A Banach space has the Banach–Saks property if every bounded sequence has a Cesàro summable subsequence ; that is, there is a such that
One says has the weak Banach–Saks property if every weakly null sequence has a subsequence such that
and has the uniform weak Banach–Saks property if there is a sequence of positive real numbers such that and such that given a weakly null sequence in the unit ball of , there are natural numbers such that
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 is an --imprimitivity bimodule, then has the uniform weak Banach–Saks property if and only if either or have the weak Banach–Saks property. Furthermore, the converse holds if either or is unital. In this article, the author proves that if or is unital and if is an --imprimitivity bimodule with the weak Banach–Saks property, then and have the weak Banach–Saks property. As a corollary, he obtains that a Hilbert module over a unital -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 -algebra has the Banach–Saks property if and only if it is finite dimensional.) For example, is an imprimitivity algebra between unital -algebras, then has the Banach-Saks property if and only if it is finite-dimensional.