\begin{thm}\label{XiaFock}
		We have
		\begin{equation*}
			C^{*} (\mathcal{WL})=\mathcal{T}^{(1)}.
		\end{equation*}
	\end{thm}

For some \(\alpha>0\), \(p>0\) and \(dV\) the standard volume measure on \(\mathbb{C}^n\), let \(L^{p}_{\alpha}(\mathbb{C}^n,dV)\) be the Lebesgue space of measurable functions \(f\) on \(\mathbb{C}^n\) such that
	\begin{equation}\label{Fnorm1}
		\left\lVert f \right\rVert^{p}_{\alpha} := \left(\frac{p\alpha}{2\pi}\right)^{n} \int_{\mathbb{C}^n} |f(z)e^{-\frac{\alpha}{2}|z|^2}|^{p} dV(z) < \infty~.
	\end{equation}
	Similarly, for \(\alpha>0\) and \(p=\infty\), we denoted by \(L^{\infty}_{\alpha}\) the space of Lebesgue measurable function \(f\) on \(\mathbb{C}^n\) such that
	\begin{equation*}
		\left\lVert f \right\rVert_{\infty,\alpha} := \text{esssup}\{|f(z)|e^{-\frac{\alpha|z|^2}{2}} : ~z \in \mathbb{C}^n\}< \infty ~.
	\end{equation*}
	The classical Fock space \(F^{p}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \( L^{p}_{\alpha}(\mathbb{C}^n, dV)\). Similarly, the Fock space \(F^{\infty}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \(L^{\infty}_{\alpha}\).

	Let \(d\mu\) be the Gaussian measure on \(\mathbb{C}^n\), \(n\ge 1\). In terms of the standard volume measure \(dV\) on \(\mathbb{C}^n\), it is given by
	\begin{equation*}
		d\mu(z) = \pi^{-n} e^{-|z|^2} dV(z)~.
	\end{equation*}

	 The Fock space \(H^{2}(\mathbb{C}^n,d\mu)\) is defined to be the subspace of the (Hilbert-) Lebesgue space \(L^{2}(\mathbb{C}^n,d\mu)\) consisting of entire functions. Notice that \(H^{2}(\mathbb{C}^n,d\mu) = F^{2}_{1} \). The symbol \(K_z\) denotes the reproducing kernel and the symbol \(k_z\) denotes the normalized reproducing kernel for \(H^{2}(\mathbb{C}^n,d\mu)\). That is,
	\begin{equation*}
		K_z(\zeta) = e^{\langle \zeta, z \rangle}, \quad k_{z}(\zeta) = e^{\langle \zeta, z\rangle}e^{-\frac{|z|^2}{2}}~, \quad z,\zeta \in \mathbb{C}^n.
	\end{equation*}

	In \cite{xia2015localization}, J. Xia showed in the case of the Bergman space on the unit ball of \(\mathbb{C}^n\) that the norm closure of \(\{T_{f}: f \in L^{\infty}(B,dv) \}\) coincides with the \(C^{*}\)-algebra of weakly localized operators. Also, he stated in \cite[Section 4 ]{xia2015localization}  that  the analogue of \cite[Theorem 1.5]{xia2015localization} on the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\) was true. In this paper, we define the notion of weakly localized operators, state Xia's theorem for the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\) and provide details of its proof. Further, we present a consequence of this theorem on the compactness of operators on \(H^{2}(\mathbb{C}^n,d\mu)\).  We begin with the following definitions and we state the main theorem, the proof of which will retain our attention in the following sections.

	\begin{defn}
		For \(f \in L^{\infty}(\mathbb{C}^n,dV) \), the \textbf{Toeplitz operator} \(T_{f}\) is defined by the formula
		\begin{equation*}
			T_{f}h = P(fh)~,\quad h \in H^{2}(\mathbb{C}^n,d\mu)~,
		\end{equation*}
		where \(P \colon L^{2}(\mathbb{C}^n,d\mu) \rightarrow H^{2}(\mathbb{C}^n,d\mu)\) is the orthogonal projection.
	\end{defn}

	The \textbf{standard lattice} in \(\mathbb{C}^n\) is denoted by
	\begin{equation*}
		\mathbb{Z}^{2n} = \{(m_{1}+il_{1}, \dots, m_{n}+il_{n}): m_{1}, l_{1}, \dots, m_{n},l_{n} \in \mathbb{Z}\}~.
	\end{equation*}
	We fix an orthonormal set \(\{e_{u}: u \in \mathbb{Z}^{2n}\}\) in \(H^{2}(\mathbb{C}^n,d\mu)\). We let \(S\) denote the \textbf{fundamental unit cube in }\(\mathbb{C}^{n}\). That is,
	\begin{equation*}
		S = \{(x_{1}+iy_{1}, \dots, x_{n}+iy_{n}) : x_{1},y_{1}, \dots , x_{n},y_{n} \in [0,1)\}.
	\end{equation*}
	With \(\mathbb{Z}^{2n}\) and \(S\), we have
	\begin{equation*}
		\cup_{u \in \mathbb{Z}^{2n}} \{S+u\} = \mathbb{C}^{n} = \cup_{u \in \mathbb{Z}^{2n}} \{u-S\} ~,
	\end{equation*}
	which is a tiling of the space, meaning that there is no overlap between \(S+u\) and \(S+v\) for \(u \neq v\) in \(\mathbb{Z}^{2n}\) (resp. between \(u-S\) and \(v-S\) for \(u \neq v \in \mathbb{Z}^{2n}\) ).

	\begin{defn}
		Let \(\mathcal{T}^{(1)}\) denote the norm closure of \(\{T_{f}: f\in L^{\infty}(\mathbb{C}^n,dV)\}\) in \(\mathcal{B}(H^2(\mathbb{C}^n,d\mu))\) with respect to the operator norm. That is
		\begin{equation*}
			\mathcal{T}^{(1)} = \{B : \lim_{k \to \infty } \norm{B-T_{b_{k}}}=0 , b_{k} \in L^{\infty}(\mathbb{C}^{n},dV)\}.
		\end{equation*}
	\end{defn}

	\begin{defn}
		We denote by \(\mathcal{S}\) the linear span of the normalized reproducing kernels \(k_z\). A linear operator \(B : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) is said to be \textbf{admissible} on \( H^{2}(\mathbb{C}^n,d\mu) \) if there exists a linear operator \( B^{*} : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) such that the duality relation
		\begin{equation}\label{R1}
			\langle Bk_{z}, k_{w} \rangle = \langle k_{z}, B^{*}k_w \rangle
		\end{equation}
		holds for all \(z,w \in \mathbb{C}^n\).
	\end{defn}
	The inner product here is with respect to \(d\mu\).

	We define below sufficiently localized operators following J. Xia and D. Zheng (XZ) in \cite{xia2013localization}.
	\begin{defn}
		A bounded linear operator \(B\) on \(H^{2}(\mathbb{C}^n,d\mu)\) is said to be XZ-\textbf{sufficiently localized} if there exist constants \(2n<\beta<\infty\) and \(0<C<\infty\) such that
		\begin{equation}\label{SL1}
			|\langle Bk_{z},k_{w}\rangle| \le \frac{C}{(1+|z-w|)^{\beta}}
		\end{equation}
		for all \(z,w\in \mathbb{C}^n\).
	\end{defn}

		\begin{defn}
		An admissible operator \(B\) on \(H^{2}(\mathbb{C}^n,d\mu)\) is said to be \textbf{weakly localized} if it satisfies the following four conditions
		\begin{equation*}
			\sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle Bk_{z}, k_{w} \rangle| dV(w) <\infty, \quad \sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle B^{*} k_{z}, k_{w} \rangle| dV(w) <\infty
		\end{equation*}

		and
		\begin{equation*}
			\lim_{r\to\infty}\sup_{z\in \mathbb{C}^n}\int_{|z-w|\ge r} |\langle Bk_{z}, k_{w} \rangle| dV(w) =0, \quad \lim_{r\to\infty}\sup_{z\in \mathbb{C}^n}\int_{|z-w|\ge r} |\langle B^{*} k_{z}, k_{w} \rangle| dV(w) =0~.
		\end{equation*}
	\end{defn}

	\begin{exa}\label{Ex1}
		If \(f\) is a \textbf{bounded measurable function} on \(\mathbb{C}^n\), then there is a positive constant \(0<C=C(f)<\infty\) such that
		\begin{equation*}
			|\langle T_{f}k_{z}, k_{w}\rangle| \le C e^{-(1/8) |z-w|^2}
		\end{equation*}
		for all \(z,w \in \mathbb{C}^n\). That is, \(T_{f} \in \text{XZ-}\mathcal{SL}\). Also, \(T_{f} \in \mathcal{L}_{p}\) for \(2<p<\frac{8}{3}\).
	\end{exa}

	\begin{proof}
		For each \(z, w, \xi \in \mathbb{C}^n\), we have
		\begin{equation*}
			|k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = |e^{\langle \xi , z \rangle - \frac{|z|^2}{2} } e^{\langle \xi, w \rangle - \frac{|w|^2}{2}} | e^{-|\xi|^2} = e^{-\frac{1}{2}\left(|z-\xi|^2 + |w-\xi|^2\right)} ~.
		\end{equation*}
		By the triangular inequality we have
		\begin{equation*}
			|z-w|^2 \le |z-\xi + \xi -w|^2 \le (|z-\xi| + |w-\xi|)^2 \le 2 (|z-\xi|^2 + |w-\xi|^2)~.
		\end{equation*}
		It follows that
		\begin{eqnarray}\label{kzw}
			|k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)}e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} \le e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} e^{-\frac{1}{8}|z-w|^2}~.
		\end{eqnarray}
		Let \(f\) be a bounded measurable function on \(\mathbb{C}^n\), then for all \(g,h\in H^{2}(\mathbb{C}^n,d\mu)\), it holds
		\begin{equation}\label{Toepfgh}
			\langle T_{f}g, h \rangle = \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
		\end{equation}
		In fact,
		\begin{eqnarray*}
			\langle T_{f}g, h \rangle &=&  \int_{\mathbb{C}^{n}} T_{f}g(w)\overline{h(w)} d\mu(w) \\
			&=& \int_{\mathbb{C}^{n}}\int_{\mathbb{C}^{n}} K(w,\xi) f(\xi)g(\xi) d\mu(\xi) \overline{h(w)} d\mu(w) \\
			&=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \int_{\mathbb{C}^{n}} K_{\xi}(w) \overline{h(w)} d\mu(w) d\mu(\xi) \\
			&=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \langle K_{\xi} , h \rangle d\mu(\xi) = \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \overline{\langle  h, K_{\xi}  \rangle} d\mu(\xi) \\
			&=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
		\end{eqnarray*}
		Therefore,
		\begin{eqnarray*}
			|\langle T_{f}k_{z}, k_{w} \rangle| &=& \left|\int_{\mathbb{C}^n}
			f(\xi)k_{z}(\xi) \overline{k_{w}(\xi)} d\mu(\xi)\right|  \\
			&\le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} \int_{\mathbb{C}^n} |k_{z}(\xi) k_{w}(\xi)| e^{-|\xi|^2} dV(\xi) \\
			& \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n}  e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)}  dV(\xi) \\
			& \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n}  e^{-\frac{1}{8}|z-w|^2} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|z-\xi|^2} dV(\xi) \right)^{\frac{1}{2}} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|w-\xi|^2}dV(\xi) \right)^{\frac{1}{2}} \\
			&=& \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n}  e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{2}|\zeta|^2} dV(\zeta) \\
			&=& (\sqrt{2})^n \left\lVert f \right\rVert_{\infty} e^{-\frac{1}{8}|z-w|^2}~.
		\end{eqnarray*}
		This completes the proof.
	\end{proof}

