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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