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Minimal primal ideal spaces and norms of inner derivations of tensor products of C*-algebras

Published online by Cambridge University Press:  24 October 2008

Eberhard Kaniuth
Eberhard Kaniuth
Affiliation:
Fachbereich Mathematik/Informatik, Universität Paderborn, D-33095 Paderborn, Germany
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Extract

An ideal I in a C*-algebra A is called primal if whenever n ≥ 2 and J1,…, Jn are ideals in A with zero product then Jk ⊆ I for at least one k. The topologized space of minimal primal ideals of A, Min-Primal (A), has been extensively studied by Archbold[3]. Very much in the spirit of Fell's work [14] it was shown in [3, theorem 5·3] (see also [5, theorem 3·4]) that if A is quasi-standard, then A is *-isomorphic to a maximal full algebra of cross-sections of Min-Primal (A). Moreover, if A is separable the fibre algebras are primitive throughout a dense subset. On the other hand, the complete regularization of the primitive ideal space of A gives rise to the space of so-called Glimm ideals of A, Glimm (A). It turned out that A is quasi-standard exactly when Min-Primal (A) and Glimm (A) coincide as sets and topologically [5, theorem 3·3].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 2 , February 1996 , pp. 297 - 308
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[1]Archbold, R. J.. On the centre of a tensor product of C*-algebras. J. London Math. Soc. (2) 10 (1975), 257–262.CrossRefGoogle Scholar
[2]Archbold, R. J.. On the norm of an inner derivation of a C*-algebra. Math. Proc. Camb. Phil. Soc. 84 (1978), 273–291.CrossRefGoogle Scholar
[3]Archbold, R. J.. Topologies for primal ideals. J. London Math. Soc. (2) 36 (1987), 524–542.Google Scholar
[4]Archbold, R. J. and Batty, C. J. K.. On factorial states of operator algebras. III. J. Operator Theory 15 (1986), 53–81.Google Scholar
[5]Archbold, R. J. and Somerset, D. W. B.. Quasi-standard C*-algebras. Math. Proc. Camb. Phil. Soc. 107 (1990), 349–360.CrossRefGoogle Scholar
[6]Batty, C. J. K.. Derivations of tensor products of C*-algebras. J. London Math. Soc. (2) 18 (1978), 129–140.CrossRefGoogle Scholar
[7]Beckhoff, F.. The minimal primal ideal space of a C*-algebra and local compactness. Canad. Math. Bull. 34 (1991), 440–446.CrossRefGoogle Scholar
[8]Blackadar, B. A.. Infinite tensor products of C*-algebras. Pacific J. Math. 72 (1977), 313–344.CrossRefGoogle Scholar
[9]Choi, M. D. and Effros, E. G.. Nuclear C*-algebras and injectivity: the general case. Indiana Univ. Math. J. 26 (1977), 443–446.CrossRefGoogle Scholar
[10]Connes, A.. Classification of injective factors. Ann. Math. 104 (1976), 73–115.CrossRefGoogle Scholar
[11]Dauns, J. and Hofmann, K. H.. Representations of rings by continuous sections. Mem. Amer. Math. Soc. 83 (Amer. Math. Soc., 1968).CrossRefGoogle Scholar
[12]Dixmier, J.. Points séparés dans le spectre d'une C*-algèbre. Acta Sci. Math. (Szeged) 22 (1961), 115–128.Google Scholar
[13]Dixmier, J.. Les C*-algèbres et leurs représentations. (Gauthier-Villars, 1964).Google Scholar
[14]Fell, J. M. G.. The structure of algebras of operator fields. Acta Math. 106 (1961), 233–280.Google Scholar
[15]Green, P.. The local structure of twisted covariance algebras. Acta Math. 140 (1978), 191–250.CrossRefGoogle Scholar
[16]Guichardet, A.. Tensor products of C*-algebras. I. Finite tensor products. (Math. Inst. Aarhus Univ. Lecture Notes 12, 1969).Google Scholar
[17]Hauenschild, W., Kaniuth, E. and Voigt, A.. *-regularity and uniqueness of C*-norm for tensor products of *-algebras. J. Fund. Anal. 89 (1990), 137–149.Google Scholar
[18]Kadison, R. V., Lance, C. E. and Ringrose, J. R.. Derivations and automorphisms of operator algebras, II. J. Funct. Anal. 1 (1967), 204–221.CrossRefGoogle Scholar
[19]Somerset, D. W. B.. The inner derivations and the primitive ideal space of a C*-algebra. J. Operator Theory 29 (1993), 307–321.Google Scholar
[20]Somerset, D. W. B.. The inner derivations and primal ideals of C*-algebras. J. London Math. Soc., to appear.Google Scholar
[21]Stampfli, J. G.. The norm of a derivation. Pacific J. Math. 33 (1970), 737–748.CrossRefGoogle Scholar
[22]Takesaki, M.. On the crossnorm of the direct product of C*-algebras. Tohoku Math. J. 16 (1964), 111–122.CrossRefGoogle Scholar
[23]Tomiyama, J.. Application of Fubini type theorem to tensor products of C*-algebras. Tohoku Math. J. 19 (1967), 213–226.CrossRefGoogle Scholar
[24]Zaki, A. M.. Primal ideals and tensor products of C*-algebras. J. Math. Phy. Sci. 24 (1990), 171–176.Google Scholar
[25]Zsido, L.. The norm of a derivation of a W*-algebra. Proc. Amer. Math. Soc. 38 (1973), 147–150.Google Scholar