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Generalized notions of amenability for a class of matrix algebras. (English) Zbl 07144888

Summary: We investigate the amenability and its related homological notions for a class of \(I\times I\)-upper triangular matrix algebra, say \(\mathrm{UP}(I,A)\), where \(A\) is a Banach algebra equipped with a nonzero character. We show that \(\mathrm{UP}(I,A)\) is pseudo-contractible (amenable) if and only if \(I\) is singleton and \(A\) is pseudo-contractible (amenable), respectively. We also study pseudo-amenability and approximate biprojectivity of \(\mathrm{UP}(I,A)\).

MSC:

46M10 Projective and injective objects in functional analysis
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
43A07 Means on groups, semigroups, etc.; amenable groups
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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References:

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