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Weak homological bidimension and its values in the class of biflat Banach algebras. (English) Zbl 0946.46059

Let \(A\) be a Banach algebra. The author introduces the weak homological bidimension (\(\text{wdb}\)) of \(A\) as the smallest integer \(n\) such that the cohomology groups \(\mathcal H^m (A,X^\ast)\) vanish for \(m > n\) and all Banach \(A\)–bimodules \(X\). \(\text{wdb} A=0\) is equivalent to amenability of \(A\). It is shown that for biflat Banach algebras [see the book of A. Ya. Khelemskij (Helemskii), ”The homology of Banach and topological algebras” (1986; Zbl 0608.46046) for this and related notions from homology] \(\text{wdb} A\) can take only the values \(0,1\) or \(2\) (depending on the existence of various types of approximate identities). Further results in this direction can be found in the author’s paper in Monatsh. Math. 128, 35-60 (1999).
Reviewer: V.Losert (Wien)

MSC:

46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
18G20 Homological dimension (category-theoretic aspects)

Citations:

Zbl 0608.46046
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