$, and $k\left( [Id]_U \right)< j_{U^*}\left( [Id]_W \right)$.
	\end{enumerate}

\end{definition}

\begin{definition}[The Lipschitz order on ultrafilters]\label{Def: Lipschitz order on ultrafilters}
Let $U,W\subseteq \mathcal{P}(\kappa)$ be ultrafilters on $\kappa$. We say that $U$ is Lipschitz below $W$, and denote $U<_L W$, if one of the following equivalent conditions hold:
\begin{itemize}
	\item player I has a winning strategy in $G_{\kappa}(W,U)$.
	\item there exists a super-Lipschitz function $f\colon \mathcal{P}(\kappa)\to \mathcal{P}(\kappa)$ that reduces $U$ to $W$, in the sense that $f^{-1}[W] = U$.
\end{itemize}

\end{definition}

\begin{abstract}
In his study of the Ultrapower Axiom (UA), Goldberg revealed a connection between UA and the determinacy of certain games that witness Lipschitz reducibility between ultrafilters. In particular, he analyzed the relationship between the Ketonen and Lipschitz orders—two natural extensions of the Mitchell order from normal measures to arbitrary $\sigma$-complete ultrafilters—and proved that the Lipschitz order extends the Ketonen order. He further observed that under UA the two orders coincide. Goldberg asked if it's consistent that the orders differ from each other. We show that the answer is positive. In fact, even the Weak Ultrapower Axiom does not imply that the Ketonen and Lipschitz orders coincide.
\end{abstract}

The study and analysis of the various connections between $\sigma$-complete ultrafilters plays a central role in the theory of large cardinals. Building on this perspective, Goldberg introduced and studied the Ultrapower Axiom (UA), which asserts
that for every pair of $\sigma$-complete ultrafilters, $U,W$, there are $\sigma$-complete ultrafilters $W^*\in M_U$ and $U^*\in M_W$ such that $M^{M_U}_{W^*} =M^{M_W}_{U^*}$ and $j^{M_U}_{W^*}\circ j_U = j^{M_{W}}_{U^*}\circ j_{W}$.\footnote{For a $\sigma$-complete ultrafilter $U$, $M_U$ denotes its ultrapower model, and $j_U\colon V\to M_U$ denotes the corresponding elementary embedding. If $U\in N$ for some inner model $N$ of $V$, we denote by $(M_U)^N$ and $j^{N}_U \colon N\to (M_U)^N$ the corresponding ultrapower and elementary embedding over $N$.}
Goldberg’s analysis of UA has led to a series of deep structural results about the set-theoretic universe. One key discovery is that, under UA, the class of $\sigma$-complete ultrafilters is well-ordered by a natural ordering called the Ketonen order (see Definition~\ref{Def: Ketonen order} below). Goldberg further showed that UA is equivalent to the linearity of the Ketonen order.

In his work, Goldberg identified that Ketonen comparability between ultrafilters implies determinacy of certain infinite games, reminiscent of the Lipschitz order used in descriptive set theory to compare subsets of the Cantor space (see Definition~\ref{Def: Lipschitz order on ultrafilters} below). He proved that, under UA, the Ketonen order and the Lipschitz order coincide when restricted to $\sigma$-complete ultrafilters, and raised the question whether it is consistent for the two orders to disagree (see \cite[Question 9.2.10]{GoldbergUABook}). Our main result gives a positive answer to this question:

The \textit{Weak Ultrapower Axiom} is obtained from UA by removing the requirement that $j^{M_U}_{W^*}\circ j_U = j^{M_{W}}_{U^*}\circ j_{W}$. Models of the Weak UA+$\neg$UA were constructed in \cite{benhamouGoldberg2024applications} and more recently in \cite{kaplan2025number}. We will rely on the construction from \cite{kaplan2025number} in the proof of Theorem \ref{theorem: intro}.

\begin{definition}[The Ketonen order\footnote{The modern formulation of the Ketonen order is due to Goldberg, building on Ketonen’s earlier work, which treated only weakly normal ultrafilters \cite{Ketonen1972strongcompactnesscardinalsins}.}]\label{Def: Ketonen order}
 Let $U,W$ be $\sigma$-complete ultrafilters. We say that $U$ is Ketonen below $W$, and denote $U<_k W$, if and only if one of the following equivalent conditions hold:
 \begin{enumerate}
 \item There exists $I\in W$ and a sequence $\la U_\xi \colon \xi\in I \ra$ of $\sigma$-complete ultrafilters, such that each $U_\xi$ concentrates on $\xi$, and, for every $X\subseteq \kappa$ (where $\kappa$ is the underlying ordinal of $U$),
 $$X\in U \iff \{ \xi\in I \colon X\cap \xi \in U_\xi \}\in W.$$
 \item There exists a $\sigma$-complete ultrafilter $U^*\in M_W$ and an elementary embedding $k\colon M_U\to M^{M_W}_{U^*}$ such that $k\circ j_U = j^{M_W}_{U^*}\circ j_W$, and $k\left( [Id]_U \right)< j_{U^*}\left( [Id]_W \right)$.
 \end{enumerate}

The study of the Ketonen order is especially interesting in the context of Goldberg's Ultrapower Axiom. Recall that the \textit{Ultrapower Axiom} (UA) is the assertion that for every pair of $\sigma$-complete ultrafilters, $U,W$, there are $\sigma$-complete ultrafilters $U^*\in M_W$ and $W^*\in M_U$ such that $M^{M_U}_{W^*} =M^{M_W}_{U^*}$, and $j^{M_U}_{W^*}\circ j_U = j^{M_{W}}_{U^*}\circ j_{W}$. The \textit{Weak Ultrapower Axiom} is obtained from UA by removing the requirement that $j^{M_U}_{W^*}\circ j_U = j^{M_{W}}_{U^*}\circ j_{W}$.

We proceed and define the Lipschitz order between $\sigma$-complete ultrafilters. For that, we describe the Lipschitz game of length $\kappa$, $G_{\kappa}(W,U)$, where $W,U$ are subsets of $\mathcal{P}(\kappa)$ for an ordinal $\kappa$. The game is being held between two players, I and II, and consists of $\kappa$ stages. On the $i$-th stage ($i<\kappa$), Player I chooses a bit $a(i)\in \{0,1\}$, and Player II responds with a bit $b(i)\in \{0,1\}$. Since Player I moves first at any stage, they are aware of the sequences $\la a(j) \colon j<i \ra, \la b(j) \colon j<i \ra$ constructed so far; Player II moves second, being aware of the values of $\la a(j) \colon j\leq i \ra, \la b(j) \colon j<i \ra$. After $\kappa$ rounds, the players have constructed a pair of subsets of $\kappa$,
$$ a = \{ i<\kappa \colon a(i) = 1 \} $$
$$ b = \{ i<\kappa \colon b(i) = 1 \}. $$
Player II wins if $a\in W\leftrightarrow b\in U$. Otherwise, Player I wins.

The Lipschitz order is a strict partial order on $\sigma$-complete ultrafilters (see \cite[Corollary 3.4.29]{GoldbergUABook}). To the best of our knowledge, it is still not known whether the Lipschitz order on $\sigma$-complete ultrafilters must be well-founded (but it is well-founded under UA, see Corollary \ref{Corollary: UA implies that Ketonen is Lipschitz} below). We remark that the Lipschitz order on arbitrary subsets of $\mathcal{P}(\kappa)$ is ill-founded in models of AC (see \cite[Theorem 3.4.30]{GoldbergUABook}); however, in this paper, we focus only on the Lipschitz order restricted to $\sigma$-complete ultrafilters. We freely refer to it as the “Lipschitz order,” without mentioning the restriction to pairs of ultrafilters.
We proceed and prove that the Lipschitz order extends the Ketonen order. We provide a proof directly from the definitions given above (for an alternative justification, see Proposition \ref{Proposition: equivalent characterization of Lipschitz and Ketonen}).

The \textit{Weak Ultrapower Axiom} is obtained from UA by removing the requirement that $j^{M_U}_{W^*}\circ j_U = j^{M_{W}}_{U^*}\circ j_{W}$. Models of the Weak UA+$\neg$UA were constructed in \cite{benhamouGoldberg2024applications} and more recently in \cite{kaplan2025number}. We will rely on the construction from \cite{kaplan2025number} in the proof of Theorem \ref{theorem: intro}.

The structure of this paper is as follows:
\begin{itemize}
	\item In Section 2 we define the Ketonen and Lipschitz orders and outline their basic properties; all the results in this section are due to Goldberg.
	\item In Section 3 we show that, consistently, the Ketonen and Lipschitz orders do not coincide (however, this is not done in a model of the Weak UA). See Theorem \ref{Theorem: Separating Lipschitz from Ketonen the Easy way}.
	\item In Section 4 we sketch the basic properties of forcing with  nonstationary support products.
	\item In Section 5, we separate the Ketonen and Lipschitz orders in a model of the Weak UA, thereby completing the proof of Theorem \ref{theorem: intro}.
\end{itemize}

\textbf{Acknowledgments:} The author deeply thanks Gabe Goldberg and Eilon Bilinsky for many conversations on the subject of this paper.

We will rely on the construction from \cite{kaplan2025number} in the proof of Theorem \ref{theorem: intro}. The paper then proceeds by: (1) defining the Ketonen and Lipschitz orders and outlining their basic properties (Section 2); (2) showing, consistently, that the Ketonen and Lipschitz orders do not coincide (not in a model of the Weak UA) (Section 3, Theorem \ref{Theorem: Separating Lipschitz from Ketonen the Easy way}); (3) sketching basic properties of forcing with nonstationary support products (Section 4); and (4) separating the Ketonen and Lipschitz orders in a model of the Weak UA, "thereby completing the proof of Theorem \ref{theorem: intro}" (Section 5).

We will rely on the construction from \cite{kaplan2025number} in the proof of the main theorem. The paper then proceeds by: (1) defining the Ketonen and Lipschitz orders and outlining their basic properties; (2) showing, consistently, that the Ketonen and Lipschitz orders do not coincide (not in a model of the Weak UA). We first prove the following theorem. \begin{theorem}\label{Theorem: Separating Lipschitz from Ketonen the Easy way}
	Assume $V= L[U]$, $\kappa$ is the unique measurable cardinal, $\po$ is the forcing notion described above and $G,G^*, U_0,U_1$ are as above. Denote $\vc = U_0$ and $\wc$ the measure derived from $j^{V[G]}_{(U_1)^2}$ using the ordinal $j_U(\kappa)+\kappa$ as a seed. Then $\vc<_{L} \wc$ but $\vc, \wc$ are Ketonen incomparable.
\end{theorem} (3) sketching basic properties of forcing with nonstationary support products; and (4) separating the Ketonen and Lipschitz orders in a model of the Weak UA, thereby completing the proof of the main theorem.

Question:
Assume the consistency of a measurable cardinal. For ultrafilters \(U,W\subseteq \mathcal P(\kappa)\) on an ordinal \(\kappa\), write \(U<_L W\) if Player I has a winning strategy in the length-\(\kappa\) game \(G_\kappa(W,U)\), where the players build sets \(a,b\subseteq\kappa\) by alternating bits and Player II wins exactly when \(a\in W\leftrightarrow b\in U\). For \(\sigma\)-complete ultrafilters \(U\) and \(W\), write \(U<_k W\) if there exist \(I\in W\) and a sequence \(\langle U_\xi:\xi\in I\rangle\) of \(\sigma\)-complete ultrafilters, each \(U_\xi\) concentrating on \(\xi\), such that for every \(X\subseteq\kappa\) (where \(\kappa\) is the underlying ordinal of \(U\)),
\[
X\in U\iff \{\xi\in I: X\cap \xi\in U_\xi\}\in W.
\]
Two ultrafilters are Ketonen-incomparable if neither is below the other in this order. The Weak Ultrapower Axiom says that for every pair of \(\sigma\)-complete ultrafilters \(U,W\), there are \(\sigma\)-complete ultrafilters \(U^*\in M_W\) and \(W^*\in M_U\) such that \(M^{M_U}_{W^*}=M^{M_W}_{U^*}\). Which existence statement is relatively consistent under these assumptions?

Relative to a measurable cardinal, it is consistent that there exist two \(\sigma\)-complete ultrafilters \(\mathcal V,\mathcal W\) such that \(\mathcal V<_L\mathcal W\) but \(\mathcal V\) and \(\mathcal W\) are Ketonen-incomparable; moreover, it is also consistent, relative to a measurable cardinal, that such ultrafilters exist in a model satisfying the Weak Ultrapower Axiom.

