On closed Lie ideals of certain tensor products of ‐algebras

Abstract

Abstract. For a simple C * -algebra A and any other C * -algebra B, it is proved that every closed ideal of A ⊗ min B is a product ideal if either A is exact or B is nuclear.

On closed Lie ideals of certain tensor products of C∗‐algebras II

“…In [12], generalizing a result of [24], all closed Lie ideals of A ⊗ min B where identified for any simple unital C * -algebra A with at most one tracial state and for any unital commutative C * -algebra B. Herein, with the help of a suitable version of Tietze Extension Theorem for functions vanishing at infinity, we prove that similar characterization holds even if A and B are both non-unital. (1) If A is unital and admits at most one tracial state, then a subspace L of A ⊗ min B is a closed Lie ideal if and only if…”

“…It was proved in [12] that if ⊗ α is the Haagerup tensor product or the operator space projective tensor product and H is an infinite dimensional separable Hilbert space, then the Banach algebra B(H) ⊗ α B(H) contains only one non-trivial central closed Lie ideal, namely C(1 ⊗ 1), and that every non-central closed Lie ideal of B(H) ⊗ α B(H) contains the product ideal K(H) ⊗ α K(H). In this article, based on some recent progress made in [13], we include the study of closed Lie ideals of the Banach space projective tensor product A ⊗ γ B of C * -algebras A and B, as well.…”

“…6. And, by [12,Remark 4.14], every closed ideal in A ⊗ α B admits a quasi-central approximate identity. Therefore, by [12,Theorem 4.5], there exists a closed ideal K in A ⊗ α B such that K ⊆ L ⊆ N (K).…”

“…However, apart from the results established in [6] and [24], not much is known about the structure of closed Lie ideals of tensor products of operator algebras. Improvising their techniques, some work on the study of closed Lie ideals of certain tensor products of C * -algebras was taken up in [12]. This article is a continuation of the same theme.…”

“…In this article, based on some recent progress made in [13], we include the study of closed Lie ideals of the Banach space projective tensor product A ⊗ γ B of C * -algebras A and B, as well. Improving upon above result of [12], we go one step ahead and identify all closed Lie ideals of B(H) ⊗ α B(H) as follows:…”