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\fancyhead[LO]{\textbf{Noncommutative von Neumann-Riesz and von Neumann-Riesz-Lorch Mean Ergodic Theorems}}
\fancyhead[RE]{\textbf{K. Mahesh Krishna}}


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{\bf\large{{Noncommutative von Neumann-Riesz and von Neumann-Riesz-Lorch Mean Ergodic Theorems}}}\\
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\textbf{K. Mahesh Krishna}\\
School of Mathematics and Natural Sciences\\
Chanakya University Global Campus\\
NH-648, Haraluru Village\\
Devanahalli Taluk, Bengaluru North District\\
Karnataka State, 562 110, India\\
Email: kmaheshak@gmail.com\\

Date: \today
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\textbf{Abstract}: We derive noncommutative von Neumann-Riesz and von Neumann-Riesz-Lorch mean ergodic theorems for adjointable contractions on a Hilbert C*-module. We crucially use a result obtained by Paschke [\textit{Trans. Amer. Math. Soc., 1973}]. We also note that the exact version of the noncommutative von Neumann–Riesz mean ergodic theorem fails.


\textbf{Keywords}: Mean Ergodic Theorem, Hilbert C*-module.


\textbf{Mathematics Subject Classification (2020)}: 46L08.\\

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\section{Introduction}
In 1931, Koopman showed that the \textit{quasi-ergodic hypothesis} of classical Hamiltonian dynamics can be interpreted in Hilbert spaces \cite{KOOPMAN}. Motivated by this, in 1932, von Neumann proved the following mean ergodic theorem for unitary operators on Hilbert spaces \cite{VONNEUMANN}.
\begin{theorem} \cite{VONNEUMANN, RIESZNAGY2, HALMOS1, HALMOS2, HALMOS3, HALMOS4, ZUND, REEDSIMON, EISNERFARKASHAASENAGEL, EINSIEDLERWARD, KRENGEL, MOORE, STEIN} \label{ME} (\textbf{von Neumann Mean Ergodic Theorem})
	Let $\mathcal{H}$ be a Hilbert space. Let $U:\mathcal{H}\to \mathcal{H}$ be a unitary operator. Then for every $h \in \mathcal{H}$, the mean ergodic sequence 
	\begin{align*}
		\left\{\frac{1}{n}\sum_{k=0}^{n-1}U^kh\right\}_{n=1}^\infty
	\end{align*}
converges in the norm on $\mathcal{H}$. 
\end{theorem}
In 1938, Riesz extended Theorem \ref{ME} to contractions on Hilbert spaces \cite{RIESZ}.
\begin{theorem} \cite{RIESZ, RIESZ2, RIESZ3, RIESZNAGY, REEDSIMON, HALMOS4, EISNERFARKASHAASENAGEL, EINSIEDLERWARD, KRENGEL, STEIN}\label{RE}
(\textbf{von Neumann-Riesz Mean Ergodic Theorem})
Let $\mathcal{H}$ be a Hilbert space. Let $T:\mathcal{H}\to \mathcal{H}$ be a contraction. Then for every $h \in \mathcal{H}$, the mean ergodic sequence 
\begin{align*}
	\left\{\frac{1}{n}\sum_{k=0}^{n-1}T^kh\right\}_{n=1}^\infty
\end{align*}
converges in the norm on $\mathcal{H}$. 	
\end{theorem}
In 1939, Birkhoff extended Theorem \ref{RE} to uniformly convex Banach spaces \cite{BIRKHOFF}.
In 1939, Lorch derived following result for power bounded operators on Hilbert spaces \cite{LORCH}.
\begin{theorem} \cite{LORCH}
(\textbf{von Neumann-Riesz-Lorch Mean Ergodic Theorem})
Let $\mathcal{H}$ be a Hilbert space. Let $T:\mathcal{H}\to \mathcal{H}$ be a bounded operator such that 
	\begin{align*}
	\sup_{n\in \mathbb{N}}\|T^n\|<\infty.