Choices:
B. Relative to a measurable cardinal, it is consistent that there exist two \(\sigma\)-complete ultrafilters \(\mathcal V,\mathcal W\) such that \(\mathcal V<_k\mathcal W\) but \(\mathcal V\not<_L\mathcal W\); moreover, it is also consistent, relative to a measurable cardinal, that such ultrafilters exist in a model satisfying the Weak Ultrapower Axiom.
C. Relative to a measurable cardinal, it is consistent that there exist two \(\sigma\)-complete ultrafilters \(\mathcal V,\mathcal W\) such that \(\mathcal V<_L\mathcal W\) and \(\mathcal V\neq \mathcal W\); moreover, it is also consistent, relative to a measurable cardinal, that such ultrafilters exist in a model satisfying the Weak Ultrapower Axiom.
D. Relative to a measurable cardinal, it is consistent that whenever \(\mathcal V,\mathcal W\) are \(\sigma\)-complete ultrafilters with \(\mathcal V<_L\mathcal W\), then \(\mathcal V<_k\mathcal W\); moreover, it is also consistent, relative to a measurable cardinal, that this implication holds in a model satisfying the Weak Ultrapower Axiom.
E. Relative to a measurable cardinal, it is consistent that there exist two \(\sigma\)-complete ultrafilters \(\mathcal V,\mathcal W\) such that \(\mathcal V<_L\mathcal W\) and \(\mathcal V,\mathcal W\) are Ketonen-incomparable; moreover, every model in which such ultrafilters exist must fail the Weak Ultrapower Axiom.

direction-of-extension between Lipschitz and Ketonen

existential separation replaced by universal coincidence

compatibility with Weak Ultrapower Axiom


\label{thmA}
Let $M$ be a type $\mathrm{III}_1$ factor with separable predual and with trivial bicentralizer. Let $\theta \in \operatorname{Aut}(M)$ be a state preserving automorphism. Then the following conditions are equivalent:
\begin{enumerate}
	\item $\theta$ has infinite order in $\mathrm{Out}(M)$;
	\item there exists $u \in \mathcal U(M)$ such that $\operatorname{Ad}(u)\circ\theta$ is state preserving and ergodic.
\end{enumerate}

\begin{thmA}\label{thmA}
Let $M$ be a type $\mathrm{III}_1$ factor with separable predual and with trivial bicentralizer. Let $\theta \in \Aut(M)$ be a state preserving automorphism. Then the following conditions are equivalent:
\begin{enumerate}
	\item $\theta$ has infinite order in $\mathrm{Out}(M)$;
	\item there exists $u \in \mathcal U(M)$ such that $\Ad(u)\circ\theta$ is state preserving and ergodic.
\end{enumerate}
\end{thmA}

Cocycle perturbations of group actions provide a useful method for modifying dynamical
properties while preserving the underlying von Neumann algebraic structure.
In the case of type $\mathrm{II}_1$ factors, Marrakchi and Vaes \cite{MV23} proved that for every
outer action of a countable amenable group, one can find a 1-cocycle such that the
perturbed action becomes ergodic, and moreover that such cocycles form a dense $G_\delta$
subset of the cocycle space.
We refer to such a cocycle as an \emph{ergodic cocycle}.
This settles the ergodicity problem for cocycle perturbations in the type $\mathrm{II}_1$ setting,
and in particular implies that for a single automorphism, the existence of an ergodic cocycle
is equivalent to the outerness of all its nonzero powers.

The purpose of this paper is to extend the above framework to the \emph{state preserving}
setting for type $\mathrm{III}$ factors, where state preserving means that the action
leaves a faithful normal state invariant.
While many concrete examples of ergodic actions on type $\mathrm{III}$ factors are known,
it has remained open whether an outer action admits an ergodic cocycle.
The state preserving assumption ensures that the action extends to an isometry on the
GNS Hilbert space, which underlies the topological arguments used in \cite{MV23} and
in the present work.
We also note that Marrakchi and Vaes \cite[Open problem~4.6]{MV23} explicitly asked whether their
results could extend to type $\mathrm{III}$ factors and suggested that the state preserving
case might be tractable.

Our main theorem, stated below, gives an affirmative answer for type $\mathrm{III}_1$ factors under the assumption of \emph{trivial bicentralizer}, see Subsection~\ref{Preliminaries}. This assumption is satisfied for the concrete examples appearing in the literature and is conjecturally expected to hold for all type~$\mathrm{III}_1$ factors. Consequently, the existence of ergodic cocycles for single automorphisms in the state preserving type $\mathrm{III}_1$ setting is settled in essentially all cases.

\label{Preliminaries}

	Let $M$ be a von Neumann algebra and $\varphi\in M_\ast$ a faithful state. The \textit{modular operator, conjugation}, and \textit{action} are denoted by $\Delta_\varphi$, $J_\

Our main theorem, stated below, gives an affirmative answer for type $\mathrm{III}_1$ factors under the assumption of \emph{trivial bicentralizer}, see Subsection~\ref{Preliminaries}. This assumption is satisfied for the concrete examples appearing in the literature and is conjecturally expected to hold for all type~$\mathrm{III}_1$ factors. Consequently, the existence of ergodic cocycles for single automorphisms in the state preserving type $\mathrm{III}_1$ setting is settled in essentially all cases.

When $M$ is amenable, the state preserving assumption can be removed by using the classification of automorphisms of the amenable type $\mathrm{III}_1$ factor \cite{KST89}.
This yields the following corollary.

\begin{thmA}\label{thmC}
Let $\alpha \colon \Gamma \actson M$ be a state preserving outer action of a countably infinite discrete group $\Gamma$ on a type $\mathrm{III}_1$ factor with separable predual.
Assume that $\Gamma$ is amenable and that $M$ has trivial bicentralizer.
If $\Gamma_{\mathrm{mod}}$ is trivial, then there exists an $\alpha$-cocycle
$u \colon \Gamma \to \mathcal U(M)$ such that the perturbed action
is state preserving and weakly mixing.
\end{thmA}

\begin{lem}\label{lem-relative-bicentralizer}
 Let $\Gamma \actson M$ be a state preserving outer action of a countable group on a von Neumann algebra. Assume that:
\begin{itemize}
 \item $\Gamma_{\mathrm{mod}}$ is trivial;
 \item $M$ is a type $\mathrm{III}_1$ factor with separable predual and has trivial bicentralizer.
\end{itemize}
Then we have $\mathrm{BC}(M \subset M\rtimes_\alpha \Gamma, \varphi)=\mathbb C$ for all faithful state $\varphi\in M_\ast$.
\end{lem}

\begin{lem}\label{lem-cocycle3.6}
 Let $\alpha\colon \Gamma \actson (M,\varphi)$ be a state preserving outer action of a countably infinite discrete group on a diffuse factor with separable predual and with a faithful normal state.
Assume that $\Gamma$ is amenable and that $\mathrm{BC}(M\subset M\rtimes \Gamma,\varphi)=\C$.
Then for every $\varepsilon>0$ and every finite subset $F\subset \Gamma$, there exist $h\in \Gamma$ and $U\in \mathcal U(M^\omega)$ such that:
\begin{itemize}
 \item $\varphi^\omega(U)=0$, \quad $\|U\varphi^\omega-\varphi^\omega U\|<\varepsilon$,\quad and \quad $\|\alpha_g^\omega(U)-U\|_{\varphi^\omega}<\varepsilon$ for all $g\in F$;
 \item $M$, $U$, and $\alpha_h^\omega(U)$ are $\ast$-free with respect to $\varphi^\omega$.
\end{itemize}
\end{lem}

\begin{lem}\label{lem-cocycle3.5}
Let $\alpha\colon \Gamma \actson (M,\varphi)$ be a state preserving outer action of a countably infinite discrete group on a diffuse factor with separable predual. Assume that $\Gamma$ is amenable and that $\mathrm{BC}(M\subset M\rtimes \Gamma,\varphi)=\C$. For any $x\in M\setminus\C$, any finite subset $E\subset M\setminus\C$, and any $\varepsilon>0$, the pair $(1,\varphi)\in\mathcal C_{\mathrm{state}}(\alpha)$ can be approximated by elements of $\mathcal U(x)\cap \mathcal V( E,\varepsilon)$, where $1\in\mathcal C(\alpha)$ is the trivial cocycle.
\end{lem}
\begin{proof}
 Fix $\varepsilon>0$, $x\in M\setminus\C$, finite subsets $F\subset\Gamma$ and $E\subset M$. We will show that for every sufficiently small $\delta>0$, there exists $u\in\mathcal U(M)$ satisfying the following conditions:
\begin{itemize}
 \item[(a)] $\|u-\alpha_g(u)\|_\varphi<\delta$ for all $g\in F$, and $\|u\varphi-\varphi u\|<\delta$;
 \item[(b)] there exists $h\in\Gamma$ such that
 \[
 \|\alpha_h(uxu^*)-uxu^*\|_\varphi>\|uxu^*-\varphi(uxu^*)\|_\varphi;
 \]
 \item[(c)] there exists $h\in\Gamma$ such that
 \[
 \sum_{a,b\in E}\bigl|
 \varphi(\alpha_h(uau^*)ubu^*)-\varphi(uau^*)\varphi(ubu^*)
 \bigr|<\varepsilon.
 \]
\end{itemize}
If such a unitary $u$ exists, define an $\alpha$-cocycle $v_g:=u^*\alpha_g(u)$ for $g\in\Gamma$ and set $\psi:=u^*\varphi u$. Then $ \alpha^v$ preserves $\psi$, and by (a) the distance between $(1,\varphi)$ and $(v,\psi)$ is small. Moreover, (b) and (c) ensures that $(v,\psi)\in\mathcal U(x) \cap \mathcal V(E,\varepsilon)$. Thus to prove the lemma, it suffices to construct such a unitary $u$.

\begin{thm}\label{thm-case1}
Let $\alpha \colon \Gamma \actson M$ be a state preserving outer action satisfying the following assumptions:
\begin{itemize}
 \item $\Gamma$ is amenable and countably infinite;
 \item $\Gamma_{\mathrm{mod}} = \{e\}$;
 \item $M$ is a type ${\rm III_1}$ factor with separable predual and has trivial bicentralizer.
\end{itemize}
Then both $\mathcal C_{\mathrm{erg}}(\alpha)$ and $\mathcal C_{\mathrm{wm}}(\alpha)$ are dense $G_\delta$ subsets of $\mathcal C_{\mathrm{state}}(\alpha)$.
\end{thm}

\begin{cor}
Let $M$ be a type $\mathrm{III}_\lambda$ factor $(0<\lambda<1)$ with separable predual, and let
$\varphi\in M_\ast$ be a faithful state such that $\sigma^\varphi$ has period
$T$, where $T=-2\pi/\log(\lambda)$.
Let $\alpha \colon \Gamma \actson (M,\varphi)$ be a state preserving outer action
satisfying the following assumptions:
\begin{itemize}
 \item $\Gamma$ is amenable and countably infinite;
 \item $\Gamma_{\mathrm{mod}} = \{e\}$.
\end{itemize}
Then both $\mathcal C_{\mathrm{erg}}(\alpha,\varphi)$ and
$\mathcal C_{\mathrm{wm}}(\alpha,\varphi)$ are dense $G_\delta$ subsets
of $\mathcal C(\alpha,\varphi)$.
\end{cor}

\label{Preliminaries}

	Let $M$ be a von Neumann algebra and $\varphi\in M_\ast$ a faithful state. The \textit{modular operator, conjugation}, and \textit{action} are denoted by $\Delta_\varphi$, $J_\

When $M$ is amenable, the state preserving assumption can be removed by using the classification of automorphisms of the amenable type $\mathrm{III}_1$ factor \cite{KST89}.
This yields the following corollary.

\begin{corA}\label{corB}
Every automorphism of the amenable type $\mathrm{III}_1$ factor $R_\infty$ that has infinite order in $\mathrm{Out}(R_\infty)$ admits an ergodic cocycle.
\end{corA}

For a von Neumann algebra $M$ with a faithful state $\varphi \in M_\ast$, define
\[
\mathrm{Mod}(M)
:= \{\, \operatorname{Ad}(u)\circ\sigma_t^{\varphi} \in \operatorname{Aut}(M) \mid u \in \mathcal U(M),\ t \in \mathbb R \,\}.
\]
This is a normal subgroup of $\operatorname{Aut}(M)$ containing $\mathrm{Int}(M)$.
For a discrete group action $\alpha \colon \Gamma \to \operatorname{Aut}(M)$, we define the normal subgroup
\[
\Gamma_{\mathrm{mod}} := \alpha^{-1}(\mathrm{Mod}(M)) \le \Gamma.
\]
For outer actions on type $\mathrm{II}$ factors, one always has $\Gamma_{\mathrm{mod}}=\{e\}$.
One of the major differences in the type $\mathrm{III}$ setting is that $\Gamma_{\mathrm{mod}}$
may be nontrivial.
The proof of Theorem~\ref{thmA} will be divided according to whether
$\Gamma_{\mathrm{mod}}$ is trivial or not.

