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Morita equivalence for ${C}^{*}$-algebras with the weak Banach–Saks property. II. (English) Zbl 1126.46036

A Banach space $X$ has the Banach–Saks property if every bounded sequence $\left\{{x}_{n}\right\}\subset X$ has a Cesàro summable subsequence $\left\{{x}_{{n}_{k}}\right\}$; that is, there is a $y\in X$ such that

$\underset{k\to \infty }{lim}∥\frac{1}{k}\left({x}_{{n}_{1}}+\cdots +{x}_{{n}_{k}}\right)-y∥=0·$

One says $X$ has the weak Banach–Saks property if every weakly null sequence $\left\{{x}_{n}\right\}$ has a subsequence such that

$\underset{k\to \infty }{lim}\frac{1}{k}\parallel {x}_{{n}_{1}}+\cdots +{x}_{{n}_{k}}\parallel =0,$

and $X$ has the uniform weak Banach–Saks property if there is a sequence $\left\{{\delta }_{n}\right\}$ of positive real numbers such that ${\delta }_{n}\to 0$ and such that given a weakly null sequence $\left\{{x}_{n}\right\}$ in the unit ball of $X$, there are natural numbers ${n}_{1}<{n}_{2}<\cdots$ such that

$\frac{1}{k}\parallel {x}_{{n}_{1}}+\cdots +{x}_{{n}_{k}}\parallel <{\delta }_{k}\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}k\text{.}$

The uniform weak Banach–Saks property certainly implies the weak Banach–Saks property, and the converse can fail in general.

The author showed in Part I [Q. J. Math. 52, 455–461 (2001; Zbl 1020.46013)] that the weak Banach–Saks property is preserved by Morita equivalence, and that if $𝖷$ is an $A$-$B$-imprimitivity bimodule, then $𝖷$ has the uniform weak Banach–Saks property if and only if either $A$ or $B$ have the weak Banach–Saks property. Furthermore, the converse holds if either $A$ or $B$ is unital. In this article, the author proves that if $A$ or $B$ is unital and if $𝖷$ is an $A$-$B$-imprimitivity bimodule with the weak Banach–Saks property, then $A$ and $B$ have the weak Banach–Saks property. As a corollary, he obtains that a Hilbert module over a unital ${C}^{*}$-algebra has the weak Banach–Saks property if and only if it has the uniform weak Banach–Saks property.

The author also proves a number of results about the Banach–Saks property. (A ${C}^{*}$-algebra has the Banach–Saks property if and only if it is finite dimensional.) For example, $𝖷$ is an imprimitivity algebra between unital ${C}^{*}$-algebras, then $𝖷$ has the Banach-Saks property if and only if it is finite-dimensional.

MSC:
 46L05 General theory of ${C}^{*}$-algebras