Tensor products of operator spaces

Equivalence of norms on operator space tensor products of $C^\ast $-algebras

“…The theorem in this section summarises and modifies a result of Blecher and Paulsen [4] and Effros and Ruan [9] in a suitable way for subsequent use here. The similarity between the Haagerup tensor product results here and those for the projective tensor product in the next section will also be clearer.…”

“…The projective operator space norm ||·|| ∧ is greater than the spatial tensor product norm || · || * of the two C * -algebras [4]. Hence…”

“…β is a decomposition as above}, and the operator space projective tensor product A ⊗ B is the completion of A ⊗ B in this norm [4], [10]. Recall that the Banach space projective (tensor) norm || · || γ is defined on the algebraic tensor product X ⊗ Y of two Banach spaces X and Y by…”

“…Note that the same technique but including the "canonical" shuffle exploited by Paulsen [19] may be used to prove the well known commutativity of ⊗ [4], [9].…”

“…Section 2 contains the notation and definitions used in the paper, and section 3 has the matrix calculations and a helpful observation concerning an isometric involution on the operator space projective tensor product A ⊗ B of two C * -algebras A and B. The well known self-duality and injectivity of the Haagerup tensor product [4], [9], [3] is presented in a slightly modified form in Theorem 4.1. This forms a crucial basis of sections 6 and 7, and is a motivation for proving the corresponding result (Theorem 5.1) for the Banach space projective tensor.…”