\bigskip
\noindent
{\bf Case 1: $\Gamma_{\mathrm{mod}}$ is trivial.}
In the case where $\Gamma_{\mathrm{mod}}$ is trivial, we prove the following theorem,
which is a direct generalization of \cite{MV23} to the type III setting.

\begin{thmA}\label{thmC}
Let $\alpha \colon \Gamma \curvearrowright M$ be a state preserving outer action of a countably infinite discrete group $\Gamma$ on a type $\mathrm{III}_1$ factor with separable predual.
Assume that $\Gamma$ is amenable and that $M$ has trivial bicentralizer.
If $\Gamma_{\mathrm{mod}}$ is trivial, then there exists an $\alpha$-cocycle
$u \colon \Gamma \to \mathcal U(M)$ such that the perturbed action
is state preserving and weakly mixing.
\end{thmA}

Up to stabilizing by another type $\mathrm{III}_1$ factor, we obtain the following consequence.

\begin{corA}\label{corD}
Let $\alpha \colon \Gamma \curvearrowright M$ be a state preserving outer action of a countably infinite discrete amenable group on a type $\mathrm{III}_1$ factor with separable predual. Let $N$ be another type $\mathrm{III}_1$ factor with separable predual. Then the action $\alpha \,\overline\otimes\, \id_N \colon \Gamma \curvearrowright M \,\overline\otimes\, N$ admits a cocycle perturbation that is state preserving and weakly mixing.
\end{corA}

The original proof of \cite{MV23} in the case of $\mathrm{II}_1$ factors relies heavily on Popa's free independence result \cite{Po95} (and its variant in \cite{PSV18}). This free independence theorem has been generalized to the setting of type III factors under appropriate hypotheses in \cite{HI14}. Therefore, if a state preserving action $\alpha\colon \Gamma \curvearrowright (M,\varphi)$ in Theorem~\ref{thmC} further satisfies $M_\varphi' \cap M=\mathbb C$, then by making use of \cite{HI14}, the proof becomes a rather direct adaptation of the argument in \cite{MV23}.

However, the theorem also covers the case where $M_\varphi=\mathbb C$, in which \cite{HI14} cannot be applied directly. The key point of our proof is the Connes--St\o rmer transitivity theorem for type $\mathrm{III}_1$ factors, which ensures that all faithful states become unitarily conjugate at the level of ultrapowers. As a consequence, $(M^\omega)_{\varphi^\omega}$ is sufficiently large for every $\varphi$, and this fact is the key technical ingredient of our proof.

\bigskip
\noindent
{\bf Case 2: $\Gamma_{\mathrm{mod}}$ is nontrivial.}
When $\Gamma_{\mathrm{mod}}$ is nontrivial, it is not clear how to deal with this situation in full generality. We therefore restrict ourselves to the case $\Gamma=\mathbb Z$, that is, to single automorphisms. In this case $\Gamma_{\mathrm{mod}}$ has the form $p\mathbb Z \leq \mathbb Z$ for some $p>0$, and the action restricted to $p\mathbb Z$ agrees with a modular action. By \cite{MV23}, one obtains an ergodic cocycle for $p\mathbb Z$. We then analyze how such a cocycle can be lifted appropriately to the entire group $\mathbb Z$. This step does not appear in \cite{MV23} and requires a new argument, which will be carried out in Theorem~\ref{thm-case2}.
\bigskip

In the final section, we collect several related results, including the case of type $\mathrm{III}_\lambda$ factors and additional properties of actions and cocycles on ultrapowers.

\tableofcontents

The proof of Theorem~\ref{thmA} is split depending on whether \(\Gamma_{\mathrm{mod}}:=\alpha^{-1}(\mathrm{Mod}(M))\) is trivial or not.

\noindent\textbf{Case 1: \(\Gamma_{\mathrm{mod}}\) is trivial.} One proves Theorem~\ref{thmC}, described as “a direct generalization of \cite{MV23} to the type III setting,” producing an \(\alpha\)-cocycle \(u\colon\Gamma\to\mathcal U(M)\) so that the perturbed action is “state preserving and weakly mixing.” The discussion explains that if additionally \(M_\varphi'\cap M=\mathbb C\), then using the type III free independence result \cite{HI14}, “the proof becomes a rather direct adaptation of the argument in \cite{MV23}.” In the remaining situation “where \(M_\varphi=\mathbb C\), in which \cite{HI14} cannot be applied directly,” the “key point” is the Connes--St\o rmer transitivity theorem for type \(\mathrm{III}_1\) factors: it “ensures that all faithful states become unitarily conjugate at the level of ultrapowers,” so “\((M^\omega)_{\varphi^\omega}\) is sufficiently large for every \(\varphi\), and this fact is the key technical ingredient of our proof.”

\noindent\textbf{Case 2: \(\Gamma_{\mathrm{mod}}\) is nontrivial.} The text says this is not handled “in full generality,” so it restricts to \(\Gamma=\mathbb Z\) (single automorphisms). Then \(\Gamma_{\mathrm{mod}}=p\mathbb Z\) for some \(p>0\) and the restriction to \(p\mathbb Z\) “agrees with a modular action.” Using \cite{MV23}, one “obtains an ergodic cocycle for \(p\mathbb Z\),” and then “analyze[s] how such a cocycle can be lifted appropriately to the entire group \(\mathbb Z\).” This lifting step “does not appear in \cite{MV23} and requires a new argument,” to be carried out later (Theorem~\ref{thm-case2}).

The proof of the main theorem is split depending on whether \(\Gamma_{\mathrm{mod}}:=\alpha^{-1}(\mathrm{Mod}(M))\) is trivial or not.

\noindent\textbf{Case 1: \(\Gamma_{\mathrm{mod}}\) is trivial.} We first prove the following theorem.
\begin{thmA}\label{thmC}
Let $\alpha \colon \Gamma \actson M$ be a state preserving outer action of a countably infinite discrete group $\Gamma$ on a type $\mathrm{III}_1$ factor with separable predual.
Assume that $\Gamma$ is amenable and that $M$ has trivial bicentralizer.
If $\Gamma_{\mathrm{mod}}$ is trivial, then there exists an $\alpha$-cocycle
$u \colon \Gamma \to \mathcal U(M)$ such that the perturbed action
is state preserving and weakly mixing.
\end{thmA}
This theorem is described as “a direct generalization of \cite{MV23} to the type III setting,” producing an \(\alpha\)-cocycle \(u\colon\Gamma\to\mathcal U(M)\) so that the perturbed action is “state preserving and weakly mixing.” The discussion explains that if additionally \(M_\varphi'\cap M=\mathbb C\), then using the type III free independence result \cite{HI14}, “the proof becomes a rather direct adaptation of the argument in \cite{MV23}.” In the remaining situation “where \(M_\varphi=\mathbb C\), in which \cite{HI14} cannot be applied directly,” the “key point” is the Connes--St\o rmer transitivity theorem for type \(\mathrm{III}_1\) factors: it “ensures that all faithful states become unitarily conjugate at the level of ultrapowers,” so “\((M^\omega)_{\varphi^\omega}\) is sufficiently large for every \(\varphi\), and this fact is the key technical ingredient of our proof.”

\noindent\textbf{Case 2: \(\Gamma_{\mathrm{mod}}\) is nontrivial.} The text says this is not handled “in full generality,” so it restricts to \(\Gamma=\mathbb Z\) (single automorphisms). Then \(\Gamma_{\mathrm{mod}}=p\mathbb Z\) for some \(p>0\) and the restriction to \(p\mathbb Z\) “agrees with a modular action.” Using \cite{MV23}, one “obtains an ergodic cocycle for \(p\mathbb Z\),” and then “analyze[s] how such a cocycle can be lifted appropriately to the entire group \(\mathbb Z\).” This lifting step “does not appear in \cite{MV23} and requires a new argument,” to be carried out later. We next prove the following theorem.
\begin{thm}\label{thm-case2}
Let $M$ be a type ${\rm III_1}$ factor with separable predual, $\varphi\in M_\ast$ a faithful state, and $\alpha\colon \mathbb Z \actson (M,\varphi)$ a state preserving outer action such that $\Gamma_{\mathrm{mod}} = p\mathbb Z$ for some $p\geq 1$.
Then there exists a unitary $u\in \mathcal U(M)$ such that $\alpha^u$ is an ergodic state preserving action.
\end{thm}

Question: Let \(M\) be a type \(\mathrm{III}_1\) factor with separable predual and trivial bicentralizer, and let \(\theta\in\operatorname{Aut}(M)\) be state preserving, meaning that \(\theta\) leaves some faithful normal state on \(M\) invariant. Write \(\mathrm{Out}(M)=\operatorname{Aut}(M)/\operatorname{Inn}(M)\), so that saying \(\theta\) has infinite order in \(\mathrm{Out}(M)\) means that \([\theta]^n\neq e\) for every nonzero integer \(n\). Also, \(\mathcal U(M)\) denotes the unitary group of \(M\), \(\operatorname{Ad}(u)(x)=uxu^*\), and an automorphism \(\beta\) of \(M\) is ergodic if its fixed-point algebra \(\{x\in M:\beta(x)=x\}\) is \(\mathbb C1\). Which statement holds?

The following are equivalent: (i) \(\theta\) has infinite order in \(\mathrm{Out}(M)\); (ii) there exists \(u\in\mathcal U(M)\) such that \(\operatorname{Ad}(u)\circ\theta\) is state preserving and ergodic.

Choices:
B. The following are equivalent: (i) \(\theta\) has finite order in \(\mathrm{Out}(M)\); (ii) there exists \(u\in\mathcal U(M)\) such that \(\operatorname{Ad}(u)\circ\theta\) is state preserving and ergodic.
C. If there exists \(u\in\mathcal U(M)\) such that \(\operatorname{Ad}(u)\circ\theta\) is state preserving and ergodic, then \(\theta\) has infinite order in \(\mathrm{Out}(M)\).
D. The following are equivalent: (i) \(\theta\) has infinite order in \(\mathrm{Out}(M)\); (ii) there exists \(u\in\mathcal U(M)\) such that \(\operatorname{Ad}(u)\circ\theta\) is ergodic.
E. The following are equivalent: (i) every nonzero power \(\theta^n\) is outer; (ii) for every faithful normal state \(\psi\) on \(M\), there exists \(u\in\mathcal U(M)\) such that \(\operatorname{Ad}(u)\circ\theta\) preserves \(\psi\) and is ergodic.

dropped the converse implication from infinite order to existence of an ergodic state-preserving perturbation

state-preserving requirement in the perturbation conclusion

existential state in the hypothesis replaced by uniform preservation for every faithful normal state


\begin{theorem}\label{thm:main}
		Let $\F$ be an algebraically closed field of characteristic different from $2$, and let $n,k \in \N$ with $n \ge 2$. A map $\phi : M_n(\F) \to M_n(\F)$
		satisfies the mixed Jordan--power identity
		\begin{equation}\label{eq:a certain product}
			\phi(A^k \circ B) = \phi(A)^k \circ \phi(B), \qquad \text{for all } A,B \in M_n(\F),
		\end{equation}
		if and only if one of the following holds:
		\begin{itemize}
			\item[(a)] $\phi$ is constant, taking a fixed $(k+1)$-potent value;
			\item[(b)] there exist an invertible matrix $T \in M_n(\F)$, a ring monomorphism $\omega : \F \to \F$, and a $k$-th root of unity $\varepsilon \in \F$ such that
			\begin{equation}\label{eq:form of phi}
				\phi(X) = \varepsilon\, T\, \omega(X)\, T^{-1}
				\quad \text{ or } \quad
				\phi(X) = \varepsilon\, T\, \omega(X)^t\, T^{-1}.
			\end{equation}
		\end{itemize}
		In particular, every nonconstant solution of \eqref{eq:a certain product} is additive.
	\end{theorem}

\begin{remark}\label{rem:k-potents not diagonalizable}
		Suppose that $n \ge 2$. All matrices in $\pot_k(M_n)$ are diagonalizable if and only if $\chr(\F)$ does not divide $k-1$. Indeed, suppose that $\chr(\F)$ does not divide $k-1$, and let $P \in \pot_k(M_n)$ be arbitrary. We have
		$$P^{k} = P \, \implies \, P(P^{k-1}-I) = 0,$$
		which implies that the minimal polynomial $m_P \in \F[x]$ of $P$ necessarily divides
		$$q(x):=x(x^{k-1}-1).$$
		Since $\F$ is algebraically closed, to show that $P$ is diagonalizable in $M_n$, it suffices to show that $q$ (and consequently $m_P$) is square-free, meaning that it decomposes into a product of distinct linear factors. We have
		$q'(x) = kx^{k-1}-1$. Assuming that $x_0 \in \F$ satisfies $q(x_0) = q'(x_0) = 0$ would lead to $k-1=0$, which is a contradiction.

