On Banach space projective tensor product of $C^*$-algebras
Ved Prakash Gupta, Ranjana Jain

TL;DR
This paper investigates the algebraic and structural properties of the Banach space projective tensor product of $C^*$-algebras, revealing its injectivity, ideal structure, and center identification, and comparing it with known tensor products.
Contribution
It provides new insights into the structure of the Banach space projective tensor product of $C^*$-algebras, including ideal structure and center characterization, extending previous understanding.
Findings
Injectivity of the tensor product on $C^*$-algebras
Detailed description of closed ideals in the tensor product
Identification of the center of the tensor product as $Z(A) ensor_{g} Z(B)
Abstract
We analyze certain algebraic structures of the Banach space projective tensor product of -algebras which are comparable with their known counterparts or the Haagerup tensor product and the operator space projective tensor product of -algebras. Highlights of this analysis include (a) injectivity of the Banach space projective tensor product when restricted to the tensor products of -algebras, (b) detailed structure of closed ideals of in terms of those of and , (c) identification of certain spaces of ideals of in terms of those of and from the perspective of hull-kernel topology, and (d) identification of the center of with , where and are -algebras.
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