Tensor products operator algebras over subalgebras

Abstract

We establish natural links between minimal C*-tensor products of C*-algebras over abelian C*-algebras, whose definition is based on a decomposition in fields of C*-algebras, and spatial W*-tensor products of W*-algebras over abelian W*-algebras, defined up to *-isomorphism by using appropriate nomai *-representations. In particular, we obtain that if C is a unital abelian C*-algebra, A[subscript 1], A[subscript 2] are C*-algebras over C and [pai 1], [pai 2] are appropriately faithful non-degenerate *-representations of A [subscript 1], respectively A[subscript 2], which coincide on C, then the minimal C*-tensor product of A[subscript 1], and A[subscript 2], over C can be identified with the C*-algebra generated by the images [pai 1(A [subscript 1], and [pai 2] (A[subscript 2]), in the spatial W*-tensor product of their weak operator closures with respect to the weak operator closure of the image of C.