		Conversely, suppose that $\chr(\F) \mid (k-1)$. For $n=2$, the matrix
		$$
		A := \begin{bmatrix}
			1  & 1 \\ 0 & 1
		\end{bmatrix} \in M_2
		$$
		satisfies $$A^{k-1} = \begin{bmatrix}
			1  & k-1 \\ 0 & 1
		\end{bmatrix} = I \implies A^k = A$$
		so $A$ is a $k$-potent in $M_2$. On the other hand, the minimal polynomial of $A$ is $m_A(x) = (x-1)^2 \in \F[x]$, which is not square-free in $\F[x]$. Therefore, $A$ is not diagonalizable in $M_2$. When $n > 2$, a similar conclusion follows by considering the matrix $\diag(A,I_{n-2}) \in M_n$.

		Additionally, the matrix $A$ demonstrates other interesting properties of the order $\preceq$. It is not difficult to show directly that any matrix $B \in M_2$ satisfies $AB=BA=B^2$ if and only if $B \in \{0,A\}$. In particular, $A$ is a minimal element of the poset $(\pot_k(M_2)\setminus\{0\}, \preceq)$, even though its rank is $2$. It is also a consequence that $A$ \emph{cannot} be written in the form $A = P+Q$ for some nonzero mutually orthogonal $k$-potents $P,Q \in \pot_k(M_2)$. On the other hand, by Remark \ref{re:maximal element}, $A$ is also a maximal element of the poset $(\pot_k(M_2), \preceq)$.
	\end{remark}

The study of multiplicative maps between rings and algebras has a long and substantial history, centered on the question of when multiplicativity alone enforces stronger algebraic behavior, most notably additivity. Foundational contributions were made by Rickart~\cite{Rickart} in 1948 and Johnson~\cite{Johnson} in 1958, who established positive results under suitable structural assumptions and highlighted the connection with the rigidity of the additive structure of rings. A major advance was achieved by Martindale~\cite{Martindale} in 1969, who proved that every bijective multiplicative map from a prime ring containing a nontrivial idempotent onto an arbitrary ring is automatically additive. In the same year, Jodeit and Lam~\cite{JodeitLam} classified all nondegenerate multiplicative self-maps of the matrix rings $M_n(\mathcal{R})$ over a principal ideal domain~$\mathcal{R}$, where nondegeneracy means that the map is not identically zero on the set of matrices with zero determinant. Specifically, for each such map $\phi: M_n(\ca{R}) \to M_n(\ca{R})$, either there exists a nonzero idempotent matrix $P \in M_n(\ca{R})$ such that $\phi - P$ is multiplicative and degenerate, or there exists an invertible matrix $T \in M_n(\ca{R})$ and a ring endomorphism $\omega$ of $\ca{R}$ such that $\phi$ takes one of the following two distinct forms:
	$$
	\phi(X) = T \, \omega(X) \, T^{-1} \quad \text{ or } \quad \phi(X) = T \, \omega(X)^* \, T^{-1},
	$$
	where $\omega(X)$ denotes the matrix obtained by applying $\omega$ entrywise, and $(\cdot)^*$ is the corresponding cofactor matrix. In particular, all bijective multiplicative self-maps on $M_n(\mathcal{R})$ are automatically additive and hence ring automorphisms. Taken together, these results expose a fundamental phenomenon: multiplicativity, when coupled with bijectivity, interacts strongly with the ambient ring structure, leaving little room for pathological nonadditive behavior. This rigidity principle later emerged as a central theme in preserver theory, especially in the context of matrix and operator algebras.

	This viewpoint was later extended from ordinary ring homomorphisms to \emph{Jordan homomorphisms}, which originate as homomorphisms in the category of \emph{Jordan algebras}, introduced by Jordan, von Neumann, and Wigner \cite{Jordan} in the 1930s as algebraic models for quantum observables.
	A \emph{Jordan algebra} is a nonassociative algebra $\ca{A}$ over a field equipped with a commutative multiplication $\diamond$ satisfying the \emph{Jordan identity}
	$$
	a^2 \diamond (a \diamond b) = a \diamond (a^2 \diamond b), \quad \text{ for all } a,b \in \ca{A}.
	$$
	Jordan rings are similarly defined. Every associative algebra (or ring) $\mathcal{A}$ naturally becomes a Jordan algebra (ring) when equipped with the \emph{Jordan product}
	$$
	a \diamond b := ab + ba.
	$$
	A Jordan algebra that is isomorphic to a Jordan subalgebra of an associative algebra is called a \emph{special Jordan algebra} (see e.g. \cite[Section~2.3]{HancheStormer}). Not all Jordan algebras are special; the Albert algebra~\cite{Albert} provides the classical example of an \emph{exceptional Jordan algebra}.

	In an associative setting, a linear (or additive) map $\phi: \ca{A} \to \ca{B}$ between associative algebras (or rings) $\ca{A}$ and $\ca{B}$ is called a \emph{Jordan homomorphism} if it satisfies
	$$
	\phi(a \diamond b) = \phi(a) \diamond \phi(b), \quad \text{ for all } a, b \in \ca{A}.
	$$
	If the algebras (rings) are $2$-torsion-free, this is equivalent to requiring that $\phi$ preserves squares with respect to the underlying product of $\mathcal{A}$, that is,
	$$
	\phi(a^2) = \phi(a)^2, \quad \text{ for all } a \in \ca{A}.
	$$
	Moreover, if $\ca{A}$ is an algebra over a field $\F$ with $\chr(\F) \ne 2$, the Jordan structure is commonly (esp.\ in quantum mechanics contexts) induced by the \emph{normalized Jordan product}
	\begin{equation}\label{eq:normJprodukt}
		a\circ b :=\frac12(ab+ba),
	\end{equation}
	and Jordan homomorphisms are equivalently defined as $\circ$-multiplicative linear maps. The study of Jordan homomorphisms in associative settings is well established and constitutes a major research direction in abstract algebra, operator theory, and the theory of preservers.

	The problem of automatic additivity for \emph{Jordan multiplicative maps} has been studied extensively since the early 2000s. This line of research focuses on maps that preserve the Jordan product, $\circ$ or $\diamond$, viewed as a natural analogue of the classical automatic additivity problem for multiplicative maps. A seminal contribution was made by Moln\'ar \cite{Molnar}, who characterized $\circ$-preserving bijections $\phi:\mathcal{A} \to \mathcal{B}$ between standard operator algebras $\mathcal{A}$ and $\mathcal{B}$ with $\dim(\mathcal{A})>1$, that is, subalgebras of bounded linear operators on a complex Banach space containing all finite-rank operators. In particular, all such maps are automatically additive. The finite-dimensional variant of Moln\'ar's result  asserts that any $\circ$-preserving bijection $\phi : M_n(\C)\to M_n(\C)$, $n \ge 2$, takes the form
	\begin{equation}\label{eq:glavnieq}
		\phi(X)=T\,\omega(X)\,T^{-1} \quad \text{ or } \quad \phi(X)=T\,\omega(X)^t\,T^{-1},
	\end{equation}
	where $T\in M_n(\C)$ is invertible, $\omega$ is a ring automorphism of $\C$ applied entrywise, and $(\cdot)^t$ denotes matrix transposition. Lu \cite{Lu} subsequently extended this result to additional classes of associative algebras. Subsequently, An and Hou \cite{AnHou} studied Jordan multiplicative bijections on the real Jordan algebra of all self-adjoint operators on a complex Hilbert space of dimension greater than $1$, proving that any $\diamond$- or $\circ$-preserving bijection is automatically additive and implemented via unitary or anti-unitary conjugation. The problem of automatic additivity has also been investigated in the context of abstract Jordan algebras. In particular, Ji \cite{Ji} showed that if $\mathcal{A}$ is a Jordan algebra containing an idempotent satisfying suitable Peirce-type structural conditions, then any bijection $\phi:\mathcal{A}\to \mathcal{B}$ to another Jordan algebra $\mathcal{B}$ that preserves the Jordan product is necessarily additive, thereby unifying a range of operator-algebraic and ring-theoretic results within a single framework.

	In the setting of full matrix algebras, our recent work \cite{GogicTomasevic-AM} provides a complete classification of all  $\circ$-preserving maps $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$, where $\chr(\mathbb{F}) \neq 2$. We show that every such map is either constant (and hence equal to a fixed idempotent) or a Jordan ring monomorphism of the form~\eqref{eq:glavnieq}, where $T \in M_n(\mathbb{F})$ is invertible and $\omega : \mathbb{F} \to \mathbb{F}$ is a ring monomorphism \cite[Theorem~1.1]{GogicTomasevic-AM}.   This generalizes the finite-dimensional version of Molnár's theorem. Moreover, if $\phi : \mathcal{A} \to \mathcal{B}$ is any $\diamond$-preserving map between $\mathbb{F}$-algebras (with $\chr(\mathbb{F}) \neq 2$), then the simple normalization
	$$
	\psi(x) := 2\,\phi\left(\frac{x}{2}\right), \quad x \in \mathcal{A},
	$$
	transforms $\phi$ into a $\circ$-preserving map $\psi$.  This reduction allows us to obtain a corresponding classification of $\diamond$-preserving maps on $M_n(\mathbb{F})$,
	with the only difference that the constant map now takes the value $P/2$ for some idempotent $P \in M_n(\mathbb{F})$. In particular, this result shows that Jordan multiplicative maps on $M_n(\mathbb{F})$ are even more rigid than general multiplicative maps: every nonconstant map is necessarily additive. For the fields $\mathbb{R}$ and $\mathbb{C}$, this classification, together with the Jodeit--Lam theorem, was further extended in \cite{GogicTomasevic-LAMA} to injective multiplicative and Jordan multiplicative maps on \emph{structural matrix algebras}, i.e.\ subalgebras of $M_n(\mathbb{F})$ containing all diagonal matrices \cite{vanWyk} (see also \cite[Proposition~3.1]{GogicTomasevic-LAA}). Further details on structural matrix algebras, their automorphisms, and their Jordan monomorphisms can be found in \cite{Coelho, GogicTomasevic-LAA}.

	A substantial body of literature also investigates multiplicative maps with respect to products other than the Jordan product, particularly various symmetrized or triple-product variants such as $(a,b) \mapsto aba$, $(a,b,c) \mapsto abc + cba$, and many others (see, e.g., \cite{AnHou, Kuzma, LesnjakSze, Lu2, Molnar2} and references therein). More recently, preservers of the product
	\begin{equation}\label{eq:the product}
		a^{2} \circ b = \frac12(a^{2} b + b a^{2})
	\end{equation}
	were studied by Ghorbanipour and Hejazian~\cite{GhorbanipourHejazian}, who showed that any bijection $\phi : \mathcal{A} \to \mathcal{A}$ preserving \eqref{eq:the product}, where $\mathcal{A}$ is a real standard Jordan operator algebra acting on a Hilbert space of dimension at least $2$, is of the form $\phi = \pm \psi$, where $\psi$ is a Jordan automorphism of $\mathcal{A}$.

	The product $(a,b)\mapsto a^2 \circ b$ plays an important role in the theory of JB-algebras. Recall that a \emph{JB-algebra} is a (typically real) Jordan algebra $(\mathcal{A},\circ)$ equipped with a complete submultiplicative norm $\|\cdot\|$  (that is, a Jordan Banach algebra) which additionally satisfies
	$$
	\|a\|^2 \le \|a^2 + b^2\|, \quad \text{ for all }  a,b \in \mathcal{A}.
	$$
	A central feature of JB-algebras is that many aspects of their structure, such as order, spectral theory, and the theory of ideals, are encoded in the associated quadratic operators (see, for example, \cite{Battaglia}). In particular, the fundamental quadratic mapping $U_a : \mathcal{A} \to \mathcal{A}$,
	$$
	U_{a}(b) := 2(a \circ b) \circ a - a^{2} \circ b,
	$$
	plays a key role in the analysis of orthogonality, annihilators, and quadratic ideals. Moreover, the product $(a,b)\mapsto a^2 \circ b$ was used in \cite{Battaglia} to define orthogonality in JB-algebras: elements $a,b \in \mathcal{A}$ are said to be orthogonal if $a^{2} \circ b = 0$. In the setting of special JB-algebras, such as the self-adjoint part of a $C^{*}$-algebra, the Jordan product is given by \eqref{eq:normJprodukt}. In this case, the expression $a^{2} \circ b$ agrees exactly with the symmetrized product appearing in \eqref{eq:the product}.

