Operator space tensor products of C*-algebras

Abstract

For C * -algebras A and B, the identity map from A ⊗B into A⊗ λ B is shown to be injective. Next, we deduce that the center of the completion of the tensor product A ⊗ B of two C * -algebras A and B with centers Z (A) and Z (B) under operator space projective norm is equal to Z (A) ⊗Z (B). A characterization of isometric automorphisms of A ⊗B and A⊗ h B is also obtained.

IDEALS IN OPERATOR SPACE PROJECTIVE TENSOR PRODUCT OF C*-ALGEBRAS

“…We use the facts that π A ⊗ π B is faithful (see [22,Theorem IV.4.9]) and that i is injective (see [11,Corollary 1]) to deduce our claim. Finally, the relation…”

“…Symmetry in group algebras has been investigated by various authors (see, for instance, [17,19]). One can easily verify that a Banach [11] Ideals in operator space projective tensor products 285 is nontrivial, where B α (v, w) := B(vα, wα) for all v, w ∈ A (see [19]). It is well known that every C * -algebra is symmetric (see [20]).…”

Projective Tensor Products of <i> C*</i>-Algebras

“…So it is equal to I ⊗J for some closed ideal J in B by ( [14], Theorem 3.8). Consider the closed ideal K h , the closure of i(K) in · h , where i : A ⊗B → A ⊗ h B is an injective map( [13], Theorem 1). Then K h ∩ (I ⊗ h B) = I ⊗ hJ for some closed idealJ in B by ( [1], Proposition 5.2).…”

“…For unital C * -algebras A and B, isometric automorphism of A ⊗B is either of the form φ ⊗ψ or ν ⊗ρ • τ , where φ : A → A, ψ : B → B, ν : B → A and ρ : A → B are isometric isomorphisms ( [13], Theorem 4). In the following, we characterize the isometric inner * -automorphisms of A ⊗B completely.…”

The Operator Space Projective Tensor Product: Embedding into the Second Dual and Ideal Structure

“…Let J min be the min-closure of J in A ⊗ min B. Then J min is a non-zero closed ideal of A ⊗ min B[14] and it is proper since J min = A⊗ min B would imply J = A ⊗ B[16].Let π : A⊗ min B → B(H) be an irreducible representation annihilating J min . Since the canonical map i : A ⊗ B → A ⊗ min B is a bounded * -homomorphism, we get a * -representationπ = π • i of A ⊗ B on H such that π(J) = {0}.…”