\end{align*}
 Then for every $h \in \mathcal{H}$, 
the mean ergodic sequence 
\begin{align*}
	\left\{\frac{1}{n}\sum_{k=0}^{n-1}T^kh\right\}_{n=1}^\infty
\end{align*}
converges in the norm on $\mathcal{H}$. 	
\end{theorem}
In 1953, Sz.-Nagy gave a new proof of Theorem \ref{RE} using his unitary dilation of contraction \cite{NAGY}.

We are fundamentally concerned with the question: What is the  noncommutative analogue of Theorem  \ref{RE}? For this, we need the noncommutative version of Hilbert spaces known as Hilbert C*-modules which  were first introduced by Kaplansky \cite{KAPLANSKY} for modules over commutative C*-algebras and later developed for modules over arbitrary C*-algebras by Paschke  \cite{PASCHKE} and Rieffel \cite{RIEFFEL}. 
\begin{definition}\cite{KAPLANSKY, PASCHKE, RIEFFEL}
	Let $\mathcal{A}$ be a  unital C*-algebra. A right module 	 $\mathcal{E}$  over $\mathcal{A}$ is said to be a (right)  \textbf{Hilbert C*-module} if there exists a  map $ \langle \cdot, \cdot \rangle: \mathcal{E}\times \mathcal{E} \to \mathcal{A}$ such that the following hold. 
	\begin{enumerate}[\upshape(i)]
		\item $\langle x, x \rangle  \geq 0$, $\forall x \in \mathcal{E}$. If $x \in  \mathcal{E}$ satisfies $\langle x, x \rangle  =0 $, then $x=0$.
		\item $\langle x,y+z \rangle  =\langle x, y \rangle+\langle x, z \rangle$, $\forall x,y,z \in \mathcal{E}$.
		\item  $\langle x, ya \rangle  =\langle x, y \rangle a$, $\forall x,y \in \mathcal{E}$, $\forall a \in \mathcal{A}$.
		\item $\langle x, y \rangle=\langle y,x \rangle^*$, $\forall x,y \in \mathcal{E}$.
		\item $\mathcal{E}$ is complete w.r.t. the norm $\|x\|\coloneqq \sqrt{\|\langle x, x \rangle\|}$,  $\forall x \in \mathcal{E}$.
	\end{enumerate}
\end{definition}


\section{Noncommutative von Neumann-Riesz and von Neumann-Riesz-Lorch Mean Ergodic Theorems}
We need the following important result of Paschke \cite{PASCHKE}.
\begin{theorem}\cite{PASCHKE}\label{PR}
Let $\mathcal{E}$ be a Hilbert C*-module. Let $T:\mathcal{E}\to \mathcal{E}$ be a bounded adjointable morphism. Then 
\begin{align*}
	\langle Tx, Tx \rangle\leq \|T\|\langle x, x \rangle, \quad \forall x \in \mathcal{E}.
\end{align*}
\end{theorem}
Following is a noncommutative version of Theorem \ref{RE}.
\begin{theorem}\label{NVR}
	(\textbf{Noncommutative von Neumann-Riesz Mean Ergodic Theorem})
	Let $\mathcal{E}$ be a Hilbert C*-module. Let $T:\mathcal{E}\to \mathcal{E}$ be an adjointable contraction. Then for every 
	 \begin{align*}
		x \in \overline{(I_\mathcal{E}-T)(\mathcal{E})}\oplus \operatorname{ker}(I_\mathcal{E}-T^*),
	\end{align*}
	 the mean ergodic sequence 
	\begin{align*}
		\left\{\frac{1}{n}\sum_{k=0}^{n-1}T^kx\right\}_{n=1}^\infty
	\end{align*}
	converges in the norm on $\mathcal{E}$. 	
\end{theorem}
\begin{proof}
	Our proof is motivated from the arguments by Riesz for the Hilbert space \cite{REEDSIMON, EISNERFARKASHAASENAGEL, HALMOS4, EINSIEDLERWARD, KRENGEL, STEIN}. Define 
\begin{align*}
	S_n \coloneqq \frac{1}{n}\sum_{k=0}^{n-1}T^k, \quad \forall n \in \mathbb{N}.
