Tensor Products of C*‐Algebras Over Abelian Subalgebras

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 ⊗ C B , the algebraic tensor product of A with B over C as follows: A ⊗ B is the algebraic tensor product over C, I C = {∑ n i −1 ( f i ⊗ 1−1⊗ f i ) x | f i ∈ C , x ∈ A ⊗ B } is the ideal in A ⊗ B generated by 〈 f ⊗1−1⊗ f | f ∈ C 〉, and A ⊗ C B = A ⊗ B / I C . Then A ⊗ C B is an involutive algebra over C, and we shall be interested in deciding when A ⊗ C B is a pre‐C*‐algebra; that is, when is there a C*‐norm on A ⊗ C B ? 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 ⊗ C B has a C*‐norm, then ‖·‖ C ‐min is a C*‐norm on A ⊗ C B . 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.