	 \begin{pro}
	 	Any XZ-sufficiently localized operator is weakly localized.
	 \end{pro}
	\begin{proof}
	Let \(B\) be a XZ-sufficiently localized operator, then  for all \(z\in \mathbb{C}^n\), we have
	\begin{eqnarray*}
		\int_{\mathbb{C}^n} |\langle Bk_z, k_w \rangle| dV(w) &\le& \frac{C}{\pi^n} \int_{\mathbb{C}^n} \frac{1}{(1+|\zeta|)^{\beta}} dV(\zeta) \\
		&=& \frac{Cc_n}{\pi^n} \int_{0}^{\infty} \frac{r^{2n-1}}{(1+r)^{\beta}} dr \\
		&=& C_n \left[ \int_{0}^{1} \frac{r^{2n-1}}{(1+r)^{\beta}} dr  + \int_{1}^{\infty } \frac{r^{2n-1}}{(1+r)^{\beta}} dr\right] \\
		&\le & C_n \left[ \int_{0}^{1} \frac{1}{(1+r)^{\beta}} dr +  \int_{1}^{\infty } \frac{r^{2n-1}}{r^{\beta}} dr\right] \\
		&=& C_n \left\{\frac{1}{\beta-1}\left(1-\frac{1}{2^{\beta}-1}\right) + \frac{1}{\beta-2n}\right\} \\
		&:=& C(n,\beta)~,
	\end{eqnarray*}
	which does not depend on \(z\). Therefore,
	\begin{equation*}
		\sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle Bk_z, k_w \rangle| dV(w) \le C(n,\beta)  <\infty~.
	\end{equation*}
	And in the same way, we also have
	\begin{equation*}
		\sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle B^*k_z, k_w \rangle| dV(w) \le C(n,\beta) <\infty~.
	\end{equation*}
	We now show that \(B\) satisfies the third condition of weakly localized operator.

	Let \(z\in \mathbb{C}^n\), using the change of variables \(\zeta=z-w\) and the spherical coordinates, it holds
	\begin{eqnarray*}
		\int_{|z-w|\ge r} |\langle Bk_z, k_w \rangle dV(w) &\le& C \int_{|z-w|\ge r} \frac{1}{(1+|z-w|)^{\beta}} dV(w) \\
		&=& C \int_{|\zeta|\ge r} \frac{1}{(1+|\zeta|)^{\beta}} dV(\zeta) \\
		&=& C c_n \int_{\rho\ge r} \frac{\rho^{2n-1}}{(1+\rho)^{\beta}} d \rho \\
		&\le& C c_n \int_{\rho\ge r} \frac{\rho^{2n-1}}{\rho^{\beta}} d \rho =  C c_n \int_{\rho\ge r} \frac{1}{\rho^{\beta-2n+1}} d \rho \\
		&=& \frac{Cc_n}{\beta-2n}\frac{1}{r^{\beta-2n}}~.
	\end{eqnarray*}
	It follows that,
	\begin{equation*}
		\lim_{r\to\infty} \sup_{z\in \mathbb{C}^n} \int_{|z-w|\ge r} |\langle Bk_z, k_w \rangle dV(w) \le \frac{Cc_n}{\beta-2n} \lim_{r\to\infty} \frac{1}{r^{\beta-2n}} =0~.
	\end{equation*}
	Using the duality relation (\ref{R1}) of an admissible operator and similarly like with \(B\), we also have:
	\begin{equation*}
		\lim_{r\to\infty} \sup_{z\in \mathbb{C}^n} \int_{|z-w|\ge r} |\langle B^*k_z, k_w \rangle dV(w) \le \frac{Cc_n}{\beta-2n} \lim_{r\to\infty} \frac{1}{r^{\beta-2n}} =0~.
	\end{equation*}
\end{proof}

	We denote by \(\mathcal{WL}\) the collection of weakly localized operators on \(H^{2}(\mathbb{C}^n,d\mu)\). We recall the following definition from \cite{zhu2007}.

		\begin{defn}
		A \textbf{Banach algebra} is a complex algebra together with a complete norm satisfying the condition \(\left\lVert xy \right\rVert \le \left\lVert x \right\rVert\left\lVert y \right\rVert\).
		A \(C^{*}\)\textbf{-algebra} is a Banach algebra \(\mathcal{A}\) with an involution \(x \mapsto x^{*}\) on it satisfying the following conditions:
		\begin{enumerate}[label=$\blacktriangleright$]
			\item \(x^{**}=x\) for all vectors \(x \in \mathcal{A} \).
			\item \((ax + by)^{*} = \bar{a}x^{*} + \bar{b}y^{*} \) for all vectors \(x,y \in \mathcal{A}\) and \(a,b \in \mathbb{C}\).
			\item \((xy)^{*}=y^{*}x^{*}\) for all vectors \(x,y \in \mathcal{A}\).
			\item \(\norm{xx^{*}} = \left\lVert x \right\rVert^{2}\) for all vector \(x \in \mathcal{A}\).
		\end{enumerate}
	\end{defn}

	\begin{defn}
		A Banach algebra \(\mathcal{A}\) is called a \textbf{\(\star\)-algebra} if for every \(A\in \mathcal{A}\), we have \(A^{*} \in \mathcal{A} \).
	\end{defn}

	\begin{defn}
		We denote by \(C^{*}(\mathcal{WL})\) the \(C^{*}\)-algebra generated by weakly localized operators on \(H^{2}(\mathbb{C}^n, d\mu)\). Also, \(C^{*}(\mathcal{WL})\) is actually the norm closure of \(\mathcal{WL}\) since \(\mathcal{WL} \) is a \(\star\)-algebra.
	\end{defn}

	We will prove the following main result.

For some \(\alpha>0\), \(p>0\) and \(dV\) the standard volume measure on \(\mathbb{C}^n\), let \(L^{p}_{\alpha}(\mathbb{C}^n,dV)\) be the Lebesgue space of measurable functions \(f\) on \(\mathbb{C}^n\) such that
 \begin{equation}\label{Fnorm1}
 \left\lVert f \right\rVert^{p}_{\alpha} := \left(\frac{p\alpha}{2\pi}\right)^{n} \int_{\mathbb{C}^n} |f(z)e^{-\frac{\alpha}{2}|z|^2}|^{p} dV(z) < \infty~.
 \end{equation}
 Similarly, for \(\alpha>0\) and \(p=\infty\), we denoted by \(L^{\infty}_{\alpha}\) the space of Lebesgue measurable function \(f\) on \(\mathbb{C}^n\) such that
 \begin{equation*}
 \left\lVert f \right\rVert_{\infty,\alpha} := \text{esssup}\{|f(z)|e^{-\frac{\alpha|z|^2}{2}} : ~z \in \mathbb{C}^n\}< \infty ~.
 \end{equation*}
 The classical Fock space \(F^{p}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \( L^{p}_{\alpha}(\mathbb{C}^n, dV)\). Similarly, the Fock space \(F^{\infty}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \(L^{\infty}_{\alpha}\).

\begin{defn}
 We denote by \(\mathcal{S}\) the linear span of the normalized reproducing kernels \(k_z\). A linear operator \(B : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) is said to be \textbf{admissible} on \( H^{2}(\mathbb{C}^n,d\mu) \) if there exists a linear operator \( B^{*} : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) such that the duality relation
 \begin{equation}\label{R1}
 \langle Bk_{z}, k_{w} \rangle = \langle k_{z}, B^{*}k_w \rangle
 \end{equation}
 holds for all \(z,w \in \mathbb{C}^n\).
 \end{defn}
 The inner product here is with respect to \(d\mu\).

\begin{exa}\label{Ex1}
 If \(f\) is a \textbf{bounded measurable function} on \(\mathbb{C}^n\), then there is a positive constant \(0<C=C(f)<\infty\) such that
 \begin{equation*}
 |\langle T_{f}k_{z}, k_{w}\rangle| \le C e^{-(1/8) |z-w|^2}
 \end{equation*}
 for all \(z,w \in \mathbb{C}^n\). That is, \(T_{f} \in \text{XZ-}\mathcal{SL}\). Also, \(T_{f} \in \mathcal{L}_{p}\) for \(2<p<\frac{8}{3}\).
 \end{exa}

