‎Quotient Ideal Amenabiliy in Tensor Banach Algebras

Abstract

A. Jabbari and O.T. Mewomo have studied the ideal amenability property‏ in ‎tensor Banach ‎algebras and have proven several results. In this paper, we explore a different form of ideal amenability within tensor Banach algebras, specifically examining the concept of quotient ideal amenability. This concept applies to tensor Banach algebras constructed from the tensor product of two Banach algebras. Let ‎$‎A‎$‎ be a Banach algebra, so $‎A \hat{\otimes} A‎$ is a tensor Banach algebra. W‎e consider closed ideal generated in the projective tensor product $‎A \hat{\otimes} A‎$, then we generated closed ideal for ‎$‎A‎$‎ and investigate the quotient ideal amenability of ‎both $‎A‎$‎ and $‎A \hat{\otimes} A‎$‏‎. Additionally, we examine the hereditary properties of quotient ideal amenability for tensor Banach algebras and explore the relationship between the quotient ideal amenability of ‎$‎A‎$‎ and its projective tensor product.