Mean ergodic theorems for multipliers on Banach algebras (Q666645)

Let \(A\) be a commutative Banach algebra. A~multiplier on \(A\) is a bounded linear operator \(T:A\to A\) such that \(T(ab)=aTb\) for all \(a,b\in A\). A~multiplier \(T\) is said to be Cesàro bounded if \(\sup_{n\in\mathbb{N}}\left\|\frac{1}{n}\sum_{k=0}^{n-1}T^k\right\|<\infty\). The author proves various general results concerning the mean ergodic property of Cesàro bounded multipliers; some of these results are stated in even a more general setting. These are then applied to Fourier and Fourier-Stieltjes algebras on locally compact groups. A~sample result is the following theorem. {Theorem}. Let \(G\) be a locally compact group and let \(u\in B(G)\) be Cesàro bounded, satisfying \[ \lim_{n\to\infty}\|u^nv\|/n\to 0 \] for all \(v\in A(G)\). Then the sequence \[ \frac{1}{n}\sum_{k=0}^{n-1}u^kv \] converges in \(A(G)\)-norm for every \(v\in A(G)\) if and only if \[ \mathscr{F}_u := \{t\in G: u(t)=1\} \] is an open set in \(G\). In that case, \[\frac{1}{n}\sum_{k=0}^{n-1}u^kv\to \1_{\mathscr{F}_u}v. \]