notes on the module homological properties of triangular banach algebras

In the current investigation, we find some necessary and sufficient conditions for the triangular banach algebra t = [ a m 0 b ] to be module amenable, module biprojective and module biflat (as u ⊕u-module), where a and b are banach u-bimodules and m is a banach a, b-module and banach u-module with compatible actions. as an application, we show that for an inverse semigroup s with the set of idempotents e, under what conditions [ l 1 (s) l 1 (s) ] is module amenable, module biprojective and module biflat (as l 1 (e) ⊕ l 1 (e)-module).