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Zbl 1126.46036
Kusuda, Masaharu
Morita equivalence for $C^{*}$-algebras with the weak Banach--Saks property. II. (English)
[J] Proc. Edinb. Math. Soc., II. Ser. 50, No. 1, 185-195 (2007). ISSN 0013-0915; ISSN 1464-3839/e

A Banach space $X$ has the {{Banach--Saks property}} if every bounded sequence $\{x_{n}\} \subset X$ has a Cesàro summable subsequence $\{x_{n_{k}}\}$; that is, there is a $y\in X$ such that $$ \lim_{k\to \infty} \bigl\Vert \frac1k(x_{n_{1}}+\dots+x_{n_{k}})-y\bigr\Vert =0.$$ One says $X$ has the weak Banach--Saks property if every weakly null sequence $\{x_{n}\}$ has a subsequence such that $$ \lim_{k\to\infty}\frac1k\Vert x_{n_{1}}+\dots+x_{n_{k}}\Vert =0,$$ and $X$ has the uniform weak Banach--Saks property if there is a sequence $\{\delta_{n}\}$ of positive real numbers such that $\delta_{n}\to0$ and such that given a weakly null sequence $\{x_{n}\}$ in the unit ball of $X$, there are natural numbers $n_{1}<n_{2}<\dots$ such that $$\frac1k\Vert x_{n_{1}}+\dots+x_{n_{k}}\Vert <\delta_{k}\quad\text{for all $k$.}$$ 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 $\mathsf{X}$ is an $A$-$B$-imprimitivity bimodule, then $\mathsf{X}$ 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 $\mathsf{X}$ 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, $\mathsf{X}$ is an imprimitivity algebra between unital $C^{*}$-algebras, then $\mathsf{X}$ has the {Banach-Saks property}{} if and only if it is finite-dimensional.
[Dana P. Williams (Hanover)]

MSC 2000:: *46L05 General theory of C*-algebras

Keywords: Morita equivalence; weak Banach-Saks property

Citations: Zbl 1020.46013

Cited in: Zbl 1206.46009

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