\begin{proof}
 For each \(z, w, \xi \in \mathbb{C}^n\), we have
 \begin{equation*}
 |k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = |e^{\langle \xi , z \rangle - \frac{|z|^2}{2} } e^{\langle \xi, w \rangle - \frac{|w|^2}{2}} | e^{-|\xi|^2} = e^{-\frac{1}{2}\left(|z-\xi|^2 + |w-\xi|^2\right)} ~.
 \end{equation*}
 By the triangular inequality we have
 \begin{equation*}
 |z-w|^2 \le |z-\xi + \xi -w|^2 \le (|z-\xi| + |w-\xi|)^2 \le 2 (|z-\xi|^2 + |w-\xi|^2)~.
 \end{equation*}
 It follows that
 \begin{eqnarray}\label{kzw}
 |k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)}e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} \le e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} e^{-\frac{1}{8}|z-w|^2}~.
 \end{eqnarray}
 Let \(f\) be a bounded measurable function on \(\mathbb{C}^n\), then for all \(g,h\in H^{2}(\mathbb{C}^n,d\mu)\), it holds
 \begin{equation}\label{Toepfgh}
 \langle T_{f}g, h \rangle = \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
 \end{equation}
 In fact,
 \begin{eqnarray*}
 \langle T_{f}g, h \rangle &=& \int_{\mathbb{C}^{n}} T_{f}g(w)\overline{h(w)} d\mu(w) \\
 &=& \int_{\mathbb{C}^{n}}\int_{\mathbb{C}^{n}} K(w,\xi) f(\xi)g(\xi) d\mu(\xi) \overline{h(w)} d\mu(w) \\
 &=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \int_{\mathbb{C}^{n}} K_{\xi}(w) \overline{h(w)} d\mu(w) d\mu(\xi) \\
 &=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \langle K_{\xi} , h \rangle d\mu(\xi) = \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \overline{\langle h, K_{\xi} \rangle} d\mu(\xi) \\
 &=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
 \end{eqnarray*}
 Therefore,
 \begin{eqnarray*}
 |\langle T_{f}k_{z}, k_{w} \rangle| &=& \left|\int_{\mathbb{C}^n}
 f(\xi)k_{z}(\xi) \overline{k_{w}(\xi)} d\mu(\xi)\right| \\
 &\le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} \int_{\mathbb{C}^n} |k_{z}(\xi) k_{w}(\xi)| e^{-|\xi|^2} dV(\xi) \\
 & \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} dV(\xi) \\
 & \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} e^{-\frac{1}{8}|z-w|^2} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|z-\xi|^2} dV(\xi) \right)^{\frac{1}{2}} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|w-\xi|^2}dV(\xi) \right)^{\frac{1}{2}} \\
 &=& \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{2}|\zeta|^2} dV(\zeta) \\
 &=& (\sqrt{2})^n \left\lVert f \right\rVert_{\infty} e^{-\frac{1}{8}|z-w|^2}~.
 \end{eqnarray*}
 This completes the proof.
 \end{proof}

\begin{defn}
 We denote by \(C^{*}(\mathcal{WL})\) the \(C^{*}\)-algebra generated by weakly localized operators on \(H^{2}(\mathbb{C}^n, d\mu)\). Also, \(C^{*}(\mathcal{WL})\) is actually the norm closure of \(\mathcal{WL}\) since \(\mathcal{WL} \) is a \(\star\)-algebra.
 \end{defn}

We will prove the following main result.

\begin{defn}
 We denote by \(C^{*}(\mathcal{WL})\) the \(C^{*}\)-algebra generated by weakly localized operators on \(H^{2}(\C^n, d\mu)\). Also, \(C^{*}(\mathcal{WL})\) is actually the norm closure of \(\mathcal{WL}\) since \(\mathcal{WL} \) is a \(\star\)-algebra.
 \end{defn}

The organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \(H^{2}(\C^n, d\mu)\) which are the analogue of those given by Xia in \cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \ref{XiaFock} and present a consequence.

\begin{defn}\label{normstar}
 For an entire function \(h\) in \(\C^{n}\), we write
 \begin{equation*}
 \norm{h}_{*} = \left(\int_{\C^{n}} |h(\zeta)|^{2} e^{-\frac{1}{2}|\zeta|^2} dV(\zeta)\right)^{\frac{1}{2}}~.
 \end{equation*}
 We denote by \(\mathcal{H}_{*}\) the collection of entire functions \(h\) on \(\C^n\) satisfying \(\norm{h}_{*}<\infty\).
 \end{defn}
 \begin{rem}
 The norm \(\norm{\cdot}_{*}\) is equivalent to the norm on the Fock space \(F^{2}_{1/2}\) given in relation (\ref{Fnorm1}) which is an Hilbert space. More precisely, \(\norm{\cdot}_{*} = (2\pi)^{n/2} \norm{\cdot}_{F^{2}_{1/2}}\). This ensures the continuity of \(\norm{\cdot}_{*}\).
 \end{rem}
 In what follows, we will use the operator \(U_{z}\) defined by
 \begin{equation*}
 U_{z}f(w) = f(z-w) k_{z}(w)~, \quad f \in H^{2}(\C^n, d\mu)~.
 \end{equation*}

\begin{pro}\label{proYz}
 Suppose \(\{c_{u}: u \in \Z^{2n}\}\) are complex numbers satisfying the condition \mbox{\(\sup_{u\in \Z^{2n}}|c_{u}| < \infty\)}. Then for each \(z\in \C^{n}\), the operator
 \begin{equation}\label{Yz}
 Y_{z} = \sum_{u \in \Z^{2n}} c_{u} k_{u-z} \otimes k_{u-z}
 \end{equation}
 is bounded on \(H^{2}(\C^n, d\mu)\). Moreover, the map \(z \mapsto Y_{z}\) from \(\C^{n}\) to \(\mathcal{B}(H^{2}(\C^{n},d\mu))\) is continuous with respect to the operator norm.
 \end{pro}
 \begin{proof}
 Let \(\{c_{u} : u \in \Z^{2n}\}\) be complex numbers such that \mbox{\(\sup_{u\in \Z^{2n}}|c_{u}| < \infty\)}. For \(z\in \C^{n}\), we define
 \begin{equation*}
 A_{z} = \sum_{u \in \Z^{2n}} c_{u} (U_{u}k_{z}) \otimes e_{u} \quad \text{ and } \quad B_{z} = \sum_{u \in \Z^{2n}}(U_{u}k_{z})\otimes e_{u}~.
 \end{equation*}
 Using the relation (\ref{Uu2}), we have \(Y_{z} = A_{z}B^{*}_{z}\). Set \(h_{u} = c_{u}k_{z}\), then using Definition \ref{normstar}, we have
 \begin{equation*}
 \norm{h_{u}}_{*} = |c_{u}|\norm{k_{z}}_{*} = (2\pi)^{n/2} e^{\frac{|z|^2}{2}} |c_{u}|~.
 \end{equation*}
 Therefore, \(\sup_{u\in \Z^{2n}} \norm{h_{u}}_{*} < \infty\). Applying Lemma \ref{LemUh} to \(h_{u} = c_{u}k_{z}\), we see that \(\norm{A_{z}} \le C \sup_{u\in \Z^{2n}}|c_{u}|\). Thus each \(A_{z}\) is bounded. Since \(B_z\) is just a special case of \(A_{z}\) (with \(c_{u}=1\) for all \(u \in \Z^{2n}\)), then it is also bounded and hence \(Y_{z} = A_{z}B^{*}_{z}\) is bounded.

\begin{lem}\label{Lim=0}
 Let \(B \in \mathcal{WL} \), then for every \(z,w \in \C^n \), we have
 \begin{equation*}
 \lim_{R\to \infty} \sup_{u\in \Z^{2n}} \sum_{\underset{|u-v|>R}{v\in \Z^{2n} }} |\langle B k_{u-z}, k_{v-w} \rangle| =0 \quad \text{ and } \quad \lim_{R\to \infty} \sup_{u\in \Z^{2n}} \sum_{\underset{|u-v|>R}{v\in \Z^{2n} }} |\langle k_{u-z}, Bk_{v-w}\rangle| = 0~.
 \end{equation*}
 \end{lem}
 \begin{proof}
 By \cite[Lemma 2.32]{zhu2012}, for any entire function \(f\) on \(\C^n\), we have
 \begin{equation*}
 \left|f(z)e^{-\frac{\alpha}{2}|z|^2}\right|^p \le C \int_{B(z,\delta)} |f(w)e^{-\frac{\alpha}{2}|w|^2}|^p dV(w) \quad \text{ for } z \in \C^n.
 \end{equation*}
 Hence for \(\alpha=p=1\) and \(\delta\) small such that the balls \(\{B(v-w,\delta): v \in \Z^{2n} \}\) are mutually disjoint, we have
 \begin{equation*}
 |\langle B k_{u-z}, k_{v-w} \rangle| = |Bk_{u-z} (v-w) |e^{-\frac{|v-w|^2}{2}} \le C \int_{B(v-w,\delta)} |Bk_{u-z}(\zeta) | e^{-\frac{|\zeta|^2}{2}} dV(\zeta).
 \end{equation*}
 Indeed, for \(\delta < \frac{1}{2} \), the balls \(\{B(v-w,\delta) : v \in \Z^{2n} \}\) are mutually disjoint. Otherwise, there would exist \(v,v' \in \Z^{2n} \) such that \(v \neq v' \), and a point \(\xi\) such that \(\xi \in B(v-w,\delta) \cap B(v'-w,\delta) \). In other words:
 \begin{equation*}
 |v-w-\xi| < \delta \quad \text{ and } \quad |v'-w - \xi| < \delta~.
 \end{equation*}
 This implies that
 \begin{equation*}
 |v-v'|=|(v-w-\xi)-(v'-w-\xi)|\le |v-w-\xi| + |v'-w-\xi| < \delta + \delta= 2 \delta < 1~.
 \end{equation*}
 That is \(|v-v'|<1\). This contradicts the well-known fact that \(|v-v'|\ge 1\). This result actually implies that there exists \(N\in \N\) such that each \(\zeta\in \C^n \) belongs to at most \(N\) balls in \(\{B(v-w,\delta): v\in \Z^{2n}\}\). That is \(\sum_{v \in \Z^{2n}} \chi_{B(v-w,\delta)}(\zeta) \le N \) for each \(\zeta \in \C^n \).

