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

Abstract

The Banach ${}^{\ast }$-algebra $A\hat{\otimes }B$, the operator space projective tensor product of $C^{\ast }$-algebras $A$ and $B$, is shown to be ${}^{\ast }$-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 $C^{\ast }$-cross norm, respectively. However, $A\hat{\otimes }B$ has a unique $C^{\ast }$-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 $B$.

References in corpus (1)

• Spectral synthesis for operator space projective tensor product of $C^{\ast }$-algebras