	\smallskip

	This motivates our study of \emph{mixed Jordan--power preservers}, namely, maps that preserve a more general Jordan-type product
	$$
	a^k \circ b= \frac{1}{2}(a^k b + b a^k),
	$$
	for arbitrary $k \in \N$. Our main result provides a complete classification of mixed Jordan--power preservers on full matrix algebras.

In an associative setting, a linear (or additive) map $\phi: \ca{A} \to \ca{B}$ between associative algebras (or rings) $\ca{A}$ and $\ca{B}$ is called a \emph{Jordan homomorphism} if it satisfies
 $$
 \phi(a \diamond b) = \phi(a) \diamond \phi(b), \quad \text{ for all } a, b \in \ca{A}.
 $$
 If the algebras (rings) are $2$-torsion-free, this is equivalent to requiring that $\phi$ preserves squares with respect to the underlying product of $\mathcal{A}$, that is,
 $$
 \phi(a^2) = \phi(a)^2, \quad \text{ for all } a \in \ca{A}.
 $$
 Moreover, if $\ca{A}$ is an algebra over a field $\F$ with $\chr(\F) \ne 2$, the Jordan structure is commonly (esp.\ in quantum mechanics contexts) induced by the \emph{normalized Jordan product}
 \begin{equation}\label{eq:normJprodukt}
 a\circ b :=\frac12(ab+ba),
 \end{equation}
 and Jordan homomorphisms are equivalently defined as $\circ$-multiplicative linear maps. The study of Jordan homomorphisms in associative settings is well established and constitutes a major research direction in abstract algebra, operator theory, and the theory of preservers.

The problem of automatic additivity for \emph{Jordan multiplicative maps} has been studied extensively since the early 2000s. This line of research focuses on maps that preserve the Jordan product, $\circ$ or $\diamond$, viewed as a natural analogue of the classical automatic additivity problem for multiplicative maps. A seminal contribution was made by Moln\'ar \cite{Molnar}, who characterized $\circ$-preserving bijections $\phi:\mathcal{A} \to \mathcal{B}$ between standard operator algebras $\mathcal{A}$ and $\mathcal{B}$ with $\dim(\mathcal{A})>1$, that is, subalgebras of bounded linear operators on a complex Banach space containing all finite-rank operators. In particular, all such maps are automatically additive. The finite-dimensional variant of Moln\'ar's result asserts that any $\circ$-preserving bijection $\phi : M_n(\C)\to M_n(\C)$, $n \ge 2$, takes the form
 \begin{equation}\label{eq:glavnieq}
 \phi(X)=T\,\omega(X)\,T^{-1} \quad \text{ or } \quad \phi(X)=T\,\omega(X)^t\,T^{-1},
 \end{equation}
 where $T\in M_n(\C)$ is invertible, $\omega$ is a ring automorphism of $\C$ applied entrywise, and $(\cdot)^t$ denotes matrix transposition. Lu \cite{Lu} subsequently extended this result to additional classes of associative algebras. Subsequently, An and Hou \cite{AnHou} studied Jordan multiplicative bijections on the real Jordan algebra of all self-adjoint operators on a complex Hilbert space of dimension greater than $1$, proving that any $\diamond$- or $\circ$-preserving bijection is automatically additive and implemented via unitary or anti-unitary conjugation. The problem of automatic additivity has also been investigated in the context of abstract Jordan algebras. In particular, Ji \cite{Ji} showed that if $\mathcal{A}$ is a Jordan algebra containing an idempotent satisfying suitable Peirce-type structural conditions, then any bijection $\phi:\mathcal{A}\to \mathcal{B}$ to another Jordan algebra $\mathcal{B}$ that preserves the Jordan product is necessarily additive, thereby unifying a range of operator-algebraic and ring-theoretic results within a single framework.

In the setting of full matrix algebras, our recent work \cite{GogicTomasevic-AM} provides a complete classification of all $\circ$-preserving maps $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$, where $\chr(\mathbb{F}) \neq 2$. We show that every such map is either constant (and hence equal to a fixed idempotent) or a Jordan ring monomorphism of the form~\eqref{eq:glavnieq}, where $T \in M_n(\mathbb{F})$ is invertible and $\omega : \mathbb{F} \to \mathbb{F}$ is a ring monomorphism \cite[Theorem~1.1]{GogicTomasevic-AM}. This generalizes the finite-dimensional version of Molnár's theorem. Moreover, if $\phi : \mathcal{A} \to \mathcal{B}$ is any $\diamond$-preserving map between $\mathbb{F}$-algebras (with $\chr(\mathbb{F}) \neq 2$), then the simple normalization
 $$
 \psi(x) := 2\,\phi\left(\frac{x}{2}\right), \quad x \in \mathcal{A},
 $$
 transforms $\phi$ into a $\circ$-preserving map $\psi$. This reduction allows us to obtain a corresponding classification of $\diamond$-preserving maps on $M_n(\mathbb{F})$,
 with the only difference that the constant map now takes the value $P/2$ for some idempotent $P \in M_n(\mathbb{F})$. In particular, this result shows that Jordan multiplicative maps on $M_n(\mathbb{F})$ are even more rigid than general multiplicative maps: every nonconstant map is necessarily additive. For the fields $\mathbb{R}$ and $\mathbb{C}$, this classification, together with the Jodeit--Lam theorem, was further extended in \cite{GogicTomasevic-LAMA} to injective multiplicative and Jordan multiplicative maps on \emph{structural matrix algebras}, i.e.\ subalgebras of $M_n(\mathbb{F})$ containing all diagonal matrices \cite{vanWyk} (see also \cite[Proposition~3.1]{GogicTomasevic-LAA}). Further details on structural matrix algebras, their automorphisms, and their Jordan monomorphisms can be found in \cite{Coelho, GogicTomasevic-LAA}.

\smallskip

This motivates our study of \emph{mixed Jordan--power preservers}, namely, maps that preserve a more general Jordan-type product
 $$
 a^k \circ b= \frac{1}{2}(a^k b + b a^k),
 $$
 for arbitrary $k \in \N$. Our main result provides a complete classification of mixed Jordan--power preservers on full matrix algebras.

\smallskip

\section{Notation and Preliminaries}\label{sec:prel}
 We begin by introducing notation that will be used throughout the paper. Let $\F$ be a fixed algebraically closed field of characteristic different from $2$. We denote by $\mathbb{F}^\times$ the multiplicative group of nonzero elements of $\mathbb{F}$, and by $\mathbb{F}[x]$ the polynomial algebra in one variable over $\mathbb{F}$.

\begin{lemma}\label{le:h exists}
 Let $\phi : M_n \to M_n$ be a nonzero map which satisfies \eqref{eq:a certain product} and $\phi(0) = 0$. There exists a unique multiplicative map $\omega : \F \to \F$ such that
 \begin{equation}\label{eq:homogeneity}
 \phi(\lambda X) = \omega(\lambda) \phi(X), \quad \text{ for all } \lambda \in \F \text{ and } X \in M_n.
 \end{equation}
 Moreover, there exists an invertible matrix $T \in M_n$ and $\varepsilon\in\Sph^1_k$ such that the map $$\psi:= T^{-1}\phi(\cdot)T$$ satisfies $$\psi(E_{jj}) = \varepsilon E_{jj}, \quad \text{ for each } j \in [n].$$
 Further, fix some distinct $i,j \in [n]$. Then $\psi(E_{ij})$ is a nonzero scalar multiple of $E_{ij}$ or $E_{ji}$. If $\psi(E_{ij}) = \beta_{ij} E_{ij}$ for some $\beta_{ij} \in \F^\times$, then for each $x,y \in \F$ we have
 \begin{equation}\label{eq: Eii+Eij 1}
 \psi(E_{ii} + x E_{ij}) = \varepsilon E_{ii} + \omega(x) (\beta_{ij} E_{ij}), \qquad \psi(E_{jj} + y E_{ij}) =
 \varepsilon E_{jj} + \omega(y) (\beta_{ij} E_{ij}).
 \end{equation}
 If, on the other hand, $\psi(E_{ij}) = \gamma_{ij} E_{ji}$ for some $\gamma_{ij} \in \F^\times$, then
 \begin{equation}\label{eq: Eii+Eij 2}
 \psi(E_{ii} + x E_{ij}) = \varepsilon E_{ii} + \omega(x) (\gamma_{ij} E_{ji}), \qquad \psi(E_{jj} + y E_{ij}) =
 \varepsilon E_{jj} + \omega(y) (\gamma_{ij} E_{ji}).
 \end{equation}
 Finally, if $n \ge 2$, the map $\omega : \F \to \F$ is a ring monomorphism. In particular, $\phi$ is $\K$-homogeneous.
 \end{lemma}
 \begin{proof}
 In view of Lemma \ref{le:basic properties II} (c) and (e), $\phi(E_{11}), \ldots ,\phi(E_{nn})$ are mutually orthogonal rank-one $(k+1)$-potents and therefore can be simultaneously diagonalized (see e.g.\ \cite[Theorem~8 of \S6.5]{HoffmanKunze}). Hence, by passing to map $T^{-1}\phi(\cdot)T$, for a suitable invertible matrix $T \in M_n$, without loss of generality we can assume that
 \begin{equation}\label{eq:phi fiksira dijagonalne}
 \phi(E_{jj}) = \varepsilon_j E_{jj}, \quad \text{ for some $\varepsilon_j \in \Sph^1_k$ for each } j \in [n].
 \end{equation}
 Note that by Lemma \ref{le:basic properties II} (f), for each diagonal $(k+1)$-potent $D \in \ca{D}_n$ we have
 \begin{equation}\label{eq:diagonal tripotents}
 \phi(D) =
 \sum_{(j,j) \in \supp D} \varepsilon_j E_{jj}.
 \end{equation}
 By Lemma \ref{le:zero map}, we have
 \begin{equation}\label{eq:lambda times ejj}
 \phi(\lambda E_{jj}) \ne 0, \quad\text{ for all } j \in [n] \text{ and } \lambda \in \F^\times.
 \end{equation} Further, for each $X \in M_n$ and $S \subseteq [n]$ we have
 \begin{equation}\label{eq:preserves support}
 \supp X \subseteq S \times S \implies \supp \phi(X) \subseteq S \times S.
 \end{equation}
 Indeed, denote the diagonal idempotent
 $P := \sum_{j \in [n]\setminus S} E_{jj}$ and note that a matrix $X\in M_n$ is supported in $S \times S$ if and only if $XP=PX=0$. In that case, obviously $X \circ P^k = 0$, so
 $$0 = \phi(X \circ P^k) = \phi(X) \circ \phi(P)^k \stackrel{\eqref{eq:diagonal tripotents}}= \phi(X) \circ P$$
 and hence Lemma \ref{le:Jordan product calculations} (a) implies the claim.
 \smallskip