\begin{proof}[\textbf{Proof of Proposition \ref{EwBEzD} }]
 From (\ref{EwBEz}), we have
 \begin{equation*}
 E_{w} BE_{z} = \frac{1}{\pi^{2n}} \sum_{u,v \in \Z^{2n}} \langle Bk_{u-z}, k_{v-w} \rangle k_{v-w} \otimes k_{u-z}~.
 \end{equation*}
 Thus for any \(R>0\), we can write \(E_{w}BE_{z} = V_{R} + W_{R}\), where
 \begin{equation*}
 V_{R} = \frac{1}{\pi^{2n}} \sum_{\underset{|u-v|\le R}{u,v \in \Z^{2n}}} \langle Bk_{u-z}, k_{v-w} \rangle k_{v-w} \otimes k_{u-z} \quad \text{ and }
 \end{equation*}
 \begin{equation*}
 \quad W_{R} = \frac{1}{\pi^{2n}} \sum_{\underset{|u-v|>R}{u,v \in \Z^{2n}}} \langle Bk_{u-z}, k_{v-w} \rangle k_{v-w} \otimes k_{u-z}~.
 \end{equation*}
 To complete the proof, it suffices to prove that:
 \begin{enumerate}[label=(\alph*)]
 \item \label{itm:first1} \(\lim_{R\to \infty} \norm{W_{R}}=0\).
 \item \label{itm:second1} \(V_{R} \in \) span(\(\mathcal{D}_{0}\)) for every \(R>0\).
 \end{enumerate}

\begin{pro}\label{DoT1}
 We have \(\mathcal{D}_{0} \subset \mathcal{T}^{(1)}\).
 \end{pro}
 To establish the proof of this proposition, we will need the next three propositions.
 \begin{pro}\label{YzT1}
 Suppose that \(\{c_{u}: u \in \Z^{2n}\}\) is a bounded set of complex coefficients. Then for each \(z \in \C^{n}\), the operator \(Y_{z}\) defined in (\ref{Yz}) belongs to \(\mathcal{T}^{(1)}\).
 \end{pro}
 \begin{proof}
 \begin{enumerate}[label=(\alph*)]
 \item Let us first show that \(Y_{0} \in \mathcal{T}^{(1)}\). We have \(|u-v|\ge 1\) for all \(u \neq v \in \Z^{2n}\). Hence \(B(u,\frac{1}{2}) \cap B(v,\frac{1}{2}) = \emptyset \) for \(u\neq v\). For each \(0 < \varepsilon < \frac{1}{2}\), define the operator
 \begin{equation*}
 A_{\varepsilon} = \frac{1}{|B(0,\varepsilon)|} \int_{B(0,\varepsilon)} Y_{z} dV(z) .
 \end{equation*}
 From Proposition \ref{proYz}, we have the norm continuity of the map \(z\mapsto Y_{z}\) and it implies that
 \begin{equation*}
 \lim_{\varepsilon \to 0} \norm{Y_{0}-A_{\varepsilon}} = 0.
 \end{equation*}
 This comes from the fact that
 \begin{eqnarray*}
 \norm{Y_{0}-A_{\varepsilon}} &=& \norm{\frac{1}{|B(0,\varepsilon)|} \int_{B(0,\varepsilon)} (Y_{0}-Y_{z}) ~dV(z)} \le \frac{1}{|B(0,\varepsilon)|} \int_{B(0,\varepsilon)} \norm{Y_{0}-Y_{z}} ~dV(z)
 \end{eqnarray*}
 and \(\lim_{z\to 0}\norm{Y_{z}-Y_{0}}=0\) .

The organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\) which are the analogue of those given by Xia in \cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \ref{XiaFock} and present a consequence.

To establish the proof of Theorem~\ref{XiaFock}, the paper first "give[s] propositions in the case of the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\)" in Section 2, as analogues of results of Xia \cite{xia2015localization} for the Bergman space of the unit ball. Then, "using these propositions in Section 3," it "establish[es] the proof of Theorem~\ref{XiaFock}."

To establish the proof of the main theorem, the paper first "give[s] propositions in the case of the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\)" in Section 2, as analogues of results of Xia \cite{xia2015localization} for the Bergman space of the unit ball. Then, "using these propositions in Section 3," it "establish[es] the proof of the main theorem."

Question: In the Fock space \(H^{2}(\mathbb{C}^{n},d\mu)\), let \(C^{*}(\mathcal{WL})\) denote the \(C^{*}\)-algebra generated by all weakly localized operators on \(H^{2}(\mathbb{C}^{n},d\mu)\). Which algebra is exactly equal to \(C^{*}(\mathcal{WL})\)?

Choices:
B. the norm closure of the weakly localized operators themselves, i.e. \(\overline{\mathcal{WL}}^{\|\cdot\|}\).
C. a subalgebra of \(\mathcal{T}^{(1)}\).
D. the algebra of all admissible operators on \(H^{2}(\mathbb{C}^{n},d\mu)\).
E. \(\mathcal{T}^{(p)}\) for every \(p>0\).

equality_to_\mathcal{T}^{(1)} dropped to mere inclusion

specific index 1 replaced by uniform all-p statement

The stem does not explicitly reveal the answer and does not contain obvious verbal overlap uniquely pointing to \(\mathcal{T}^{(1)}\). The respondent must already know or infer the identification.

This is essentially a direct restatement of the theorem-level fact being tested: that \(C^{*}(\mathcal{WL})\) equals \(\mathcal{T}^{(1)}\). It asks for recall of the exact equality rather than applying it in a new setting.

There is some pressure to distinguish exact equality from nearby statements such as mere inclusion or a stronger uniform-in-\(p\) claim, but the item mainly tests recognition/recall rather than substantive mathematical generation.

The distractors are plausible and mathematically targeted: one weakens equality to inclusion, one confuses generated algebra with norm closure of the class itself, one swaps in a broader operator class, and one overgeneralizes in \(p\). These reflect realistic failure modes.

A solid recall-based MCQ with strong distractors, but it is largely theorem restatement rather than a genuine reasoning task.


\begin{equation}
		\label{eq:orth_decomp}
		\{f\in\ell^2(E)\colon \partial f=0\}= \ell_{0,c}^2(E)\oplus_{\ell^2(E)} \{du\mid u\colon G\to \R, \, \Delta_0 u = 0, \, du\in \ell^2(E)\}.
	\end{equation}

\label{sec:Intro}

Let $G=(V,E)$ be countable oriented graph with degrees of vertices bounded from the above. Let $\tilde G$ be the non-oriented graph obtained from $G$ by forgetting the orientation of edges. Pick $D\in\mathbb N$ large enough. Consider all cycles in $\tilde G$ having lengths $\le D$. In $G$, glue each such cycle with a polygon. Choose any orientation of the latter polygons. We arrive to oriented $2$-dimensional complex, denote it by $G^{(2)}$ with implicit dependence on~$D$. Denote by $F$ the set of $2$-dimensional faces in $G^{(2)}$ which are polygons. Sometimes we write $F=FG$ and also $E=EG$ to indicate the dependence of these sets on $G$. Any of sets $V, E, F$ is endowed with counting measure which we denote by $\card$. In graph $G$, we define graph metric $\dist_G$ at $V\cup E$ along edges in $E$ so that any edge has length $1$.

If $\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\Cay(\Gamma,S)$, Cayley graph of $\Gamma$; then $V=\Gamma$. If $\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\Cay^{(2)}(\Gamma,S)$ is $\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that
\begin{equation*}\mbox{$G^{(2)}$ is simply connected}
\end{equation*}
(and that such $D$ does exist).

A \emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\mathbb R}$. We often understand  cochains as chains.
Discrete differentials (coboundaries)
$$
\{0\mbox{-cochains}\}\xrightarrow{d}\{1\mbox{-cochains}\}\xrightarrow{d}\{2\mbox{-cochains}\}
$$
and boundary operators
$$
\{2\mbox{-cochains}\}\xrightarrow{\partial}\{1\mbox{-cochains}\}\xrightarrow{\partial}\{0\mbox{-cochains}\}
$$
are introduced in the standard way with respect to the orientation of edges and faces.  Since valencies of vertices are bounded, all these operators are also bounded with respect to $\ell^2$-norms on cochains. We have $(d|_{\ell^2(V)})^*=\partial|_{\ell^2(E)}$, $(d|_{\ell^2(E)})^*=\partial|_{\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for  $\ell^2$-cochains by $\ell^2$-approximation.
If $\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\gamma$: for $e\in E$, let $f_\gamma(e)$ be the number of passes of $\gamma$ through $e$ in its direction minus number of passes of $\gamma$ over $e$ in its reversed direction. Then we have $\partial f_\gamma=0$.

Our space of interest is
\begin{equation*}
	\ell_{0,c}^2(E) := \clos_{\ell^2(E)}\{f\colon E\to {\mathbb R}\mid \partial f=0, \, \supp f
	\,\mbox{is finite}\}.\end{equation*}
Any of $1$-cochains at the right-hand side can be (convexly) decomposed into simple finite loops. Thus, $\ell_{0,c}^2(E)$ is $\ell^2$-closed linear span of (co)chains of the form $f_\gamma$ with $\gamma$ a finite loop in $G$.

Laplace operator on $0$-cochains is
$$
-\Delta_0=\partial d\colon \left(\mbox{functions on }V\right)\to \left(\mbox{functions on }V\right).
$$
A discrete integration by parts leads to the following Hodge-type decomposition:

\begin{predl}

	We have
	\begin{equation}
		\label{eq:orth_decomp}
		\{f\in\ell^2(E)\colon \partial f=0\}= \ell_{0,c}^2(E)\oplus_{\ell^2(E)} \{du\mid u\colon G\to {\mathbb R}, \, \Delta_0 u = 0, \, du\in \ell^2(E)\}.
	\end{equation}
\end{predl}

The second summand in the right-hand side of the latter relation is \emph{$\ell^2$-cohomology} of $G$. It is known to be invariant with respect to change of generating system in a group: the factorspace nature of cohomology allows to implement "discrete change of variables" from one to another set of generators. Cohomology is invariant with respect to more general quasiisometries.
Now, we pass to spectral estimates for $1$-cochains. Define non-negative Laplacian operator $\Delta_1:=\partial d+d\partial \colon \ell^2(E)\to \ell^2(E)$. On $\ell^2_{0,c}(E)$, our space of interest, this reduces to $\partial d$.