\end{align*}
\begin{enumerate}[{Case} 1:]
	\item  Let $x \in \operatorname{ker}(I_\mathcal{E}-T^*)$. Then $T^*x=x$. Since $T$ is a contraction, using Theorem \ref{PR}, we get
\begin{align*}
	\langle x-Tx, x-Tx \rangle &=\langle x, x \rangle-\langle T^*x, x \rangle-\langle x, T^*x\rangle+\langle Tx, Tx \rangle\\
	&=\langle x, x \rangle-\langle x, x \rangle-\langle x, x\rangle+\langle Tx, Tx \rangle\\
	&=-\langle x, x\rangle+\langle Tx, Tx \rangle\\
	&\leq -\langle x, x\rangle+\|T\|\langle x, x \rangle\\
	&\leq -\langle x, x\rangle+\langle x, x \rangle\\
	&\leq 0.
\end{align*}
Therefore $Tx=x$. So we have $S_nx=nx/n=x$ for all $n$. Therefore $\{S_nx\}_{n=1}^n$ converges.
\item  Let $x \in (I_\mathcal{E}-T)(\mathcal{E})$. Choose $y \in \mathcal{E}$ such that $x=y-Ty$. We then have 
\begin{align*}
	S_nx&= \frac{1}{n}\sum_{k=0}^{n-1}T^kx\\
	&= \frac{1}{n}\sum_{k=0}^{n-1}T^k(y-Ty)\\
	&=\frac{1}{n}\sum_{k=0}^{n-1}T^ky-\frac{1}{n}\sum_{k=0}^{n-1}T^{k+1}y\\
	&=\frac{1}{n}(y-T^ny), \quad \forall n \in \mathbb{N}.
\end{align*}
Therefore 
\begin{align*}
	\lim_{n\to \infty} \|S_n x\| &= \lim_{n\to \infty} \frac{1}{n} \|y - T^n y\|\\
	& \leq \lim_{n\to \infty} \frac{1}{n} \|I_{\mathcal{E}} - T^n\| \|y\|\\
	&\leq \lim_{n\to \infty} \frac{1}{n} (\|I_{\mathcal{E}}\| +\|T^n\|) \|y\|\\
	&\leq  \lim_{n\to \infty} \frac{1}{n} (\|I_{\mathcal{E}}\| +\|T\|^n) \|y\|\\
	&\leq 2\lim_{n\to \infty} \frac{1}{n}  \|y\|=0.
\end{align*}
\item   Let $x \in \overline{(I_\mathcal{E}-T)(\mathcal{E})}\setminus(I_\mathcal{E}-T)(\mathcal{E})$. Let $\varepsilon >0$. Choose $y \in (I_\mathcal{E}-T)(\mathcal{E})$ such that $\|y-x\|<\varepsilon$. Let $ z\in \mathcal{E}$ be such that $y=(I_\mathcal{E}-T)z$. Since $\{S_ny\}_{n=1}^\infty$ converges to zero,  there exists $N$ such that $\|S_ny\|<\varepsilon$ for all $n \geq N$. Then 
\begin{align*}
\|S_nx\|&\leq \|S_nx-S_ny\|+\|S_ny\|\\
&\leq \|S_n\|\|x-y\|+\|S_ny\|\\
&\leq \|x-y\|+\varepsilon\\
&<2\varepsilon, \quad \forall n \geq N.
\end{align*}
Therefore $	\lim_{n\to \infty} \|S_n x\|=0$.
\end{enumerate}
\end{proof}
\begin{corollary}
	Let $\mathcal{E}$ be a Hilbert C*-module. Let $T:\mathcal{E}\to \mathcal{E}$ be an adjointable contraction. For a given $x \in \mathcal{E}$, 
	\begin{align*}
		Tx=x \iff T^*x=x.
