$^{*}$-Regularity of Operator Space Projective Tensor Product of C$^{*}$-Algebras

TL;DR

This paper investigates the conditions under which the operator space projective tensor product of C*-algebras is *-regular, focusing on properties like Tomiyama's property (F) and the uniqueness of C*-norms.

Contribution

It establishes criteria for *-regularity of the tensor product based on property (F) and norm equality, and explores the property (F) for related tensor products.

Findings

A*hat*B is *-regular if property (F) holds for A⊗_min B and A⊗_min B = A⊗_max B.

A*hat*B has a unique C*-norm if and only if A⊗ B does.

Discussion of property (F) for A*hat*B and A⊗_h B, the Haagerup tensor product.

Abstract

The Banach $^{*}$-algebra $A\hat{\otimes}B$, the operator space projective tensor product of $C^{*}$-algebras $A$ and $B$, is shown to be $^{*}$-regular if Tomiyama's property ($F$) holds for $A\otimes_{\min}B$ and $A \otimes_{\min}B=A \otimes_{\max}B$, where $\otimes_{\min}$ and $\otimes_{\max}$ are the injective and projective $C^{*}$-cross norm, respectively. However, $A\hat{\otimes}B$ has a unique $C^{*}$-norm if and only if $A\otimes B$ has. We also discuss the property ($F$) of $A\hat{\otimes}B$ and $A\otimes_{h}B$, the Haagerup tensor product of $A$ and $B$.