We have one more Hodge-type decomposition:
$$
\ell^2(E) = \clos_{\ell^2(E)}\{du\mid u \colon V\to {\mathbb R}, \, \supp u \mbox{ is finite}\} \oplus_{\ell^2(E)} \{f\in\ell^2(E)\colon \partial f=0\}.$$
Spectral questions for $\Delta_1$ on the first summand are generally reduced to the same for $\Delta_0$ on $\ell^2(V)$.   What concerns decomposition (\ref{eq:orth_decomp}) for $\{f\in\ell^2(E)\colon \partial f=0\}$, operator $\Delta_1$ vanishes at the second summand of its right-hand side, $\ell^2$-cohomology. Also, by the definition of $\ell^2_{0,c}(E)$ and by $\ell^2$-approximation, we see that $\Delta_1(\ell^2_{0,c}(E))\subset\ell^2_{0,c}(E)$.

\begin{define}
	We say that $\Delta_1$ \emph{has a spectral gap at $\ell^2_{0,c}(E)$} \emph{(}or just that graph $G$ has \emph{{coexact $1$-Laplacian spectral gap}}\emph{)} if $$\spec\left(\Delta_1|_{\ell^2_{0,c}(E)}\right)\cap [0,\varepsilon)=\varnothing$$ for some $\varepsilon>0$ small enough.
\end{define}

Applying discrete integration by parts, we conclude that this is equivalent to the estimate
\begin{equation}
	\label{eq:rot_estim}
\langle f, f \rangle_{\ell^2(E)}	\le 1/\varepsilon\cdot \langle df, df\rangle_{\ell^2(F)}
\end{equation}
for $f\in \ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner.
 We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way.
 \begin{predl}
 	\label{predl:general_invariance}
 	Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\subset E_2$.  	Assume the following:
 	\begin{enumerate}
 		\item $G_1$ is connected;

 		\item degrees of vertices in $G_2$ are bounded from the above;

 		\item $\sup\limits_{e\in E_2\setminus E_1} \dist_{G_1}(\beg e, \ennd e) < +\infty$ with the obvious notation.
 	\end{enumerate}
	\emph{(}In other words, metrics $\dist_{G_1}$ and $\dist_{G_2}$ on $V$ are bilipshitz equivalent.\emph{)} Then, $G_1$ has a coexact $1$-Laplacian spectral gap  \emph{(}with some $D$ implied at the construction of $G_1^{(2)}$\emph{)}
	if and only if  $G_2$ has such a spectral gap \emph{(}with some $D'$ for $G_2^{(2)}$\emph{)}.
 \end{predl}

 \noindent Proof of this quasiinvariance  is similar to the proof of quasiiinvariance of $\ell^2$-cohomology; both are based on orthogonal projection. We give a detailed argument in the Appendix. Notice also that the proof is constructive: we may estimate $D'$ via $D$ and the supremum from the third assumption of Proposition \ref{predl:general_invariance}, and vice versa.

 \begin{sled}
 	For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them  are equivalent.
 \end{sled}

Our main result is

If $\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\Cay(\Gamma,S)$, Cayley graph of $\Gamma$; then $V=\Gamma$. If $\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\Cay^{(2)}(\Gamma,S)$ is $\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that
\begin{equation*}\mbox{$G^{(2)}$ is simply connected}
\end{equation*}
(and that such $D$ does exist).

A \emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\mathbb R}$. We often understand cochains as chains.
Discrete differentials (coboundaries)
$$
\{0\mbox{-cochains}\}\xrightarrow{d}\{1\mbox{-cochains}\}\xrightarrow{d}\{2\mbox{-cochains}\}
$$
and boundary operators
$$
\{2\mbox{-cochains}\}\xrightarrow{\partial}\{1\mbox{-cochains}\}\xrightarrow{\partial}\{0\mbox{-cochains}\}
$$
are introduced in the standard way with respect to the orientation of edges and faces. Since valencies of vertices are bounded, all these operators are also bounded with respect to $\ell^2$-norms on cochains. We have $(d|_{\ell^2(V)})^*=\partial|_{\ell^2(E)}$, $(d|_{\ell^2(E)})^*=\partial|_{\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for $\ell^2$-cochains by $\ell^2$-approximation.
If $\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\gamma$: for $e\in E$, let $f_\gamma(e)$ be the number of passes of $\gamma$ through $e$ in its direction minus number of passes of $\gamma$ over $e$ in its reversed direction. Then we have $\partial f_\gamma=0$.

\begin{define}
 We say that $\Delta_1$ \emph{has a spectral gap at $\ell^2_{0,c}(E)$} \emph{(}or just that graph $G$ has \emph{{coexact $1$-Laplacian spectral gap}}\emph{)} if $$\spec\left(\Delta_1|_{\ell^2_{0,c}(E)}\right)\cap [0,\varepsilon)=\varnothing$$ for some $\varepsilon>0$ small enough.
\end{define}

Applying discrete integration by parts, we conclude that this is equivalent to the estimate
\begin{equation}
 \label{eq:rot_estim}
\langle f, f \rangle_{\ell^2(E)} \le 1/\varepsilon\cdot \langle df, df\rangle_{\ell^2(F)}
\end{equation}
for $f\in \ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner.
 We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way.
 \begin{predl}
 \label{predl:general_invariance}
 Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\subset E_2$. Assume the following:
 \begin{enumerate}
 \item $G_1$ is connected;

\begin{sled}
 For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them are equivalent.
 \end{sled}

Our main result is

\renewcommand{\abstractname}{}
\begin{abstract}
Let $\Gamma$ be a discrete finitely presented group. Pick any system $S$ of generators in $\Gamma$. In Cayley graph $\Cay(\Gamma)=\Cay(\Gamma, S)$ with edge set~$E$, glue with oriented polygons all the group relations translated to all the points of $\Gamma$; denote the obtained simply connected complex by $\Cay^{(2)}(\Gamma)$.
We study non-negative \emph{Hodge--Laplace operator $\Delta_1$} on edge functions which is defined via complex $\Cay^{(2)}(\Gamma)$; $\Delta_1$ acts on
$$
\ell^2_{0,c}(E):= \clos_{\ell^2(E)}\left\{\mbox{finitely supported closed $1$-(co)chains in }\Cay^{}(\Gamma)\right\}.
$$

We prove the following implication in the spirit of Kesten Theorem: \emph{if $\Delta_1|_{\ell_{0,c}^2(E)}$ has a spectral gap then $\Gamma$ either has exponential growth or is virtually $\mathbb Z$}. \end{abstract}

Our main result is

Let us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:

\begin{theorem}[\cite{K59}] \label{th:Kesten}
 Let $\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\Cay(\Gamma)$, $$0\in \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$$ if and only if $\Gamma$ is amenable.
\end{theorem}

If we assume the contrary to Theorem \ref{th:exp_growth}, then, first, $\Gamma$ cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic, see, e.g., \cite{Me08}. Second, $\Gamma$ cannot have infinitely many ends because then $\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end.

Now, let us briefly recall the proof of Kesten Theorem \ref{th:Kesten} to compare it to our argument. Non-amenability of $\Gamma$ means that
\begin{equation}
\label{eq:isoperimetr}
\|du\|_{\ell^1(E)} \gtrsim \|u\|_{\ell^1(\Gamma)}
\end{equation}
for $u=\mathds 1_E$, $E$ ranges all finite subsets in $\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\colon \Gamma\to\R$.
The spectral gap condition $0\notin \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$ means that
$$
\langle u, u\rangle_{\ell^2(\Gamma)} \lesssim \langle du, du\rangle_{\ell^2(E)}, ~~~ u\colon\Gamma\to\R \mbox{ is finitely supported}.
$$
To obtain this from (\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.

This paper is organized as follows. In Section \ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \ref{th:exp_growth}. In Section~\ref{sec:loop}, we prove Lemma \ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \ref{sec:spectra_homology}, we conclude the proof of Theorem~\ref{th:exp_growth}. This is done by approximating the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$ provided that $0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\gamma$ from Lemma \ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\Cay(\Gamma)$ and also make use of homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$. Finally, in Section \ref{sec:examples} we check the most natural examples of Cayley graphs.

\begin{lemma}[on loop embedding]
\label{lemma:embed}
Suppose that $\Gamma$ has subexponential growth and just one end.

Let  $C>20$, $x\ge 1$. There exist $\mathscr L>2DCx$ and injective naturally parametrized $\gamma\colon \mathbb T_{\mathscr L}\to \Cay(\Gamma)$ such that, for $t_1, t_2\in\mathbb T_{\mathscr L}$ with $\gamma(t_1), \gamma(t_2)\in \Gamma$,
\emph{
\begin{equation}
	\label{eq:x_bilip}
\mbox{if }\dist_{\Cay(\Gamma)}(\gamma(t_1), \gamma(t_2)) \le x
\mbox{ then }
\dist_{\mathbb T_{\mathscr L}}(t_1, t_2) \le Cx.
\end{equation}
}
\end{lemma}

\begin{theorem}[\cite{K59}]	\label{th:Kesten}
	Let $\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\Cay(\Gamma)$,  $$0\in \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$$ if and only if $\Gamma$ is amenable.
\end{theorem}

\begin{theorem}
	\label{th:exp_growth}
	Let $\Gamma$ be a countable finitely presented group.
	If $\Delta_1$ has a spectral gap on 	$\ell_{0,c}^2(E)$ 	then either $\Gamma$ has exponential growth, 	 or $\Gamma$ is virtually infinite cyclic. \end{theorem}

Let us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:

\begin{theorem}[\cite{K59}]	\label{th:Kesten}
	Let $\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\Cay(\Gamma)$,  $$0\in \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$$ if and only if $\Gamma$ is amenable.
\end{theorem}

Non-amenability of a group, that is, the existence of a spectral gap for $-\Delta_0$, easily implies exponential growth. The reverse is not true, in general. Thus, it is natural to ask, for example, whether Baumslag--Solitar groups $\langle\mathscr a, \mathscr b\mid \mathscr b^{-1}\mathscr a\mathscr b=\mathscr a^n\rangle$, $n\in\mathbb N$, have a spectral gap for $1$-Laplacian. These groups are \emph{non-elementary amenable} but have exponential growth. Such groups are not covered by Theorem \ref{th:exp_growth}, and it is still unclear for the author whether $1$-Laplacian has a spectral gap on them.

If we assume the contrary to Theorem \ref{th:exp_growth}, then, first, $\Gamma$ cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic, see, e.g., \cite{Me08}. Second, $\Gamma$ cannot have infinitely many ends because then $\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end.

For $\mathscr L>0$, denote by $\mathbb T_{\mathscr{L}}$ a circle of length $\mathscr L$. On $\mathbb T_{\mathscr L}$, one may measure distances along this loop. To prove Theorem \ref{th:exp_growth}, we need the following

\begin{lemma}[on loop embedding]
\label{lemma:embed}
Suppose that $\Gamma$ has subexponential growth and just one end.

Let  $C>20$, $x\ge 1$. There exist $\mathscr L>2DCx$ and injective naturally parametrized $\gamma\colon \mathbb T_{\mathscr L}\to \Cay(\Gamma)$ such that, for $t_1, t_2\in\mathbb T_{\mathscr L}$ with $\gamma(t_1), \gamma(t_2)\in \Gamma$,
\emph{
\begin{equation}
	\label{eq:x_bilip}
\mbox{if }\dist_{\Cay(\Gamma)}(\gamma(t_1), \gamma(t_2)) \le x
\mbox{ then }
\dist_{\mathbb T_{\mathscr L}}(t_1, t_2) \le Cx.
\end{equation}
}
\end{lemma}

In fact, we are able to make $\mathscr L$ arbitrarily large with fixed $x$.

