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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 {x n }X has a Cesàro summable subsequence {x n k }; that is, there is a yX such that

lim k 1 k (x n 1 ++x n k ) - y=0·

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

lim k 1 kx n 1 ++x n k =0,

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

1 kx n 1 ++x n k <δ k forallk.

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:
46L05General theory of C * -algebras