	\end{align*}
	In particular, 
	\begin{align*}
		\operatorname{Fix}(T)\coloneqq \{x \in \mathcal{E}: Tx=x\}= \{x \in \mathcal{E}: T^*x=x\}=	\operatorname{Fix}(T^*).
	\end{align*}	
\end{corollary}
\begin{corollary}
	Let $\mathcal{E}$ be a Hilbert C*-module. Let $T:\mathcal{E}\to \mathcal{E}$ be an adjointable contraction such that $\overline{(I_\mathcal{E}-T)(\mathcal{E})}$ or $ \operatorname{ker}(I_\mathcal{E}-T^*)$ is orthogonally complementable in $\mathcal{E}$. Then for every $x \in \mathcal{E}$, the mean ergodic sequence 
\begin{align*}
	\left\{\frac{1}{n}\sum_{k=0}^{n-1}T^kx\right\}_{n=1}^\infty
\end{align*}
converges in the norm on $\mathcal{E}$. 		
\end{corollary}
\begin{proof}
	$\mathcal{E}=\overline{(I_\mathcal{E}-T)(\mathcal{E})}\oplus \operatorname{ker}(I_\mathcal{E}-T^*)$.
\end{proof}
A careful inspection of the proof of Theorem \ref{NVR} gives the following result for contractions which need not be adjointable. 
\begin{theorem}
	(\textbf{Noncommutative von Neumann-Riesz Mean Ergodic Theorem})
Let $\mathcal{E}$ be a Hilbert C*-module. Let $T:\mathcal{E}\to \mathcal{E}$ be a contraction (need not be adjointable). Then for every 
\begin{align*}
	x \in \overline{(I_\mathcal{E}-T)(\mathcal{E})}\oplus \operatorname{ker}(I_\mathcal{E}-T),
\end{align*}
the mean ergodic sequence 
\begin{align*}
	\left\{\frac{1}{n}\sum_{k=0}^{n-1}T^kx\right\}_{n=1}^\infty
\end{align*}
converges in the norm on $\mathcal{E}$. 	
\end{theorem}
By seeing the proof of Theorem \ref{NVR} we also get following result for power bounded morphisms. 
\begin{theorem} (\textbf{Noncommutative von Neumann-Riesz-Lorch Mean Ergodic Theorem})
	Let $\mathcal{E}$ be a Hilbert C*-module. Let $T:\mathcal{E}\to \mathcal{E}$ be a bounded morphism such that 
	\begin{align*}
		\sup_{n\in \mathbb{N}}\|T^n\|<\infty.
	\end{align*}
 Then for every 
\begin{align*}
	x \in \overline{(I_\mathcal{E}-T)(\mathcal{E})}\oplus \operatorname{ker}(I_\mathcal{E}-T),
\end{align*}
the mean ergodic sequence 
\begin{align*}
	\left\{\frac{1}{n}\sum_{k=0}^{n-1}T^kx\right\}_{n=1}^\infty
\end{align*}
converges in the norm on $\mathcal{E}$. 	
\end{theorem}
Following example shows that exact noncommutative version of von Neumann–Riesz mean ergodic theorem fails.
\begin{example}\label{CE}
Let $\mathbb{T}$ be the standard unit circle and $\mathcal{C}(\mathbb{T})$ be the C*-algebra of continuous complex valued functions on $\mathbb{T}$ in the sup norm. Define $\mathcal{A}\coloneqq \mathcal{C}(\mathbb{T})$. Define $\mathcal{E}\coloneqq \mathcal{A}$. Define 
\begin{align*}
	U: \mathcal{E} \ni f \mapsto Uf \in \mathcal{E};~ (Uf)(z)\coloneqq zf(z), \quad \forall z \in \mathbb{T}.
\end{align*}
Then $U$ is unitary. Now 
\begin{align*}
		(S_n\chi_\mathbb{T})(z) = \frac{1}{n}\sum_{k=0}^{n-1}(U^k\chi_\mathbb{T})(z)=\frac{1}{n}\sum_{k=0}^{n-1}z^k=\frac{1}{n}\sum_{k=0}^{n-1}z^k, \quad \forall n \in \mathbb{N}, \quad \forall z \in \mathbb{T}.