Now, let us briefly recall the proof of Kesten Theorem \ref{th:Kesten} to compare it to our argument. Non-amenability of $\Gamma$ means that
\begin{equation}
\label{eq:isoperimetr}
\|du\|_{\ell^1(E)} \gtrsim \|u\|_{\ell^1(\Gamma)}
\end{equation}
for $u=\mathds 1_E$, $E$ ranges all finite subsets in $\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\colon \Gamma\to{\mathbb R}$.
The spectral gap condition $0\notin \spec\left(-\Delta_0|_{\ell^2(\Gamma)}\right)$ means that
$$
\langle u, u\rangle_{\ell^2(\Gamma)} \lesssim \langle du, du\rangle_{\ell^2(E)}, ~~~ u\colon\Gamma\to{\mathbb R} \mbox{ is finitely supported}.
$$
To obtain this from (\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.

In the first step of the latter argument, we assemble a function $u\colon \Gamma\to{\mathbb R}$, say, non-negative one, from its super-level sets $\mathds 1_{\{u\ge \mathscr t\}}$, $\mathscr t$ ranges $[0,+\infty)$; we also assemble $du$ from $d\mathds 1_{\{u\ge \mathscr t\}}$. (Both decompositions are $\ell^1$-convex.) Thus, in Kesten Theorem, we deal with "sets of codimensions $0$ and $1$". At least, we will have such genuine codimensions in the case of a manifold instead of a group, the corresponding result linking spectra and isoperimetry is known as Cheeger--Yau inequality, see \cite{Cheeger}, \cite{Yau}.

Unlike this, in our argument we work with dimension $1$ sets --- loops, in particular, as in Lemma \ref{lemma:embed}. Also, in Section \ref{sec:spectra_homology} we bound $1$-cycles with $2$-dimensional surfaces.

Notice also that an analogue of Cheeger--Yau inequality for $1$-forms was obtained in \cite{BC22} in the case of manifolds. Coexact $1$-Laplacian spectrum is indeed related to appropriate isoperimetric ratio, namely, to $\sup_{\gamma}\inf_h|h|/\length\gamma$ with $\gamma$ ranging homologicaly trivial loops at a manifold $\mathscr M$ and $h$ be a $2$-dimensional chain in $\mathscr M$ bounding $\gamma$; here, $|h|$ is area of $h$. Some Poincar\'e-type estimates for operator $d$ on coclosed $1$-forms are  possible if isoperimetric ratios as above are bounded from the below. But, in \cite{BC22}, authors impose the condition of finite diameter of $\mathscr M$  which is not our case; also, \cite{BC22} does not deal with effects of negative curvature.

What concerns spaces with negative curvature, let us mention recent works \cite{AAGLZ24}, \cite{R23} devoted to $3$-dimensional hyperbolic manifolds. It turns out that, first, $1$-coexact spectral gap is related to exponential growth of torsion $1$-homology of the manifolds; second, there are relations  between the spectral gap and isoperimetric ratios. The latter estimates from \cite{R23} are also volume-dependent, as in \cite{BC22}.

\medskip

This paper is organized as follows. In Section \ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \ref{th:exp_growth}. In Section~\ref{sec:loop}, we prove Lemma \ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \ref{sec:spectra_homology}, we conclude the proof of Theorem~\ref{th:exp_growth}. This is done by approximating the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$ provided that $0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\gamma$ from Lemma \ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\Cay(\Gamma)$ and also  make use of homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$.  Finally, in Section \ref{sec:examples} we check the most natural examples of Cayley graphs.

\medskip

\noindent {\bf Some notation.} For a set $A$ we denote by $\card A$ the number of its elements.
If $v_1, v_2$ are vertices of some oriented graph then we denote by $\edge(v_1, v_2)$ the oriented edge in the graph under consideration provided that the edge exists. If $e$ is an edge in a oriented graph or $\gamma$ is an oriented path in a metric space then we write $\beg e$, $\beg\gamma$ for their beginnings and $\ennd e$, $\ennd\gamma$ for their endpoints, respectively. The notation $\length\gamma$ is obvious.

We write $\mathcal B_X(x, \rho)$ for the open ball in a metric space $X$ centered in a point $x\in X$ and having radius $\rho\ge 0$.

Assuming the contrary to Theorem~\ref{th:exp_growth}, one first reduces the possibilities for the number of ends: $\Gamma$ "cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic"; it also "cannot have infinitely many ends because then $\Gamma$ has exponential growth". Hence, "by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end."

Under the hypothesis that $\Gamma$ has subexponential growth and one end, one proves Lemma~\ref{lemma:embed} ("on loop embedding"), producing for given $C>20$, $x\ge1$ an injective loop $\gamma\colon\mathbb T_{\mathscr L}\to\Cay(\Gamma)$ with $\mathscr L>2DCx$ and the metric control (\ref{eq:x_bilip}): if $\dist_{\Cay(\Gamma)}(\gamma(t_1),\gamma(t_2))\le x$ then $\dist_{\mathbb T_{\mathscr L}}(t_1,t_2)\le Cx$ (and "we are able to make $\mathscr L$ arbitrarily large with fixed $x$"). The lemma is proved "by dropping lots of geodesic perpendiculars in a branching way."

The proof of Theorem~\ref{th:exp_growth} is then completed in Section~\ref{sec:spectra_homology}: assuming a spectral gap, i.e. "$0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$", one "approximat[es] the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$" and applies this to the "$1$-cochain given by curve $\gamma$ from Lemma~\ref{lemma:embed}$." One also "put[s] some metric control on the approximating process" in $\Cay(\Gamma)$ and uses the "homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$", together with the fact that "we bound $1$-cycles with $2$-dimensional surfaces."

Assuming the contrary to the main theorem, one first reduces the possibilities for the number of ends: $\Gamma$ "cannot have two \emph{ends} since in this case $\Gamma$ is virtually cyclic"; it also "cannot have infinitely many ends because then $\Gamma$ has exponential growth". Hence, "by Freudenthal--Hopf Theorem we may assume that $\Gamma$ has one end."

Under the hypothesis that $\Gamma$ has subexponential growth and one end, one proves the following lemma.

\begin{lemma}[on loop embedding]
\label{lemma:embed}
Suppose that $\Gamma$ has subexponential growth and just one end.

Let  $C>20$, $x\ge 1$. There exist $\mathscr L>2DCx$ and injective naturally parametrized $\gamma\colon \mathbb T_{\mathscr L}\to \Cay(\Gamma)$ such that, for $t_1, t_2\in\mathbb T_{\mathscr L}$ with $\gamma(t_1), \gamma(t_2)\in \Gamma$,
\emph{
\begin{equation}
	\label{eq:x_bilip}
\mbox{if }\dist_{\Cay(\Gamma)}(\gamma(t_1), \gamma(t_2)) \le x
\mbox{ then }
\dist_{\mathbb T_{\mathscr L}}(t_1, t_2) \le Cx.
\end{equation}
}
\end{lemma}

This produces for given $C>20$, $x\ge1$ an injective loop $\gamma\colon\mathbb T_{\mathscr L}\to\Cay(\Gamma)$ with $\mathscr L>2DCx$ and the metric control given by the displayed equation above (and "we are able to make $\mathscr L$ arbitrarily large with fixed $x$"). The lemma is proved "by dropping lots of geodesic perpendiculars in a branching way."

In completing the proof of the main theorem, assuming a spectral gap, i.e. "$0\notin\spec\Delta_1|_{\ell_{0,c}^2(E)}$", one "approximat[es] the resolvent $\Delta^{-1}_1$ by polynomials of $\Delta_1$" and applies this to the "$1$-cochain given by curve $\gamma$ from the lemma above." One also "put[s] some metric control on the approximating process" in $\Cay(\Gamma)$ and uses the "homological nature of $\Delta_1$: this operator is divisible by $\partial\colon \ell^2(F)\to\ell^2(E)$ at $\ell_{0,c}^2(E)$", together with the fact that "we bound $1$-cycles with $2$-dimensional surfaces."

\label{th:exp_growth}
	Let $\Gamma$ be a countable finitely presented group.
	If $\Delta_1$ has a spectral gap on 	$\ell_{0,c}^2(E)$ 	then either $\Gamma$ has exponential growth,	 or $\Gamma$ is virtually infinite cyclic.,

Question: Let Γ be a countable finitely presented group, choose a generating set S, and form its Cayley graph Cay(Γ,S) with edge set E. Glue oriented polygons along all translates of the defining relations to obtain the associated simply connected 2-complex, and let Δ1 be the corresponding Hodge–Laplace operator on 1-cochains. Write ℓ^2_{0,c}(E) for the ℓ^2-closure of the finitely supported closed 1-cochains on E. If Δ1 has a spectral gap on ℓ^2_{0,c}(E), meaning spec(Δ1|_{ℓ^2_{0,c}(E)}) avoids some interval [0, ε) with ε > 0, what must be true of Γ?

Γ either has exponential growth or is virtually infinite cyclic.

Choices:
B. Γ either is non-amenable or is virtually infinite cyclic.
C. Γ is not a torsion group; in particular, Γ has an element of infinite order.
D. Γ either has exponential growth or is virtually cyclic.
E. If Δ_1 has a spectral gap on \ell^2_{0,c}(E), then for every choice of generating set S the group Γ has exactly one end unless it is virtually infinite cyclic.

replace growth conclusion by amenability-type conclusion in Kesten spirit

dropped the dichotomy 'exponential growth or virtually infinite cyclic' to the weaker consequence 'contains an infinite order element'

strength of the exceptional case: 'virtually infinite cyclic' broadened to 'virtually cyclic'

one-end reduction used only under contradiction turned into a global conclusion for every generating set

The stem gives the hypothesis and asks for a necessary conclusion, but it does not explicitly or implicitly reveal the correct choice. There is no direct phrasing overlap with the answer.

This is essentially a direct theorem-recall item: the stem states the full technical hypothesis and asks for the theorem's conclusion. It is close to a restatement rather than a novel inference task.

Some reasoning is needed to distinguish the strongest valid conclusion from nearby alternatives, especially against weaker true or overgeneralized statements. However, for a prepared student, the item is mostly recognition/application of a known result rather than substantial generation.

The distractors are mathematically plausible and target realistic confusions: amenability vs growth, weaker true consequences, overbroad virtually cyclic exceptions, and an unjustified ends conclusion. They are distinct and well-designed.

