Schur tensor product of operator spaces

Abstract

We develop a systematic study of the schur tensor product both in the category of operator spaces and in that of C * -algebras.

Polynomials in operator space theory: matrix ordering and algebraic aspects

“…In particular, we have the following well known result (see [14,17]): Corollary 5.2. ⊗ ⊗ A i , the projective tensor product and ⊗ ⊙ A i , the Schur tensor product are Banach * -algebras with a bounded approximate identity, however, ⊗ • A i , the Haagerup tensor product is a Banach algebra.…”

“…It is thus natural to ask for appropriate matrix ordering and algebraic structure that is compatible with this generalized λ-theory. In [9] and [17], the projective and Schur operator space tensor product of matrix ordered operator spaces are shown to be matrix ordered respectively. Further, Han in [8] successfully introduced cones at each matrix level of the tensor product of operator spaces that are closely related to projective and injective operator space tensor norms thereby, constructing two extremal tensor products of matrix regular operator space.…”

Arens regularity of projective tensor products

“…From all the results above, the Arens regularity of all Banach algebras A ⊗ s B, the Schur tensor product of C * -algebras A and B [20] Recall that an operator space X is exact if and only if it is locally embeds into a nuclear C * -algebra (say A), i.e. there is a constant C such that for any finite dimensional E ⊆ X, there is a subspace Ẽ ⊆ A and an isomorphism u : E → Ẽ with u cb u −1 cb ≤ C. Now using this definition of exact operator space and the fact that direct sum of two nuclear C * -algebras is nuclear if and only if each one of them is, one can easily verify that if V and W are exact operator algebras then V ⊕ W with sup-norm is also exact.…”

Variations of completely bounded maps on operator spaces

“…β} and u s = inf{ α e f β u = α(e • f )β}, which gives three operator space structures to E ⊗ F , whose completion with respect to these norms are named as the operator space projective tensor product, the Haagerup tensor product and the Schur tensor product respectively [3,8]. All necessary details on operator space tensor products can be seen in [1,5] and [7].…”

“…On the other hand, if φ 1 ⊗ φ 2 turn out to be completely bounded whenever φ 1 and φ 2 are completely bounded with φ 1 ⊗φ 2 cb ≤ φ 1 cb φ 2 cb , then we say µ is functorial. The operator space injective and the Haagerup tensor products are well known examples for injective tensor products while most of the tensor products that we consider including the operator space projective and the Schur tensor product are functorial [5,8]. An operator space tensor norm ⋅ µ is said to be matrix subcross if for any e ∈ M n (E) and f ∈ M n (F ), e ⊗ f µ ≤ e f and if equality holds then we call it matrix cross.…”