One More Pathology of C-algebraic Tensor Products

Following E. Kirchberg, [3], we call a bifunctor (A,B) → A⊗α B a C -algebraic tensor product functor if it is obtained by completing of the algebraic tensor product A ⊙ B of C-algebras in a functional way with respect to a suitable C-norm ‖ · ‖α. We call such a functor symmetric if the standard isomorphism A ⊙ B ∼= B ⊙ A extends to an isomorphism A⊗αB ∼= B⊗αA. Similarly, we call it associative if the standard isomorphism A⊙(B⊙C) ∼= (A⊙B)⊙C extends to an isomorphism A⊗α (B⊗αC) ∼= (A⊗αB)⊗αC for any C-algebras A, B, C. It is well known that both the minimal tensor product functor ⊗min and the maximal tensor product functor ⊗max are symmetric and associative. In this paper we construct a collection of symmetric C-algebraic tensor product functors related to asymptotic homomorphisms of C-algebras. For technical reasons we restrict ourselves to the category of separable C-algebras. Using C-algebras related to property T groups [9] we show that some of these tensor product functors are not associative. Recall that asymptotic homomorphisms of C-algebras were first defined and studied in [2] in relation to topological properties of C-algebras. The most important and the best known case is the case of asymptotic homomorphisms from a suspended C-algebra SA to the C-algebra K of compact operators, since the homotopy classes of those are the K-homology of A, the E-theory. Asymptotic homomorphisms to other C-algebras are less known. For example, it is known that any asymptotic homomorphism to the Calkin algebra is homotopic to a genuine homomorphism [4, 6]. Even less is known about asymptotic homomorphisms to B(H), where there is no topological obstruction (recall that the K-groups of B(H) are trivial). Such asymptotic homomorphisms are called asymptotic representations and were first studied in relation to the asymptotic tensor product C-algebras [7] and to semi-invertibility of C-algebra extensions [8].