A technically strong MCQ with no answer leakage and very good distractors, but it mainly tests recall of a specific theorem rather than deeper generative reasoning.


\label{thmintro:delocalised-trace-identification-simplified}
  Let $\cH$ be an Iwahori Hecke algebra of right-angled type $(W,S)$.  Let $\cO \subseteq W$ be a conjugacy class.  Then there is a unique trace $\vphi_\cO$ on $\cH$ that satisfies $\vphi_\cO(T_w) = \mathbb{1}_\cO(w)$ for every element $w \in W$ that is of minimal length in its conjugacy class.

\begin{remark}
  \label{rem:finite-order-element-cliques}
  Let $\Gamma$ be the commutation graph of a right-angled Coxeter system $(W, S)$.  Denote by $\mathrm{Cliq}(\Gamma)$ the set of cliques in $\Gamma$, that is the set of complete subgraphs of $\Gamma$ (including the empty graph).  Identifying vertices of $\Gamma$ with $S$, we consider the map $C \mapsto \prod_{s \in \rV(C)} s$.  By Remark~\ref{rem:finite-order-element-conjugacy} every finite order element is conjugate to an element in the image of this map.  Further, if $\prod_{s \in \rV(C)} s$ and $\prod_{s \in \rV(D)} s$ are conjugate, they must be equal by Tits solution to the word problem.  It follows that $C \mapsto \prod_{s \in \rV(C)} s$ defines a bijection between $\mathrm{Cliq}(\Gamma)$ and the set of conjugacy classes of finite order elements of $W$.
\end{remark}

\begin{theorem}
  \label{thm:right-angled-cocentre-basis}
  Let $\cH$ be a generic Hecke algebra of right-angled type $(W,S)$ over an algebraically closed field $K$ of characteristic different from $2$. Let $(a_s)_{s\in S}$, $(b_s)_{s\in S}$ be the deformation parameters of $\cH$ and assume that $b_s \neq -\frac{a_s^2}{4}$ for each $s \in S$.  Then $(T_{\cO})_{\cO}$ where $\cO$ runs through the conjugacy classes of $W$ is a basis of the cocentre of $\cH$.
\end{theorem}

\label{sec:introduction}

The study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \cite{lott1999}, who investigated for example delocalised $\Ltwo$-Betti numbers and delocalised $\Ltwo$-torsion of hyperbolic manifolds. The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint \cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts. More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.

While this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory. They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively. Delorme-Opdam's work in \cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context. Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed. These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \cite{opdamsolleveld2010-discrete-series} and generalised principal series \cite{delormeopdam2011} of affine Hecke algebras.

Hecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \cite{davisdymarajanusykiewiczokun07} around two decades ago. Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field. Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \cite{caspersklisselarsen2021}, simplicity results \cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \cite{raumskalski2022} for Hecke C*-algebras. Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \cite{klisseperovic25}. Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups. See e.g.\ \cite[Corollary B]{raumskalski2023}.

The cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \cite{baumnistor2002} in the purely algebraic setup and by Solleveld \cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.

In this work we consider Hecke algebras of right-angled, hyperbolic type. In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras. This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\ref{thm:delocalised-trace-identification}. For simplicity we state it in a simplified form for Iwahori-Hecke algebras.

\begin{theorem}
  \label{thm:delocalised-trace-identification}
  Let $\cH$ be a generic Hecke algebra of right-angled type $(W,S)$.  For every conjugacy class $\cO \subseteq W$, the map $\vphi_\cO$ is a trace on $\cH$.  If $(T_{\cU})_{\cU}$ denotes the basis of the cocentre of $\cH$ satisfying $T_{\cU} = T_w + [\cH, \cH]$ for every element of minimal length $w \in \cU$, then $\vphi_\cO(T_\cU) = \delta_{\cO, \cU}$.
\end{theorem}

The next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.
\begin{introtheorem}
 \label{introthm:Schwartz-algebra-existence}
 Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \in S} \in \RR_{> 0}^S$. There is a subalgebra $\cS(W, S, q) \subseteq \Cstarred(W, S, q)$, which
 \begin{itemize}
 \item is dense and closed under holomorphic functional calculus,
 \item and such that for every conjugacy class $\cO \subseteq W$, the trace $\vphi_\cO$ continuously extends to $\cS(W, S, q)$.
 \end{itemize}
\end{introtheorem}

The proof of \cite[Theorem 5.3]{henie2014} applies verbatim to show the following result: the collection $(T_\cO)_{\cO}$ linearly generates the cocentre of any generic Hecke algebra. We will use the following description of the coefficients appearing in linear combinations of these elements.
\begin{lemma}
 \label{lem:non-minimal-element-cocentre}
 Let $\cH$ be a generic Hecke algebra of type $(W,S)$. Let $\cO \subseteq W$ be a conjugacy class and let $w \in \cO$. Then
 \begin{gather*}
 T_w + [\cH, \cH]
 =
 \sum_\cK c_{\cK, w} T_\cK
 \end{gather*}
 for certain coefficients $c_{\cK, w}$, where $\cK$ runs over all conjugacy classes of elements that are Bruhat subordinate to $w$.
\end{lemma}
\begin{proof}
 Let $\ell$ be the minimal length of an element in $\cO$. We prove the claim by induction on $|w| - \ell = k \in \NN_0$. For $k = 0$ there is nothing to prove.

\begin{notation}
 \label{not:delocalised-trace}
 Let $(W, S)$ be a Coxeter system of right-angled type and let $\cH$ be a generic Hecke algebra of type $(W, S)$ defined over $R$. For a conjugacy class $\cO \subseteq W$, we denote by $\Sigma_\cO\colon \cH \lra R$ the sum over all coefficients of minimal length elements in $\cO$, that is
 \begin{gather*}
 \Sigma_\cO(T_w)
 =
 \begin{cases}
 1 & \text{ if $w \in \cO$ and $w$ is of minimal length in $\cO$} \\
 0 & \text{ otherwise}\eqstop
 \end{cases}
 \end{gather*}
 Combining this with Notation~\ref{not:cyclic-reduction-map}, we will write $\vphi_\cO = \Sigma_\cO \circ \Phi$.
\end{notation}

\begin{theorem}
 \label{thm:delocalised-trace-identification}
 Let $\cH$ be a generic Hecke algebra of right-angled type $(W,S)$. For every conjugacy class $\cO \subseteq W$, the map $\vphi_\cO$ is a trace on $\cH$. If $(T_{\cU})_{\cU}$ denotes the basis of the cocentre of $\cH$ satisfying $T_{\cU} = T_w + [\cH, \cH]$ for every element of minimal length $w \in \cU$, then $\vphi_\cO(T_\cU) = \delta_{\cO, \cU}$.
\end{theorem}
\begin{proof}
 In order to show that $\vphi_\cO$ is a trace, it suffices to show that for all $w \in W$ and all $s \in S$ the equality $\vphi_\cO(T_s T_w) = \vphi_\cO(T_w T_s)$ holds. We prove this statement by induction on $|w| \in \NN_0$ and observe that the case $|w| = 0$ is clear. We distinguish several cases.

\begin{theorem}
 \label{thm:delocalised-trace-existence}
 Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and let $\cH$ be the Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \in S} \in \RR_{> 0}^S$. Let $\cS'(W, S, q), \cS(W, S, q) \subseteq \Cstarred(W, S, q)$ be smooth subalgebras, such that $\cS'(W, S, q)$ is an unconditional completion of $\CC(W, S, q)$ and the natural quasi-derivation
 \begin{gather*}
 \Delta\colon T_w \mapsto \sum_{\substack{w = w_1w_2 \\ \text{ reduced decompositon}}} T_{w_1} \ot T_{w_2}
 \end{gather*}
 continuously extends to a map into the unconditional tensor product $\cS(W, S, q) \lra \cS'(W, S, q) \ot \cS'(W, S, q)$. Then for every conjugacy class $\cO \subseteq W$ the trace $\vphi_\cO$ continuously extends to $\cS(W, S, q)$.
\end{theorem}
\begin{proof}
 Let us first observe that the map $\Sigma_\cO$ summing over the finitely many coefficients of elements of minimal length in $\cO$ extends to a continuous map on $\cS'(W, S, q)$, as $\cS'(W, S, q)$ is an unconditional completion of $\CC(W, S, q)$. Consider maps $\gamma, \rho\colon W \lra W$ defined as in Lemma~\ref{lem:cyclic-reduction-hecke-algebra}. Since $\Delta$ extends continuously to a map $\cS(W, S, q) \lra \cS'(W, S, q) \ot \cS'(W, S, q)$ and since $\cS'(W, S, q)$ is an unconditional completion of $\CC(W, S, q)$, the map $\Delta'\colon \cS(W, S, q) \lra \cS'(W, S, q) \ot \cS'(W, S, q)$ satisfying $\Delta'(T_w) = T_{\gamma(w) \rho(w)} \ot T_{\gamma(w)^{-1}}$ is well-defined and continuous. Let $\Phi\colon \CC(W, S, q) \lra \CC(W, S, q)$ be the unique linear map satisfying $\Phi(T_w) = T_{\gamma(w)^{-1}} T_{\gamma(w)} T_{\rho(w)}$. Then we have $\vphi_\cO = \Sigma_\cO \circ \Phi$ as shown in Theorem~\ref{thm:delocalised-trace-identification}. So a continuous extension of $\vphi_\cO$ to $\cS(W, S, q)$ is obtained by the composition
 \begin{multline*}
 \cS(W, S, q) \stackrel{\Delta'}{\lra}
 \cS'(W, S, q) \ot \cS'(W, S, q) \stackrel{\text{tensor flip}}{\lra} \\
 \cS'(W, S, q) \ot \cS'(W, S, q) \stackrel{\text{multiplication}}{\lra}
 \cS'(W, S, q) \stackrel{\Sigma_\cO}{\lra}
 \CC
 \eqcomma
 \end{multline*}
This finishes the proof.
\end{proof}