\end{align*}
Hence
\begin{align*}
	(S_n\chi_\mathbb{T})(z)=\frac{1-z^n}{n(1-z)}, \quad \forall n \in \mathbb{N}, \quad \forall z \in \mathbb{T}\setminus \{1\}
\end{align*}
and 
\begin{align*}
		(S_n\chi_\mathbb{T})(1)=1.
\end{align*}
We therefore have 
\begin{align*}
	\lim_{n\to \infty} (S_n\chi_\mathbb{T})(z)=0, \quad \forall z \in \mathbb{T}\setminus \{1\}
\end{align*}
and 
\begin{align*}
	\lim_{n\to \infty} (S_n\chi_\mathbb{T})(1)=1.
\end{align*}
Therefore 
\begin{align*}
	\lim_{n\to \infty} S_n\chi_\mathbb{T}=\chi_{\{1\}}\notin \mathcal{C}(\mathbb{T}).
\end{align*}
Hence \begin{align*}
	\left\{\frac{1}{n}\sum_{k=0}^{n-1}U^k\chi_\mathbb{T}\right\}_{n=1}^\infty
\end{align*}
does not converge in the norm on $\mathcal{E}$.
\end{example}




\section{Non-Archimedean von Neumann-Riesz Mean Ergodic Problem}
We recall the definition of non-Archimedean Banach and p-adic Hilbert spaces.
\begin{definition}\cite{SCHIKHOF}
	Let $\mathbb{K}$ be a field. A map $|\cdot|:\mathbb{K} \to [0, \infty)$ is said to be a \textbf{non-Archimedean valuation} if the following conditions hold:
	\begin{enumerate}[\upshape(i)]
		\item If $\lambda \in \mathbb{K}$ satisfies $|\lambda|=0$, then $\lambda=0$. 
		\item $|\lambda\mu|=|\lambda||\mu|$ for all $\lambda, \mu \in \mathbb{K}$.
		\item (Ultratriangle inequality) $|\lambda+\mu|\leq \max\{\lambda|, |\mu|\}$ for all $\lambda, \mu \in \mathbb{K}$.
	\end{enumerate}
	We say $\mathbb{K}$ is non-Archimedean complete valued field if it is complete w.r.t. the ultrametric $d(\lambda, \mu)\coloneqq |\lambda-\mu|$ for all $\lambda, \mu \in \mathbb{K}$.
\end{definition}
\begin{definition}\cite{SCHNEIDER}
	Let $\mathbb{K}$ be a non-Archimedean complete valued field. A vector space $\mathcal{X}$ over $\mathbb{K}$ is said to be a \textbf{non-Archimedean Banach space} if there is a map (called as norm) $\|\cdot\|:\mathcal{X} \to [0, \infty)$ satisfying the following:
	\begin{enumerate}[\upshape(i)]
		\item If $x \in \mathcal{X}$ is such that $\|x\|=0$, then $x=0$.
		\item $\|\lambda x\|=|\lambda|\|x\|$ for all $\lambda \in \mathbb{K}$, for all $x \in \mathcal{X}$.
		\item  (Ultranorm inequality) $\|x+y\|\leq \max\{\|x\|, \|y\|\}$ for all $x, y \in \mathcal{X}$.
		\item $\mathcal{X}$ is complete w.r.t. the ultrametric $d(x,y)\coloneqq \|x-y\|$ for all $x, y \in \mathcal{X}$.
	\end{enumerate}
\end{definition}
\begin{definition}\cite{KALISCH} \label{PADICDEF}
	Let $\mathbb{K}$ be a non-Archimedean  valued field (with valuation $|\cdot|$) and $\mathcal{X}$ be a non-Archimedean Banach space (with norm $\|\cdot\|$) over $\mathbb{K}$. We say that $\mathcal{X}$ is a \textbf{p-adic Hilbert space} if there is a map (called as \textbf{p-adic inner product}) $\langle \cdot, \cdot \rangle: \mathcal{X} \times \mathcal{X} \to \mathbb{K}$ satisfying the following.
	\begin{enumerate}[\upshape (i)]
		\item If $x \in \mathcal{X}$ is such that $\langle x,y \rangle =0$ for all $y \in \mathcal{X}$, then $x=0$.
