Tensor Products of C*-Algebras Over Abelian Subalgebras |
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Authors: | Giordano Thierry; Mingo James A |
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Institution: | Department of Mathematics, University of Ottawa Ottawa K1N 6N5, Canada
Department of Mathematics, Queen's University Kingston K7L 3N6, Canada |
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Abstract: | Suppose that A is a C*-algebra and C is a unital abelian C*-subalgebrawhich is isomorphic to a unital subalgebra of the centre ofM(A), the multiplier algebra of A. Letting = , so that we maywrite C = C(), we call A a C()-algebra (following Blanchard7]). Suppose that B is another C()-algebra, then we form ACB, the algebraic tensor product of A with B over C as follows:A B is the algebraic tensor product over C, IC = {ni1(fi 11fi)x|fiC, xAB} is the ideal in AB generated by f11f|fC,and A CB = AB/IC. Then ACB is an involutive algebra over C,and we shall be interested in deciding when ACB is a pre-C*-algebra;that is, when is there a C*-norm on AC B? There is a C*-semi-norm,which we denote by ||·||C-min, which is minimal in thesense that it is dominated by any semi-norm whose kernel containsthe kernel of ||·||C-min. Moreover, if A C B has a C*-norm,then ||·||C-min is a C*-norm on AC B. The problem isto decide when ||·||C-min is a norm. It was shown byBlanchard 7, Proposition 3.1] that when A and B are continuousfields and C is separable, then ||·||C-min is a norm.In this paper we show that ||·||C-min is a norm whenC is a von Neumann algebra, and then we examine some consequences. |
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