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On closed Lie ideals of certain tensor products of C ∗ ‐algebras
Author(s) -
Gupta Ved Prakash,
Jain Ranjana
Publication year - 2018
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700009
Subject(s) - mathematics , unital , pure mathematics , ideal (ethics) , commutator , tensor product , hausdorff space , norm (philosophy) , tensor product of algebras , lie algebra , lie conformal algebra , algebra over a field , tensor product of hilbert spaces , philosophy , epistemology , political science , law , tensor contraction
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.