		\item $\langle x, y \rangle =\langle y, x \rangle$ for all $x,y \in \mathcal{X}$.
		\item $\langle \alpha x, y \rangle =\alpha \langle x, y \rangle$ for all $\alpha  \in \mathbb{K}$, for all $x,y \in \mathcal{X}$.
		\item $\langle x+y, z \rangle = \langle x,  z \rangle+\langle y,z\rangle$ for all $x,y,z \in \mathcal{X}$.
		\item $|\langle x, y \rangle |\leq \|x\|\|y\|$ for all $x,y \in \mathcal{X}$.
	\end{enumerate}
\end{definition}
Let $\mathcal{X}, \mathcal{Y}$ be  p-adic Hilbert spaces. A bounded linear operator $T:\mathcal{X} \to \mathcal{Y}$ is said to be adjointable if there exists a linear operator (which is unique) $T^*:\mathcal{Y} \to \mathcal{X}$ such that 
\begin{align*}
	\langle Tx, y\rangle =\langle x, T^*y \rangle, \quad \forall x \in \mathcal{X}, \forall  y \in \mathcal{Y}.
\end{align*}
We say that an adjointable  linear operator $T:\mathcal{X} \to \mathcal{Y}$ is an isometry if $\|Tx\|=\|x\|$ for all $x \in \mathcal{X}$. An invertible isometry $T:\mathcal{X} \to \mathcal{Y}$ is said to be unitary if $\langle Tx, Ty\rangle =\langle x, y \rangle $ for all $x\in \mathcal{X}$, for all $y\in \mathcal{Y}$.
\begin{problem} (\textbf{Non-Archimedean von Neumann-Riesz Mean Ergodic Problem})
Let $\mathcal{X}$ be  p-adic Hilbert space. Let $T:\mathcal{X} \to \mathcal{X}$	 be an adjointable contraction or unitary operator. For which $x \in \mathcal{X}$, the p-adic mean ergodic sequence 
\begin{align*}
	\left\{\frac{1}{n}\sum_{k=0}^{n-1}T^kx\right\}_{n=1}^\infty
\end{align*}
converges in the norm on $\mathcal{X}$. 
\end{problem}
\begin{problem} (\textbf{Non-Archimedean von Neumann-Riesz-Lorch Mean Ergodic Problem})
	Let $\mathcal{X}$ be  p-adic Hilbert space. Let $T:\mathcal{X} \to \mathcal{X}$	 be a bounded operator such that 
		\begin{align*}
		\sup_{n\in \mathbb{N}}\|T^n\|<\infty.
	\end{align*}
	 For which $x \in \mathcal{X}$, the p-adic mean ergodic sequence 
	\begin{align*}
		\left\{\frac{1}{n}\sum_{k=0}^{n-1}T^kx\right\}_{n=1}^\infty
	\end{align*}
	converges in the norm on $\mathcal{X}$. 
\end{problem}
\section{Conclusions}
\begin{enumerate}
	\item In 1932,  von Neumann derived the mean ergodic theorem for unitary operators on Hilbert spaces \cite{VONNEUMANN}.
	\item In 1938, Riesz derived the mean ergodic theorem for contractions on Hilbert spaces \cite{RIESZ}.
	\item In 1939, Lorch derived the mean ergodic theorem for power bounded operators on Hilbert spaces \cite{LORCH}.
	\item In this article, we derived noncommutative version of von Neumann-Riesz mean ergodic theorem. 
\end{enumerate}
 \section{Acknowledgments}
AI (GPT, Claude, Gemini) is used for the literature survey. Claude AI by Anthropic (free version) was used to obtain  Example \ref{CE}.
 
 
 
 
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