Fix $(i,j), (j,p) \in [n]^2 \setminus \Delta_n$. If $i \ne p$, then by Lemma \ref{le:h exists} we have
 \begin{align*}
 \frac12 g(i,p)E_{ip} &= \phi\left(\frac12 E_{ip}\right) =\phi((E_{ii}+E_{ij})^k \circ E_{jp}) = \phi(E_{ii}+E_{ij})^k \circ \phi(E_{jp}) \\
 &\leftstackrel{\eqref{eq: Eii+Eij 1}}= (E_{ii}+g(i,j)E_{ij})^k \circ (g(j,p)E_{jp})\\
 &= \frac12 g(i,j)g(j,p)E_{ip},
 \end{align*}
 which implies $g(i,p) = g(i,j)g(j,p)$.
 On the other hand, if $i = p$, then by Lemma \ref{le:h exists} we have
 \begin{align*}
 \frac{1}{2}\phi\left(E_{ii} + E_{jj} + E_{ji}\right) &= \phi\left(\frac{1}{2}\left(E_{ii} + E_{jj} + E_{ji}\right)\right) = \phi((E_{ii}+E_{ij})^k \circ E_{ji}) \\
 &= \phi(E_{ii}+E_{ij})^k \circ \phi(E_{ji}) \\
 &\leftstackrel{\eqref{eq: Eii+Eij 1}}= (E_{ii}+ g(i,j)E_{ij})^k \circ (g(j,i)E_{ji})\\
 &= \frac12 g(j,i) \Big(g(i,j)(E_{ii}+E_{jj}) + E_{ji}\Big).
 \end{align*}
 Furthermore, we have
 \begin{align*}
 E_{ii} +\frac12g(j,i) E_{ji} \,\, &\leftstackrel{\eqref{eq: Eii+Eij 1}}= \phi\left(E_{ii} +\frac12 E_{ji}\right) =\phi\left((E_{ii} + E_{jj} + E_{ji}) \circ E_{ii}^k\right) \\
 &= \phi\left(E_{ii} + E_{jj} + E_{ji}\right) \circ \phi(E_{ii})^k\\
 &= g(j,i) \Big(g(i,j)(E_{ii}+E_{jj}) + E_{ji}\Big) \circ E_{ii} \\
 &= g(j,i)\left(g(i,j) E_{ii} +\frac12 E_{ji}\right)
 \end{align*}
 We obtain
 $$
 g(i,j)g(j,i) = 1,
 $$
 as desired. Following \cite{Coelho}, denote by $$g^* : M_n \to M_n, \qquad g^*(E_{ij}) := g(i,j)E_{ij}$$
 the induced algebra automorphism of $M_n$. Note that $g^*$ is implemented via conjugation by the invertible diagonal matrix
 $$
 D := \operatorname{diag}\big(g(1,1), g(2,1), \ldots, g(n,1)\big) \in \ca{D}_n,
 $$
 i.e.\ $g^*(\cdot)=D(\cdot)D^{-1}$. Hence, by passing to the map $(g^*)^{-1}(\phi(\cdot)) = D^{-1}\phi(\cdot)D$, without loss of generality we can assume that
 $$\phi(E_{ij}) = E_{ij}, \quad \text{ for all } (i,j) \in [n]^2.$$
 In view of Lemma \ref{le:h exists}, denote by $\omega :\F \to \F$ the induced ring monomorphism that satisfies \eqref{eq:homogeneity}. We claim that
 \begin{equation}\label{eq:phi = omega}
 \phi(X) = \omega(X), \quad \text{ for all } X \in M_n.
 \end{equation}
 First assume that $X=[x_{ij}] \in M_n$ is of the form $X = Y^k$ for some $Y \in M_n$. Fix $(i,j) \in [n]^2$ and denote $\phi(X)=[\phi(X)_{ij}]$. If $i = j$, then by Lemma \ref{le:preserves triple product} we have
 \begin{align*}
 \omega(x_{ii})E_{ii} &= \phi(x_{ii}E_{ii}) = \phi(E_{ii}Y^k E_{ii}) = \phi(E_{ii})\phi(Y)^k \phi(E_{ii}) = E_{ii}\phi(X) E_{ii} \\
 &= \phi(X)_{ii}E_{ii},
 \end{align*}
 which shows that
 \begin{equation}\label{eq:Yii}
 \phi(X)_{ii} = \omega(x_{ii}).
 \end{equation} Now assume $i \ne j$. Denote $P:= E_{ii}+E_{ji}$ and note that
 $$
 PAP = (a_{ii}+a_{ij})P, \quad \text{ for all matrices $A=[a_{ij}] \in M_n$}.
 $$
 By \eqref{eq: Eii+Eij 1} we have $\phi(P) = P$ and hence
 \begin{align*}
 \omega(x_{ii}+x_{ij})P &=\phi((x_{ii}+x_{ij})P) = \phi(PXP) \stackrel{\text{Lemma }\ref{le:preserves triple product}}= \phi(P)\phi(X)\phi(P) \\
 &= P\phi(X)P = (\phi(X)_{ii}+\phi(X)_{ij})P.
 \end{align*}
 Therefore, by the additivity of $\omega$ and \eqref{eq:Yii} we obtain
 $$\omega(x_{ii})+\omega(x_{ij}) = \phi(X)_{ii}+\phi(X)_{ij} = \omega(x_{ii})+\phi(X)_{ij}.$$
 This implies $\phi(X)_{ij} = \omega(x_{ij})$, which shows \eqref{eq:phi = omega} for all matrices $X \in M_n(\F)$ which are a $k$-th power of some matrix. Using the notation of Lemma \ref{le:iterated R}, this means
 $$\phi(X) = \omega(X), \quad \text{ for all } X \in \ca{R}_0^n.$$
 We now extend this conclusion to all matrices in $M_n$. Note that if $A,B \in M_n$ satisfy $\phi(A) = \omega(A)$ and $\phi(B) = \omega(B)$, then clearly
 $$\phi(A^k \circ B) = \phi(A)^k \circ \phi(B) = \omega(A)^k \circ \omega(B) = \omega(A^k \circ B).$$
 Again returning to the notation of Lemma \ref{le:iterated R}, this inductively implies that $$\phi(X) = \omega(X), \quad \text{ for all } X\in \ca{R}^n.$$
 By the same lemma we have $\ca{R}^n = M_n$, which completes the proof of the theorem.
 \end{proof}
 We conclude the paper with the following generalization of \cite[Corollary~3.5]{GogicTomasevic-AM}, which follows immediately from the first part of the proof of Theorem~\ref{thm:main}.
 \begin{corollary}
 Let $m < n$. A map $\phi : M_n \to M_m$ satisfies \eqref{eq:a certain product} if and only if it is constant and equal to a fixed $(k+1)$-potent.
 \end{corollary}

This result extends all previously known Jordan-type additivity results on $M_n(\F)$, at least for algebraically closed fields with $\operatorname{char}(\F) \neq 2$. The proof of Theorem~\ref{thm:main}, presented in Section~\S\ref{sec:main}, follows the general strategy of \cite[Theorem~1.1]{GogicTomasevic-AM}. The classification relies on the fact that maps satisfying \eqref{eq:a certain product} preserve $(k+1)$-potents along with their ranks and natural partial order, and that they are orthoadditive on these elements. These notions will be introduced and discussed in the next section. A notable difference from the case $k=1$, which corresponds to idempotents, is that when $\operatorname{char}(\F)$ is positive and divides $k$, $(k+1)$-potents may fail to be diagonalizable (see Remark~\ref{rem:k-potents not diagonalizable}). This phenomenon requires more refined arguments in order to handle the mixed Jordan--power preservers effectively. We also note that linear and closely related variants of $k$-potent preservers on matrix and operator algebras have been studied in papers such as \cite{HouHou,SongCao,YouWang}.

The proof of Theorem~\ref{thm:main} (given in Section~\S\ref{sec:main}) “follows the general strategy of \cite[Theorem~1.1]{GogicTomasevic-AM}.” The classification “relies on the fact that maps satisfying \eqref{eq:a certain product} preserve $(k+1)$-potents along with their ranks and natural partial order, and that they are orthoadditive on these elements.” A key complication relative to the case $k=1$ (idempotents) is that if $\operatorname{char}(\F)$ is positive and divides $k$, then “$(k+1)$-potents may fail to be diagonalizable,” which “requires more refined arguments in order to handle the mixed Jordan--power preservers effectively.”

The proof of the main theorem (given later) “follows the general strategy of Gogic--Tomasevic, Theorem~1.1 (GogicTomasevic-AM).” The classification “relies on the fact that maps satisfying
\begin{equation}\label{eq:a certain product}
			\phi(A^k \circ B) = \phi(A)^k \circ \phi(B), \qquad \text{for all } A,B \in M_n(\F),
		\end{equation}
preserve $(k+1)$-potents along with their ranks and natural partial order, and that they are orthoadditive on these elements.” A key complication relative to the case $k=1$ (idempotents) is that if $\operatorname{char}(\F)$ is positive and divides $k$, then “$(k+1)$-potents may fail to be diagonalizable,” which “requires more refined arguments in order to handle the mixed Jordan--power preservers effectively.”

		\label{thm:main}
		Let $\F$ be an algebraically closed field of characteristic different from $2$, and let $n,k \in \N$ with $n \ge 2$. A map $\phi : M_n(\F) \to M_n(\F)$
		satisfies the mixed Jordan--power identity
		\begin{equation}\label{eq:a certain product}
			\phi(A^k \circ B) = \phi(A)^k \circ \phi(B), \qquad \text{for all } A,B \in M_n(\F),
		\end{equation}
		if and only if one of the following holds:
		\begin{itemize}
			\item[(a)] $\phi$ is constant, taking a fixed $(k+1)$-potent value;
			\item[(b)] there exist an invertible matrix $T \in M_n(\F)$, a ring monomorphism $\omega : \F \to \F$, and a $k$-th root of unity $\varepsilon \in \F$ such that
			\begin{equation}\label{eq:form of phi}
				\phi(X) = \varepsilon\, T\, \omega(X)\, T^{-1}
				\quad \text{ or } \quad
				\phi(X) = \varepsilon\, T\, \omega(X)^t\, T^{-1}.
			\end{equation}
		\end{itemize}
		In particular, every nonconstant solution of the mixed Jordan--power identity
		\begin{equation}\label{eq:a certain product}
			\phi(A^k \circ B) = \phi(A)^k \circ \phi(B), \qquad \text{for all } A,B \in M_n(\F),
		\end{equation}
		is additive.,

Biconditional or Equivalence
Classification or Bijection

Question:
Let \(\mathbb F\) be an algebraically closed field with \(\operatorname{char}(\mathbb F)\neq 2\), let \(n\ge 2\) and \(k\in\mathbb N\), and write \(A\circ B:=\tfrac12(AB+BA)\) for the Jordan product on \(M_n(\mathbb F)\). Which maps \(\phi:M_n(\mathbb F)\to M_n(\mathbb F)\) satisfy
\[
\phi(A^k\circ B)=\phi(A)^k\circ \phi(B)\qquad\text{for all }A,B\in M_n(\mathbb F)?
\]

Exactly those maps \(\phi\) for which either: (i) \(\phi\) is constant, with value a fixed \((k+1)\)-potent matrix \(P\) (that is, \(P^{k+1}=P\)); or (ii) there exist an invertible matrix \(T\in M_n(\mathbb F)\), a ring monomorphism \(\omega:\mathbb F\to\mathbb F\), and a \(k\)-th root of unity \(\varepsilon\in\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=\varepsilon T\,\omega(X)\,T^{-1}\qquad\text{or}\qquad \phi(X)=\varepsilon T\,\omega(X)^t\,T^{-1},
\]
where \(\omega(X)\) means applying \(\omega\) entrywise. In particular, every nonconstant such \(\phi\) is additive.

Choices:
B.
Exactly those maps \(\phi\) for which either: (i) \(\phi\) is constant, with value a fixed \((k+1)\)-potent matrix \(P\); or (ii) there exist an invertible matrix \(T\in M_n(\mathbb F)\) and a ring monomorphism \(\omega:\mathbb F\to\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=T\,\omega(X)\,T^{-1}\qquad\text{or}\qquad \phi(X)=T\,\omega(X)^t\,T^{-1}.
\]
In particular, every nonconstant such \(\phi\) is additive.
C.
Every map \(\phi:M_n(\mathbb F)\to M_n(\mathbb F)\) of one of the following forms satisfies
\[
\phi(A^k\circ B)=\phi(A)^k\circ\phi(B)\qquad\text{for all }A,B\in M_n(\mathbb F):
\]
either \(\phi\) is constant with value a fixed \((k+1)\)-potent matrix \(P\), or there exist an invertible matrix \(T\in M_n(\mathbb F)\), a ring monomorphism \(\omega:\mathbb F\to\mathbb F\), and a \(k\)-th root of unity \(\varepsilon\in\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=\varepsilon T\,\omega(X)\,T^{-1}\qquad\text{or}\qquad \phi(X)=\varepsilon T\,\omega(X)^t\,T^{-1}.
\]
D.
Exactly those maps \(\phi\) for which either: (i) \(\phi\) is constant, with value a fixed idempotent matrix \(P\) (that is, \(P^2=P\)); or (ii) there exist an invertible matrix \(T\in M_n(\mathbb F)\), a ring monomorphism \(\omega:\mathbb F\to\mathbb F\), and a scalar \(\varepsilon\in\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=\varepsilon T\,\omega(X)\,T^{-1}\qquad\text{or}\qquad \phi(X)=\varepsilon T\,\omega(X)^t\,T^{-1},
\]
where \(\omega(X)\) is applied entrywise. In particular, every nonconstant such \(\phi\) is additive.
E.
Exactly those maps \(\phi\) for which either: (i) \(\phi\) is constant, with value a fixed \((k+1)\)-potent matrix \(P\); or (ii) there exist an invertible matrix \(T\in M_n(\mathbb F)\), a field automorphism \(\omega:\mathbb F\to\mathbb F\), and a \(k\)-th root of unity \(\varepsilon\in\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=\varepsilon T\,\omega(X)\,T^{-1}.
\]
In particular, every nonconstant such \(\phi\) is additive.

replaced classification iff by one-way sufficient condition

constant-value type and unrestricted scalar factor

transpose branch and monomorphism generality


Question:
Let $\mathbb F$ be an algebraically closed field with $\operatorname{char}(\mathbb F)\neq 2$, let $n,k\in\mathbb N$ with $n\ge 2$, and let $\circ$ denote the normalized Jordan product on $M_n(\mathbb F)$, defined by $A\circ B=\tfrac12(AB+BA)$. Which of the following statements is equivalent to saying that a map $\phi:M_n(\mathbb F)\to M_n(\mathbb F)$ satisfies
\[
\phi(A^k\circ B)=\phi(A)^k\circ\phi(B)\qquad\text{for all }A,B\in M_n(\mathbb F)?
\]

Either $\phi$ is constant with value a fixed $(k+1)$-potent matrix (that is, a matrix $P$ such that $P^{k+1}=P$), or there exist an invertible matrix $T\in M_n(\mathbb F)$, a ring monomorphism $\omega:\mathbb F\to\mathbb F$, and a $k$-th root of unity $\varepsilon\in\mathbb F$ such that for every $X\in M_n(\mathbb F)$,
\[
\phi(X)=\varepsilon\,T\,\omega(X)\,T^{-1}
\quad\text{or}\quad
\phi(X)=\varepsilon\,T\,\omega(X)^t\,T^{-1},
\]
where $\omega(X)$ is obtained by applying $\omega$ entrywise. In particular, every nonconstant such map is additive.

