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The space of ideals in the minimal tensor product of C*-algebras

Published online by Cambridge University Press:  15 January 2010

ALDO J. LAZAR
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69778, Israel. e-mail: aldo@post.tau.ac.il
Corresponding
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Abstract

For C*-algebras A1, A2 the map (I1, I2) → ker(qI1qI2) from Id′(A1) × Id′(A2) into Id′(A1minA2) is a homeomorphism onto its image which is dense in the range. Here, for a C*-algebra A, the space of all proper closed two sided ideals endowed with an adequate topology is denoted Id′(A) and qI is the quotient map of A onto A/I. This result is used to show that any continuous function on Prim(A1) × Prim(A2) with values into a T1 topological space can be extended to Prim(A1minA2). This enlarges the scope of [7, corollary 3·5] that dealt only with scalar valued functions. A new proof for a result of Archbold [3] about the space of minimal primal ideals of A1minA2 is obtained also by using the homeomorphism mentioned above. New proofs of the equivalence of the property (F) of Tomiyama for A1minA2 with certain other properties are presented.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

[1]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), 111148.Google Scholar
[2]Archbold, R. J.Topologies for primal ideals. J. London Math. Soc.(2) 37 (1987), 524542.CrossRefGoogle Scholar
[3]Archbold, R. J.Continuous bundles of C*-algebras and tensor products. Quart. J. Math. Oxford 50 (1999), 131146.CrossRefGoogle Scholar
[4]Archbold, R. J. and Batty, C. J. K.On factorial states of operator algebras, III. J. Operator Theory 15 (1986), 5381.Google Scholar
[5]Archbold, R. J., Somerset, D. W. B., Kaniuth, E. and Schlichting, G., Ideal spaces of the Haagerup tensor product of C*-algebras. International J. Math. 8 (1995), 129.CrossRefGoogle Scholar
[6]Blanchard, E. and Kirchberg, E.Non-simple purely infinite C*-algebras: the Hausdorff case. J. Funct. Anal. 207 (2004), 461513.CrossRefGoogle Scholar
[7]Brown, L. G., Stable isomorphisms of hereditary subalgebras of C*-algebras. Pacific J. Math. 71 (1977), 335348.CrossRefGoogle Scholar
[8]Dixmier, J., Sur les espaces localement quasi-compacts. Canadian J. Math. 20 (1968), 10931100.CrossRefGoogle Scholar
[9]Guichardet, A.Tensor products of C*-algebras, part I. Finite tensor products. Math. Inst. Aarhus Univ. Lecture Notes 12 (1969).Google Scholar
[10]Hauenschild, W., Kaniuth, E. and Voigt, A.*-Regularity and uniqueness of C*-norm for tensor products of *-algebras. J. Funct. Anal. 89 (1990), 137149.CrossRefGoogle Scholar
[11]Kaniuth, E.Minimal primal ideal spaces and norms of inner derivations of tensor products of C*-algebras. Math. Proc. Camb. Phil. Soc. 119 (1996), 297308.CrossRefGoogle Scholar
[12]Kirchberg, E. Dini functions on spectral spaces, SGFB487 preprint, nr. 321, (University of Münster, Münster, 2004).Google Scholar
[13]Tomiyama, J.Applications of Fubini type theorems to the tensor product of C*-algebras. Tôhoku Math. J. 19 (1967), 213226.CrossRefGoogle Scholar
[14]Wassermann, S.On tensor products of certain group C*-algebras. J. Funct. Anal. 23 (1976), 239254.CrossRefGoogle Scholar
[15]Wulfsohn, A.Produit tensoriel de C*-algèbres, Bull. Sci. Math. 87 (1963), 1327.Google Scholar

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