Approximate biprojectivity and $\phi $-biflatness of certain Banach algebras

Abstract

In this paper, we study left φ-biflatness and left φ-biprojectivity of some Banach algebras, where φ is a non-zero multiplicative linear function. We show that if the Banach algebra A * * is left φ-biprojective, then A is left φ-biflat. Using this tool we study left φbiflatness of some matrix algebras. We also study left φ-biflatness and left φ-biprojectivity of the projective tensor product of some Banach algebras. We prove that for a locally compact2010 Mathematics Subject Classification. Primary 46M10, 46H05 Secondary 43A07, 43A20.

Generalized notion of amenability for a class of matrix algebras

“…G is compact and also we show that L 1 (G, w) is approximately biprojective if and only if G is compact, provided that w ≥ 1 is a continuous weight function, see [21] and [23].…”

Generalized notions of amenability for a class of matrix algebras

“…Proof: Since A is an approximately biprojective Banach algebra with a left approximate identity, by [18,Theorem 3.9] A is left ϕ-contractible for every ϕ ∈ ∆(A). Applying [4, Corollary 2.2] one can see that ∆(A) is discrete.…”

“…Proof: Suppose that A is approximately biprojective. Using [18,Theorem 3.9], the existence of approximate identity implies that A is left and right ϕ-contractible. Then there exist m 1 and m 2 in A such that…”

On (approximate) homological notions of certain Banach algebras

“…For more information about φ−biflatness and φ−biprojectivity, the reader refers to [8] and [9]. Theorem 2.6.…”