Choices:
B.
Either $\phi$ is constant with value a fixed idempotent matrix, or there exist an invertible matrix $T\in M_n(\mathbb F)$, a field automorphism $\omega:\mathbb F\to\mathbb F$, and a scalar $\varepsilon\in\mathbb F$ such that for every $X\in M_n(\mathbb F)$,
\[
\phi(X)=\varepsilon\,T\,\omega(X)\,T^{-1}
\quad\text{or}\quad
\phi(X)=\varepsilon\,T\,\omega(X)^t\,T^{-1},
\]
with $\varepsilon^{k+1}=1$. In particular, every nonconstant such map is additive.
C.
Either $\phi$ is constant with value a fixed $(k+1)$-potent matrix, or there exist an invertible matrix $T\in M_n(\mathbb F)$, a ring monomorphism $\omega:\mathbb F\to\mathbb F$, and a scalar $\varepsilon\in\mathbb F$ such that for every $X\in M_n(\mathbb F)$,
\[
\phi(X)=\varepsilon\,T\,\omega(X)\,T^{-1}
\quad\text{or}\quad
\phi(X)=\varepsilon\,T\,\omega(X)^t\,T^{-1},
\]
where $\omega(X)$ is obtained by applying $\omega$ entrywise.
D.
Either $\phi$ is constant with value a fixed $(k+1)$-potent matrix, or for every nonconstant such map there exist an invertible matrix $T\in M_n(\mathbb F)$, a ring monomorphism $\omega:\mathbb F\to\mathbb F$, and a function $\varepsilon:M_n(\mathbb F)\to\mathbb F$ whose values are $k$-th roots of unity such that for every $X\in M_n(\mathbb F)$,
\[
\phi(X)=\varepsilon(X)\,T\,\omega(X)\,T^{-1}
\quad\text{or}\quad
\phi(X)=\varepsilon(X)\,T\,\omega(X)^t\,T^{-1}.
\]
In particular, every nonconstant such map is additive.
E.
Either $\phi$ is constant with value a fixed $(k+1)$-potent matrix, or there exist an invertible matrix $T\in M_n(\mathbb F)$ and a ring monomorphism $\omega:\mathbb F\to\mathbb F$ such that for every $X\in M_n(\mathbb F)$,
\[
\phi(X)=T\,\omega(X)\,T^{-1}
\quad\text{or}\quad
\phi(X)=T\,\omega(X)^t\,T^{-1}.
\]
In particular, every nonconstant such map is additive.

correct potent class and root-of-unity order

dropped the conclusion that every nonconstant solution is additive and weakened the restriction on $\varepsilon$

global scalar factor replaced by pointwise dependence on $X$

missing $k$-th-root-of-unity scalar factor


\begin{theorem}\label{EucTrees}
 In the Euclidean setting, with edges labeled randomly by $n-1$ colors, we have $Z_{\MST} = \Theta(\sqrt n)$ w.h.p.\ and $Z_{\MST}-Z^*_{\MST}=\Theta(\sqrt{n})$ w.h.p.\end{theorem}

The minimum length of structures like spanning trees and Hamilton cycles among random points in Euclidean space has been a topic of interest for the better part of a century. Beardwood, Halton and Hammersley \cite{BHH} showed that the minimum cost of a tour through $n$ random points in the unit square is asymptotic to $\b_{TSP} n^{1/2}$ with high probability (we say an event $\cE_n$ occurs {\em with high probability} or w.h.p.\ if $\mathbb{P}(\cE_n) \rightarrow 1$ as $n \rightarrow \infty$). Here $\b_{TSP}$ is a positive constant, still unknown after more than 60 years. Steele \cite{Steele1} generalised this result to {\em Euclidean Functionals} and showed that the minimum cost of a spanning tree on $n$ random points is asymptotic to $\b_{MST} n^{1/2}$, w.h.p. In both cases, the cost of edges is determined by the Euclidean distance between points. Steele's work shows that there are many instances of optimisation problems where the expected minimum cost grows like $\beta n^{1/2}$ for unknown constants $\beta$. Frieze and Pegden \cite{FP1} showed the constants for lengths of lower bound structures for the tour (like the minimum spanning tree, or twice a matching) really are distinct from the constant for the TSP, even though the best known explicit bounds on the constants generally overlap.

When the costs are generated independently, the situation becomes much clearer. In the case of spanning trees where the edges of the complete graph $K_n$ are given independent uniform $[0,1]$ random costs, Frieze \cite{F1} showed that $\mathbb{E}(L_n)\sim\zeta(3)=\sum_{k=1}^\infty\frac{1}{k^3}$, where $L_n$ denotes the (random) minimum cost of a spanning tree.

The expected cost of the minimum cost Hamilton cycle in the complete graph $K_{n}$ is asymptotic to $\tau=\frac12\int_{x=0}^\infty y(x)dx$ where $y$ is the positive solution to $\brac{1+\frac{x}{2}}e^{-x}+\brac{1+\frac{y}{2}}e^{-y}=1$. This was proved by Wastl\"und \cite{Wast1}. Frieze \cite{F2} showed that the expected minimum cost of a Hamilton cycle is asymptotically equal to the expected minimum cost of a 2-factor.

If we orient the edges then it follows from the work of Karp \cite{K1} and Aldous \cite{A1} that the expected minimum cost of a directed Hamilton cycle is asymptotic to $\zeta(2)=\sum_{k=1}^\infty\frac{1}{k^2}=\frac{\pi^2}{6}$.

In this work we add a spot of color to the problems. We assume that the edges of the graphs in question have been given uniformly random colors and ask for minimum cost rainbow Hamilton cycles or spanning trees. In this context, a set $S$ of edges is {\em rainbow} colored if all the colors in $S$ are distinct. We do this within two contexts: either we have $n$ random points in $[0,1]^2$ and the cost of an edge is the Euclidean distance between its endpoints, or else we have independent uniform $[0,1]$ random costs.

In the first context we let ${\mathcal X}=\left\{\bx_1,\bx_2,\ldots,\bx_n\right\}$ where the $\bx_i$ are independently chosen uniformly from the unit square $[0,1]^2$. The cost of edge $\left\{\bx_i,\bx_j\right\}$ is the Euclidean length $|\bx_i-\bx_j|$. Our first results concern the length $Z_{\mathrm{MST}}$ of the minimum rainbow spanning tree in this model, when we randomly color labeled from a set of $n-1$ colors. (Note that with fewer colors, no rainbow spanning tree would exist. Also, even if we have enough colors there is always some small probability that there is no rainbow spanning tree at all, in which case $Z_{\mathrm{MST}}$ is undefined.) We let $Z^*_{\mathrm{MST}}$ denote the length of the minimum spanning tree (without the rainbow constraint); recall that $Z^*_{\mathrm{MST}}\sim \beta_{\mathrm{MST}} \sqrt n$ w.h.p., for some constant $\beta_{\mathrm{MST}}$.

\begin{theorem}\label{EucTours}
In the Euclidean setting, with edges labeled randomly by $q=(1+\ep)n$ colors, we have (a) $Z_{\TSP}=O(n^{1/2}\log n)$, and (b) $Z_{\TSP}-Z_{\TSP}^*=\Omega(\sqrt{n})$ w.h.p. Indeed, (a) holds for any set of $n$ points with randomly colored edges.
\end{theorem}
Here like before the starred version $Z^*_{\TSP}$ represents the minimum length in the absence of the rainbow condition.

We define a weighted bipartite (multi-)graph $\G$ with bipartition $\cX_1 \cup C$ where $C=[n-1]$ is our set of colors. The edge set will be the union $E_1\cup E_2$ of sets which we define next; for both sets $E_1$ and $E_2$, an edge $(\bx,c)$ will correspond to some edge $(\bx,\bx')$ with $\xi_2(\bx) < \xi_2(\bx')$ in the embedded graph that has been assigned color $c$. This last property guarantees that a perfect matching in $\G$ corresponds to a rainbow spanning tree in the embedded graph. So to prove the first assertion in Theorem~\ref{EucTrees}, it suffices to show that w.h.p.
\begin{equation}\label{match}
\text{$\G$ contains a perfect matching,}
\end{equation}
and
\begin{equation}\label{wt}
\text{the total weight of all edges in $\G$ is $O(n^{1/2})$.}
\end{equation}

\begin{claim}\label{clm:3}
 If the event $\cG$ holds (where $\cG$ is defined above \eqref{negG}), then for any fixed $S \subseteq A$ of size $s$, (under the probability of choosing random colors for each edge)
\[
 \Pr(\cE_c(S)\mid A) \leq \bfrac{s}{|A|}^{\sqrt{K}} + O(n^{-\sqrt{K} + 2}\log^{8\sqrt{K}} n).
\]
 \end{claim}
 \begin{proof}
 Fix $\bx\in A$ and recall the definition of $\nu_\bx$ above \eqref{negG}. Let $\cA$ be the event that for each $\bx \in A$, each color appears fewer than $\sqrt{K}$ times on edges of length at most $2Bn^{-1/2}\log^4 n$. The probability of this event failing for fixed $\bx \in A$ is at most
 \[|C| {\nu_\bx \choose \sqrt{K}}|C|^{-\sqrt{K}},\]
 and taking a union bound over $A$ and using \eqref{negG} gives the second bound in the claim.

We begin with a simple lemma.
\begin{lemma}\label{repeat}
Let $S$ be a set of size $\a=\Theta(n)$ that is randomly colored using $\b=\Theta(n)$ colors. Then w.h.p. there are $\Theta(n)$ colors that are used more than once.
\end{lemma}
\begin{proof}
Let $Z$ denote the number of colors that appear more than once. Then
\[
\E(Z)=\b\brac{1-\brac{1-\frac{1}{\b}}^{\a}-\a \cdot \frac{1}{\b}\brac{1-\frac{1}{\b}}^{\a-1}}=\Theta(n).
\]
Now changing the color of one edge changes $Z$ by at most one. So, applying McDiarmid's inequality \cite{McD} we have
\[
\Pr\brac{Z\leq \E(Z)/2}\leq \exp\set{-\frac{\E(Z)^2}{2\a}}=e^{-\Omega(n)}.
\]
\end{proof}
Now back to the main argument. For notational convenience, in this section we scale $\cX$ so that we are instead working with a set $\cY_n$ of $n$ random points in a square of side length $\sqrt{n}$; note that there is thus one point on average, per unit of area.

