The higher-dimensional amenability of tensor products of Banach algebras

zbMATH Open1265.46077arXiv0904.4548MaRDI QIDQ3116315

Publication date: 22 February 2012

Abstract: We investigate the higher-dimensional amenability of tensor products

A p t p B

of Banach algebras

A

and

B

. We prove that the weak bidimension

d b_{w}

of the tensor product

A p t p B

of Banach algebras

A

and

B

with bounded approximate identities satisfies [ db_w A ptp B = db_w A + db_w B. ] We show that it cannot be extended to arbitrary Banach algebras. For example, for a biflat Banach algebra

A

which has a left or right, but not two-sided, bounded approximate identity, we have

d b_{w} A p t p A l e 1

and

d b_{w} A + d b_{w} A = 2 .

We describe explicitly the continuous Hochschild cohomology

H^{n} (A p t p B, (X p t p Y)^{*})

and the cyclic cohomology

H C^{n} (A p t p B)

of certain tensor products

A p t p B

of Banach algebras

A

and

B

with bounded approximate identities; here

(X p t p Y)^{*}

is the dual bimodule of the tensor product of essential Banach bimodules

X

and

Y

over

A

and

B

respectively.

Full work available at URL: https://arxiv.org/abs/0904.4548

zbMATH Keywords

Hochschild cohomology \(C^*\)-algebra cyclic cohomology semigroup algebra

Mathematics Subject Classification ID

Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) (46M20) Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) (46H25) Tensor products in functional analysis (46M05)