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# Journal of Operator Theory

Volume 57, Issue 1, Winter 2007  pp. 173-206.

Spatial representation of minimal $C^*$-tensor products over abelian $C^*$-algebras

Authors Somlak Utudee
Author institution: Department of Mathematics, Faculty of Science, Chulalongkorn University, Bangkok, 10330, Thailand

Summary:  We establish natural links between minimal $C^*$-tensor products of $C^*$-algebras over abelian $C^*$-algebras, whose definition is based on a natural decomposition in fields of $C^*$-algebras, and spatial $W^*$-tensor products of $W^*$-algebras over abelian $W^*$-algebras, defined up to natural $*$-isomorphism by using appropriate normal $*$-representations. In particular, we obtain that if $C$ is a unital, abelian $C^*$-algebra, $A_1 , A_2$ are unital $C^*$-algebras over $C$ and $\pi_{\substack{{}\\ 1}} , \pi_{\substack{{}\\ 2}}$ are non-degenerate $*$-representations of $A_1$ respectively $A_2$, which coincide on $C$, are separated by a type {\rm I} von Neumann algebra with centre equal to the weak operator closure of the image of $C$ and are faithful in a certain stronger sense, then the minimal $C^*$-tensor product of $A_1$ and $A_2$ over $C$ can be identified with the $C^*$-algebra generated by the images $\pi_{\substack{{}\\ 1}}(A_{\substack{{}\\ 1}})$ and $\pi_{\substack{{}\\ 2}} (A_{\substack{{}\\ 2}})$ in the spatial $W^*$-tensor product of their weak operator closures with respect to the weak operator closure of the image of $C$.

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