Let $X$ be a finite set and let $\cH$ be an $r$-bounded hypergraph on $X$. For $k \geq r$, let $X^* = X \times [k]$. Let $\cH^*$ denote the family consisting of all possible rainbow copies of $\cH$ using the color set $[k]$. Let $\xi_x\,(x\in X)$ be independent random variables, each uniform from $[0,1]$, and set
\[
\xi_{\cH}=\min_{S\in \cH^*}\sum_{x\in S}\xi_x.
\]
We prove
\begin{theorem}\label{rweight}
Let $\cH$ be a $\k$-spread, $r$-uniform hypergraph on vertex set $X$. Suppose that (i) $\k=\Omega(r)$ and (ii) $|X|\leq \k^2r/\log^5r$. If we randomly color the elements of $\cH$ with $k\geq r$ colors, then $\xi_\cH\leq 3C_0r/\k$ w.h.p. (assuming $r\to\infty$) for some absolute constant $C_0>0$.
\end{theorem}
(The paper \cite{HY} removes conditions (i) and (ii), but the proof doesn't adapt well to deal with the costs, as far as we can see.)
\begin{proof}
We let $\ell_0$ be the least $i$ such that $r/2^i<\log r$, and let $\ell$ be the least $i>\ell_0$ such that $r/2^i<1$. We sort the elements of $X=\set{x_1,x_2,\ldots,x_N}$ so that $\xi_{x_1}<\xi_{x_2}<\cdots<\xi_{x_N}$. We then let $p_i=C_i/\k$ where
\[
C_i=\begin{cases}C_0&i\leq \ell_0.\\\frac{\log r}{\log\log r}&\ell_0<i\leq \ell.\end{cases}
\]
Here $C_0$ is a large constant. Let $L_i=Np_i$ and define $n_0$ to be the smallest integer such that $2^{-n_0}\leq \log r$ and let $n$ be the least $i>n_0$ such that $r/2^i<1$. Then for $i\in[n]$ we let
\[
a_i=\sum_{j=1}^iL_i\text{ and }\e_i=\frac{2a_i}{N}.
\]
Then let $W^*_1=\set{x_1,x_2,\ldots,x_{a_1}},\,W^*_2=\set{x_{a_1+1},x_{a_1+2},\ldots,x_{a_2}}$ etc. Note that $W^*_1,W^*_2,\ldots,W^*_n$ are randomly colored random sets of size $L_i$ . The Chernoff bounds imply that
\begin{align}
\Pr(\exists i:\max\set{\xi_x:x\in W^*_i}\geq \e_i)&=\sum_{i=1}^n\Pr(Bin\brac{N,\e_i}\leq\e_i N)\nn\\
& \leq \sum_{i=1}^n e^{-\e_iN/8}\leq 2e^{-a_1/4}=2e^{-C_0N/\k}.\label{max}
\end{align}
In \cite{BFM} (as we will explain shortly), it is proved that with probability $1-\frac{\log\log r}{\log^{1/4}r}$, there is a rainbow colored edge $e$ such that $|e\setminus \bigcup_{j\leq i}W^*_j|\leq r/2^{i-1}$. We will use the main theorem (and its proof which we discuss below) from \cite{BFM}:
\begin{theorem}[Theorem 2 in \cite{BFM}]\label{thrainbow}
Let $\cH$ be an $r$-bounded, $\k$-spread hypergraph and let $X=V(\cH)$ be randomly colored from $Q=[q]$ where $q\ge r$. Suppose also that (i) $\k=\Omega(r)$, that is, there exists a constant $L>0$ such that $\kappa\ge Lr$ for all valid $r$, and that (ii) $N\leq \k^2r/\log^5r$. Then there is a constant $C>0$ such that if
\beq{mbounds}{
m\geq\frac{(C\log_2r)|X|}{\k}
}
then $X_m$ contains a rainbow colored edge of $\cH$ w.h.p.
\end{theorem}
 The proof of Theorem \ref{thrainbow} in \cite{BFM} proceeds by iteratively choosing random sets of colored elements $W^*_i$ (just as we have in our current proof). Theorem \ref{thrainbow} is proved by iteratively applying the lemma we state next. Here $\cH^*$ is the set of possible rainbow colored edges of $\cH$. For $H^* \in \cH^*$, $T^*(W^*,H^*)$ is $G^* \setminus W^*$ for some $G^* \in \cH^*$.
 \begin{lemma}[Lemma 5 in \cite{BFM}] \label{ML}
Let $H^*\in\cH^*$ be good with respect to $W^*$ if $H^*\sim W^*$ and $|T^*(W^*,H^*)|<r/2$. Let
\[
\text{$\cS$ be the event $|\set{H^*\in \cH^*:H^*\text{ is good}}|<(1-\e)|\cH^*|(1-p)^r$.}
\]
Then
\[
\Pr(\cS)\leq \exp\set{-\frac{\e^2\k^2q(1-p)}{16e^3N}} +\frac{2K}{C_0^{r/3}\e\brac{1-p}^r}
\]
\end{lemma}
In other words, so long as the unlikely event $\cS$ does not occur, we have that for each rainbow colored edge $H^*$, there is some rainbow colored edge $G^*$ that is mostly contained in $W^*$. The proof of Theorem \ref{thrainbow} proceeds by iteratively applying Lemma \ref{ML}, each time replacing $\cH^*$ with the hypergraph whose edges are the $T^*(W^*, H^*)$ (i.e. portions of edges of $\cH^*$ which are not contained in $W^*$). In \cite{BFM} they show that during all applications of the Lemma \ref{ML},
 the unlikely event $\cS$ never happens w.h.p. This means that the rainbow edge $H^*$ we obtain in the end was good for every application of Lemma \ref{ML}. In particular, we have
 \beq{forcost}{
|H^*\setminus \bigcup_{j\leq i}W^*_j|\leq \frac{r}{2^{i-1}}, \qquad \text{i=1, \ldots, $\ell$.}
}
In our setting, since $W^*_i$ was the set of vertices of cost at most $\e_i$, the rainbow edge $H^*$ we obtain from Theorem \ref{thrainbow} costs at most
\mults{
\sum_{i=1}^n\frac{r\e_i}{2^{i-1}}=\frac{r}{N}\sum_{i=1}^n\frac{1}{2^{i-1}}\sum_{j=1}^iNp_i\\
=r\brac{\sum_{i=1}^{\ell_0}\frac{C_i}{2^{i-1}\k}+\sum_{i=\ell_0+1}^\ell\frac{C_{\ell_0}+(i-\ell_0)\log r}{2^{i-1}\k}}\leq \frac{3C_0r}{\k}=O(1).
}

In particular, Theorem \ref{EucTrees} says that the minimum length doesn't change by more than a constant factor under a rainbow constraint, but that a constant-factor change does necessarily occur.  To put Theorem \ref{EucTrees} in context, there is an important sense in which it is much less obvious that the minimum rainbow spanning tree should have length $\Theta(\sqrt{n})$ than it is for the various optimization problems (matchings, 2-factors, linear programming relaxations of the TSP, etc) to which the general methods derived from Beardwood, Halton, and Hammersley and from Steele have been applied.  In particular, the rainbow condition seems to preclude a straightforward application of subadditivity, which explains, for example, why we do not yet know that $Z_{\mathrm{MST}}/\sqrt{n}$ has a limit.  In some sense, the success of subadditivity approaches to structure lengths in random point-sets reflects the sense in which the asymptotic structure lengths really depend only on the local geometry of the ambient space.  (For example, in the Beardwood-Halton-Hammersley theorem, the unit square can be replaced by any bounded open set of $\mathbb{R}^2$ of measure 1, and the asymptotic length of a tour through $n$ points remains the same, with the same unknown constant $\beta_{\mathrm{TSP}}$. But the rainbow condition we impose is a global constraint on the spanning tree that cannot be easily decomposed into noninteracting parts.

For the minimum length $Z_{\mathrm{TSP}}$ of a rainbow Hamilton cycle in the Euclidean model, we do not achieve an optimal result up to constant factors unless we allow a slight excess in the number of available colors.

\begin{theorem}\label{EucTours}
In the Euclidean setting, with edges labeled randomly by $q=(1+\varepsilon)n$ colors, we have (a) $Z_{\mathrm{TSP}}=O(n^{1/2}\log n)$, and (b) $Z_{\mathrm{TSP}}-Z_{\mathrm{TSP}}^*=\Omega(\sqrt{n})$ w.h.p. Indeed, (a) holds for any set of $n$ points with randomly colored edges.
\end{theorem}
Here like before the starred version $Z^*_{\mathrm{TSP}}$ represents the minimum length in the absence of the rainbow condition.

In the second scenario, $[0,1]^2$ is replaced by the complete graph $K_n$ and the costs $C(i,j)$ are independent uniform $[0,1]$ random variables.  We prove a general theorem (see Theorem \ref{rweight}) for cheap rainbow substructures in randomly colored structures with random costs. The proof is an adaptation of some of the recent work on thresholds in spread hypergraphs (see \cite{BFM}) based on Frankston, Kahn, Narayanan, and Park's recent proof of the fractional Kahn--Kalai conjecture \cite{FKNP}. Our Theorem \ref{rweight} implies the following:
\begin{theorem}\label{th2}
The minimum cost solution of the following problems have value $O(1)$ w.h.p. The proofs are non-constructive.
\begin{enumerate}[(a)]
\item Traveling Salesperson Problem in $K_n$ with at least $n$ colors for the edges.
\item Minimum cost perfect matching in complete $k$-uniform hypergraphs, $k\geq 2$ with at least $n/k$ colors for the edges.
\item Minimum cost classes of degree bounded spanning trees in $K_n$ with at least $n-1$ colors for the edges e.g. spanning binary trees.
\end{enumerate}
\end{theorem}
The rest of the paper is organized as follows. In Section \ref{sec:treeUB} we prove the first assertion of Theorem \ref{EucTrees}, i.e.\ that $Z_{\mathrm{MST}} = \Theta(\sqrt{n})$ w.h.p. In Section \ref{sec:tours} we prove the first assertion of Theorem \ref{EucTours}, i.e.\ that $Z_{\mathrm{TSP}}=O(\sqrt {n}\log n)$ for $n$ colors. In Section \ref{sec:separate} we prove the remaining assertions in Theorems \ref{EucTrees} and \ref{EucTours}, i.e.\ that $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=\Theta(\sqrt{n})$ for $n$ colors and that $Z_{\mathrm{TSP}}-Z_{\mathrm{TSP}}^*=\Theta(\sqrt{n})$ for $(1+\varepsilon)n$ colors. In Section \ref{sec:spread} we prove Theorem \ref{th2}. Finally, we make some observations and state open questions in Section \ref{sec:conclusion}.

The post-theorem discussion does not give a proof outline for Theorem~\ref{EucTrees} beyond noting why it is "much less obvious" (the rainbow condition is a "global constraint" that "seems to preclude a straightforward application of subadditivity"). The only structural indication is the paper roadmap: in Section~\ref{sec:treeUB} they "prove the first assertion of Theorem~\ref{EucTrees}, i.e.\ that $Z_{\mathrm{MST}}=\Theta(\sqrt n)$ w.h.p.," and in Section~\ref{sec:separate} they "prove the remaining assertions in Theorem~\ref{EucTrees}\ldots i.e.\ that $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=\Theta(\sqrt n)$ for $n$ colors."

\label{EucTrees}
 In the Euclidean setting, with edges labeled randomly by $n-1$ colors, we have $Z_{\mathrm{MST}} = \Theta(\sqrt n)$ w.h.p.\ and $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=\Theta(\sqrt{n})$ w.h.p.,

Question: Let ${\mathcal X}=\{\mathbf x_1,\dots,\mathbf x_n\}$ be $n$ points chosen independently and uniformly from the unit square $[0,1]^2$. Form the complete graph on ${\mathcal X}$, give each edge the Euclidean length of its endpoints as its cost, and independently label the edges uniformly at random with colors from a set of exactly $n-1$ colors. A spanning tree is called rainbow if all of its edges have distinct colors. Let $Z_{\mathrm{MST}}$ be the minimum total Euclidean length of a rainbow spanning tree, and let $Z^*_{\mathrm{MST}}$ be the minimum total Euclidean length of an ordinary spanning tree (with no rainbow constraint). Here “with high probability” means with probability tending to $1$ as $n\to\infty$. As $n\to\infty$, which asymptotic statement holds?

With high probability, $Z_{\mathrm{MST}}=\Theta(\sqrt n)$ and also $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=\Theta(\sqrt n)$.

Choices:
B. With high probability, $Z_{\mathrm{MST}}=\Theta(\sqrt n)$ and also $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=o(\sqrt n)$.
C. With high probability, $Z_{\mathrm{MST}}=\Theta(\sqrt n)$.
D. With high probability, $Z_{\mathrm{MST}}=\Theta(\sqrt n\log n)$ and also $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=\Theta(\sqrt n\log n)$.
E. With high probability, $Z_{\mathrm{MST}}=\Theta(\sqrt n)$ and also $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=\Theta(1)$.