A Result on the diagonal of contractible operator Banach algebras

Author
Maysam Maysami Sadr
Department of Mathematics
Abstract
A Banach algebra A is called contractible if for any Banach A-bimodule E, every continuous derivation from A into E is inner. One of the oldest unconfirmed conjectures in amenability says that every contractible Banach algebra is finite dimensional. It is well-known that a Banach algebra A is contractible if and only if its unital and has a diagonal, that is a member M in the Banach algebra A⊗πA such that satisfies in ∆(M)=1 and a⊗1M=M(1⊗a) for every a in A. In this note we show that any diagonal of a contractible Banach algebra of operators on an infinite dimensional Banach space has a specific null property.
Keywords
Banach algebra
contractibility
diagonal
algebra of bounded linear operators
Amenability
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Mathematical Research
Volume 9, Issue 4 - Serial Number 27
Autumn 2023
Pages 178-185