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

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 $B$.