TENSOR PRODUCTS OF C*-ALGEBRAS OVER ABELIAN SUBALGEBRAS

Abstract

Suppose that A is a C*-algebra and C is a unital abelian C*-subalgebra which is isomorphic to a unital subalgebra of the centre of M(A), the multiplier algebra of A. Letting Ω = Ĉ, so that we may write C = C(Ω), we call A a C(Ω)-algebra (following Blanchard [7]). Suppose that B is another C(Ω)-algebra, then we form A[otimes ]_CB, the algebraic tensor product of A with B over C as follows: A [otimes ] B is the algebraic tensor product over [Copf ], I_C = {[sum ]ⁿ_i=1 (f_i [otimes ] 1−1[otimes ] f_i)x| f_i∈C, x∈A[otimes ]B} is the ideal in A[otimes ]B generated by 〈f[otimes ]1−1[otimes ] f|f∈C〉, and A [otimes ]_CB = A[otimes ]B/I_C. Then A[otimes ]_CB is an involutive algebra over [Copf ], and we shall be interested in deciding when A[otimes ]_CB is a pre-C*-algebra; that is, when is there a C*-norm on A[otimes ]_CB? There is a C*-semi-norm, which we denote by ‖·‖_C-min, which is minimal in the sense that it is dominated by any semi-norm whose kernel contains the kernel of ‖·‖_C-min. Moreover, if A [otimes ]_CB has a C*-norm, then ‖·‖_C-min is a C*-norm on A[otimes ]_CB. The problem is to decide when ‖·‖_C-min is a norm. It was shown by Blanchard [7, Proposition 3.1] that when A and B are continuous fields and C is separable, then ‖·‖_C-min is a norm. In this paper we show that ‖·‖_C-min is a norm when C is a von Neumann algebra, and then we examine some consequences.