Before proving Theorem~\ref{thmintro:pairing}, let us clarify its context and notation beyond what we have done in the introduction already. Recall from \cite[Theorem A.2.1]{bost1990-principe-oka} that the inclusion of a smooth subalgebra into a unital C*-algebra induces an isomorphism in K-theory. We also recall the K-theory calculations for right-angled Hecke C*-algebras from \cite[Section 4]{raumskalski2022}. To this end we use the description of conjugacy classes of finite order elements from Remark~\ref{rem:finite-order-element-cliques}. Denote by $\mathrm{Cliq}(\Gamma)$ the set of cliques in the commutation graph $\Gamma$ of a right-angled Coxeter system $(W, S)$, that is subsets $C \subseteq S$ such that $st = ts$ for all $s,t \in C$. For a clique $C \in \mathrm{Cliq}(\Gamma)$, we write $p_C = \prod_{s \in C} \chi_1^s$. Here $\chi_1^s = \pi_s(\chi_1)$ denotes the image of the projection $\chi_1 \in \CC[\ZZ/2]$ associated with the trivial representation under the linear map extending the group homomorphism $\pi_s\colon \ZZ/2 \lra \CC(W, S, q)^\times$ which satisfies $\pi_s(1) = \frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. Then for every deformation parameter $q \in \RR_{> 0}^S$ the map $\mathrm{Cliq}(\Gamma) \lra \rK_0(\Cstarred(W, S, q))\colon C \mapsto [p_C]$ induces an isomorphism $\ZZ^{\mathrm{Cliq}(\Gamma)} \lra \rK_0(\Cstarred(W, S, q))$.
\begin{proof}[Proof of Theorem~\ref{thmintro:pairing}]
 As described above, for $D \in \mathrm{Cliq}(\Gamma)$ we have $p_D = \prod_{s \in D} \chi_1^s$ where $\chi_1^s$ is the projection associated with the trivial representation of the unitary representation $\ZZ/2 \mapsto \CC(W, S, q)$ mapping $1$ to $\frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. We have $\chi_1^s = \mathbb{1}_{(0,\infty)}(T_s) = \frac{1}{q_s^{1/2} + q_s^{-1/2}} (T_s + q_s^{-1/2}) =\frac{1}{q_s^{1/2} + q_s^{-1/2}} T_s + \frac{1}{1 + q_s}$. Writing $T_D = \prod_{s \in D} T_s$, we find that
 \begin{gather*}
 p_D = \sum_{C \subseteq D} \bigl ( \prod_{s \in C} \frac{1}{q_s^{1/2} + q_s^{-1/2}} \bigr ) \bigl ( \prod_{s \in D \setminus C} \frac{1}{1 + q_s} \bigr ) T_C
 \eqstop
 \end{gather*}
 Since all elements in the support of $p_D$ have minimal length and are pairwise non-conjugate, we find that
 \begin{gather*}
 \vphi_C(p_D)
 =
 \Sigma_C(p_D)
 =
 \begin{cases}
 0 & C \nsubseteq D \eqcomma \\
 \prod_{s \in C} \frac{1}{q_s^{1/2} + q_s^{-1/2}} \prod_{s \in D \setminus C} \frac{1}{1 + q_s} & C \subseteq D \eqstop
 \end{cases}
 \end{gather*}
 Now the claimed formula for $\vphi_C(p_D)$ follows.

We mention that in order to prove this result, we establish in Theorem~\ref{thm:right-angled-cocentre-basis} that the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.  The problem of describing the cocentre of Hecke algebras is well-known in the context of character theory.  See e.g. \cite{geckpfeiffer2000-characters} for spherical Coxeter types and \cite{henie2014} for affine Coxeter types.  The recent work \cite{chen2025-centralizers} explicitly states folklore conjectures about the cocentre for indefinite Coxeter types.

The next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.
\begin{introtheorem}
  \label{introthm:Schwartz-algebra-existence}
  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \in S} \in \RR_{> 0}^S$.  There is a  subalgebra $\cS(W, S, q) \subseteq \Cstarred(W, S, q)$, which
  \begin{itemize}
  \item is dense and closed under holomorphic functional calculus,
  \item and such that for every conjugacy class $\cO \subseteq W$, the trace $\vphi_\cO$ continuously extends to $\cS(W, S, q)$.
  \end{itemize}
\end{introtheorem}

From the point of view of operator algebras, one of the main purposes of delocalised traces is to determine K-theory classes via their pairing.  The K-theory of right-angled Hecke C*-algebras was calculated in \cite{raumskalski2022}.  We briefly recall that for every clique $C$ in the commutation graph of a right-angled Coxeter system there is a projection $p_C$, such that the classes $([p_C])_C$ freely generate K-theory.  In Remark~\ref{rem:finite-order-element-cliques}, we observe that these cliques are in natural bijection with finite order elements of a right-angled Coxeter system.  We obtain the following result, which determines the pairing of K-theory and delocalised traces.
\begin{introtheorem}
  \label{thmintro:pairing}
  Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $(q_s)_{s \in S} \in \RR_{> 0}^S$ a deformation parameter.  Denote by $\mathrm{Cliq}(\Gamma)$ the set of cliques of the commutation graph $\Gamma$ of $(W, S)$, identified with the set of conjugacy classes of finite order elements in $W$.  For a clique $C \in \mathrm{Cliq}(\Gamma)$ let $p_C$ be the associated projection and let $\vphi_C$ be the delocalised trace associated with $C$.  Then the pairing between K-theory of $\Cstarred(W, S, q)$ and delocalised traces of finite order elements is determined by the formula
  \begin{gather*}
    \vphi_C(p_D) =
    \begin{cases}
      0 & C \nsubseteq D\eqcomma \\
      \prod_{s \in C} \frac{1}{q_s^{1/2} + q_s^{-1/2}} \prod_{s \in D \setminus C} \frac{1}{1 + q_s} & C \subseteq D \eqstop
    \end{cases}
  \end{gather*}
  In particular, the pairing $\rK_0(\Cstarred(W,S,q)) \times \RR^{\mathrm{Cliq(\Gamma)}} \lra \RR$ is non-degenerate.
\end{introtheorem}

As mentioned above, in the last section of \cite{puschnigg2010} Puschnigg used his main results to obtain certain decompositions of cyclic cohomology for his smooth algebras and the group C*-algebras of hyperbolic groups, close in spirit to the computation of cyclic homology of group rings due to Burghelea \cite{burghelea1985, burghelea2023-erratum-for-1985}. This line of research was later continued, using the Jolissaint's rapid decay subalgebras, for example in \cite{jiogleramsey14} and in \cite{engel20}. Our results in principle open the way to obtaining similar decompositions of cyclic cohomology in the context of Hecke algebras, such as the Fr{\'e}chet algebra $\cS(W, S, q)$ appearing in Theorem \ref{introthm:Schwartz-algebra-existence}. However, the setting of right-angled hyperbolic Coxeter types remains too narrow to make such an endeavour worthwhile at present, since higher cyclic cohomology is expected to vanish.  Further work in this direction, would chime with the recent progress in understanding noncommutative geometric properties of the Iwahori-Hecke algebras such as in the work \cite{klisseperovic25} already mentioned.

This paper is organised as follows: after this introduction, in the preliminaries we describe background material and fix notation, focusing on Coxeter groups and Hecke algebras.  Then in the third section, we investigate the combinatorics of the multiplication in right-angled Hecke algebras, which plays a major role in this work.  In the fourth section, we adapt work of Blackadar-Cuntz and Puschnigg on differential seminorms and quasi-derivations to the needed generality.  In the fifth section, we use this framework in order to exhibit specific smooth subalgebras of the Hecke algebras under consideration, to which the natural quasi-derivation extends.  In the sixth section, we describe traces on right-angled Hecke algebras, and in that last section combine this with the previous results on smooth subalgebras, to obtain delocalised traces on Hecke-Schwartz algebras and calculate their pairing with K-theory.

\subsection*{Acknowledgements}

SP was partially supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 677120-INDEX) and the Knut and Alice Wallenberg Foundation (grant number KAW 2020.0252).

SR was partially supported by the Swedish Research Council (grant number 2018-04243) and the German Research Foundation (DFG project no. 550184791).

PN was supported by the National Science Center Grant Maestro-13 UMO-2021/42/A/ST1/00306.

The authors would like to thank IMPAN and its B{\c e}dlewo Conference Center as well as the University to Potsdam for their hospitality during respective visits.

To prove Theorem~\ref{thmintro:delocalised-trace-identification-simplified}, the introduction says that the authors first establish in Theorem~\ref{thm:right-angled-cocentre-basis} that “the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.”

Question: Let \(\mathcal H\) be the Iwahori Hecke algebra associated with a right-angled Coxeter system \((W,S)\), and let \(\{T_w\}_{w\in W}\) denote its standard basis indexed by \(W\). Let \(\mathcal O\subseteq W\) be a conjugacy class. A trace on \(\mathcal H\) means a linear functional \(\varphi:\mathcal H\to\mathbb C\) such that \(\varphi(xy)=\varphi(yx)\) for all \(x,y\in\mathcal H\). For \(w\in W\), say that \(w\) is of minimal length in its conjugacy class if its Coxeter length is minimal among all elements conjugate to \(w\). Which existence statement holds under these assumptions?

There exists a unique trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) that is of minimal length in its conjugacy class, \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.

Choices:
B. There exists a unique trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\), \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.
C. There exists a trace \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) that is of minimal length in its conjugacy class, \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.
D. For each conjugacy class \(\mathcal O\subseteq W\), there exists a unique linear functional \(\varphi_{\mathcal O}:\mathcal H\to\mathbb C\) such that for every \(w\in W\) that is of minimal length in its conjugacy class, \(\varphi_{\mathcal O}(T_w)=\mathbb 1_{\mathcal O}(w)\).
E. There exists a trace \(\varphi:\mathcal H\to\mathbb C\) such that for every conjugacy class \(\mathcal O\subseteq W\) and every \(w\in W\) that is of minimal length in its conjugacy class, \(\varphi(T_w)=\mathbb 1_{\mathcal O}(w)\), where \(\mathbb 1_{\mathcal O}(w)=1\) if \(w\in\mathcal O\) and \(0\) otherwise.

trace property replaced by mere linearity

dependence on the conjugacy class parameter \(\mathcal O\)

The stem gives only background definitions and asks which existence statement is valid; it does not explicitly state the theorem or directly reveal the correct option.

The item is very close to a theorem-recall question: the correct choice is essentially the precise theorem statement, though the alternatives introduce meaningful variations in quantifiers, uniqueness, and the trace condition.

Some reasoning is needed to distinguish the exact valid statement from stronger or weaker variants, but the task is mainly precise theorem recognition rather than substantial mathematical generation or derivation.

The distractors are plausible and mathematically targeted: they test common confusions about minimal-length restriction, uniqueness, dependence on the conjugacy class parameter, and trace versus arbitrary linear functional.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it remains largely a near-restatement/recall item rather than a high-generative-reasoning question.