2017/01/09 by Ranjana Jain, Ved Prakash Gupta, Jain, Ranjana +1 · 1 citation
Mathematics · #46L06 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1701.02218
openalex publication_date 2017/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. Closed commutator of a closed ideal in a Banach algebra whose every closed ideal possesses a quasi-central approximate identity is described in terms of the commutator of the Banach algebra. If α is either the Haagerup norm, the operator space projective norm or the C^*-minimal norm, then this allows us to identify all closed Lie ideals of A ⊗α B, where A and B are simple, unital C^*-algebras with one of them admitting no tracial functionals, and to deduce that every non-central closed Lie ideal of B(H) ⊗α B(H) contains the product ideal K(H) ⊗α K(H). Closed Lie ideals of A ⊗min C(X) are also determined, A being any simple unital C^*-algebra with at most one tracial state and X any compact Hausdorff space. And, it is shown that closed Lie ideals of A ⊗α K(H) are precisely the product ideals, where A is any unital C^*-algebra and α any completely positive uniform tensor norm.