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CLOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT OF CERTAIN C*-ALGEBRAS

Part of: Lie algebras and Lie superalgebras Selfadjoint operator algebras

Published online by Cambridge University Press:  03 July 2020

BHARAT TALWAR  and
RANJANA JAIN
BHARAT TALWAR
Affiliation:
Department of Mathematics, University of Delhi, Delhi, 110007, India, e-mails: [email protected], [email protected]
RANJANA JAIN
Affiliation:
Department of Mathematics, University of Delhi, Delhi, 110007, India, e-mails: [email protected], [email protected]
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Abstract

For a locally compact Hausdorff space X and a C*-algebra A with only finitely many closed ideals, we discuss a characterization of closed ideals of C0(X,A) in terms of closed ideals of A and a class of closed subspaces of X. We further use this result to prove that a closed ideal of C0(X)⊗minA is a finite sum of product ideals. We also establish that for a unital C*-algebra A, C0(X,A) has the centre-quotient property if and only if A has the centre-quotient property. As an application, we characterize the closed Lie ideals of C0(X,A) and identify all the closed Lie ideals of HC0(X)⊗minB(H), H being a separable Hilbert space.

MSC classification

Primary: 46L06: Tensor products of $C^*$-algebras
Secondary: 17B05: Structure theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 2 , May 2021 , pp. 414 - 425
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

Allen, S. D., Sinclair, A. M. and Smith, R. R., The ideal structure of the Haagerup tensor product of C*-algebras, J. Reine Angew. Math 442 (1993), 111–148.Google Scholar
Archbold, R. J., Robert, L. and Tikuisis, A., The Dixmier property and tracial states for C*-algebras, J. Funct. Anal 273(8) (2017), 2655–2718.CrossRefGoogle Scholar
Brešar, M., Kissin, E. and Shulman, V. S., Lie ideals: from pure algebra to C*-algebras, J. Reine Angew. Math 623 (2008), 73–121.Google Scholar
Gupta, V. P. and Jain, R., On closed Lie ideals of certain tensor products of C*-algebras, Math. Nachr 291(8–9) (2018), 1297–1309.CrossRefGoogle Scholar
Gupta, V. P. and Jain, R., On Banach space projective tensor product of C*-algebras, Banach J. Math. Anal 14 (2020), 524–538.CrossRefGoogle Scholar
Gupta, V. P., Jain, R. and Talwar, B., On closed Lie ideals of certain tensor products of C*-algebras II, Math. Nachr 293 (2020), 1–14.CrossRefGoogle Scholar
Jain, R. and Kumar, A., Ideals in operator space projective tensor product of C*-algebras, J. Aust. Math. Soc 91(2) (2011), 275–288.CrossRefGoogle Scholar
Kadison, R. V. and Ringrose, J. R., Fundamentals of the theory of operator algebras, vol. I, Pure and Applied Mathematics, vol. 100 (Academic Press, Inc., New York, 1983).Google Scholar
Kaniuth, E., A course in commutative Banach algebras, Graduate Texts in Mathematics, vol. 246 (Springer, New York, 2009).CrossRefGoogle Scholar
Longstaff, W. E., Strongly reflexive lattices, J. London Math. Soc. s2-11(4) (1975), 491–498.CrossRefGoogle Scholar
Luft, E., The two-sided closed ideals of the algebra of bounded linear operators of a Hilbert space, Czechoslovak Math. J 93(18) (1968), 595–605.CrossRefGoogle Scholar
Marcoux, L. W., Projections, commutators and Lie ideals in C*-algebras, Math. Proc. R. Ir. Acad., 110A(1) (2010), 31–55.Google Scholar
Naimark, M. A., Normed algebras (Groningen, 1964).Google Scholar
Rudin, W., Real and complex analysis, 3rd edition (McGraw-Hill Book Co., New York, 1987).Google Scholar
Takesaki, M., Theory of operator algebras I (Springer-Verlag, New York, 1979).CrossRefGoogle Scholar
Vesterstrøm, J., On the homomorphic image of the center of a C*-algebra, Math. Scand. 29(1971), 134–136.CrossRefGoogle Scholar
Wassermann, S., A pathology in the ideal space of L(H) ⊗ L(H), Indiana Univ. Math. J 27(6) (1978), 1011–1020.CrossRefGoogle Scholar