×

An essay on noncommutative topology. (English) Zbl 0683.54007

By “noncommutative topology” the authors mean C. J. Mulvey’s theory of quantales [Rend. Circ. Mat. Palermo., II. Ser., Suppl. 12, 99- 104 (1986; Zbl 0633.46065)]. This was originally introduced in the hope of providing a “Gelfand-type” representation for noncommutative \(C^*\)-algebras.
The present paper carries this programme forward by setting up a category of “quantum spaces” having the same relation to quantales as topological spaces do to locales, and then showing that the spectrum (i.e. the set of pure states) of a noncommutative \(C^*\)-algebra may be given the structure of a quantum space. The Gelfand-Naimark theorem is reinterpreted in this context. The result is interesting, but too complicated to quote here.
Reviewer: P.T.Johnstone

MSC:

54A05 Topological spaces and generalizations (closure spaces, etc.)
18B30 Categories of topological spaces and continuous mappings (MSC2010)
46L05 General theory of \(C^*\)-algebras
54B35 Spectra in general topology
46L30 States of selfadjoint operator algebras

Citations:

Zbl 0633.46065
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akemann, C. A., A Gelfand representation theory for \(C^∗\)-algebras, Pacific J. Math., 39 (1971) · Zbl 0203.44502
[2] Arveson, W., An Invitation to \(C^∗\)-Algebras (1976), Springer: Springer Berlin · Zbl 0344.46123
[3] Banaschewski, B., Sheaves of Banach spaces, Quaestiones Math., 2, 1-22 (1977) · Zbl 0435.46051
[5] Borceux, F.; van den Bossche, G., Quantales and their sheaves, (Rapp. de Sém. de Math. (1985), Université de Louvain) · Zbl 0595.18003
[6] Burden, C. W.; Mulvey, C. J., Banach Spaces in a Category of Sheaves, (Lecture Notes in Math., 753 (1978), Springer: Springer Berlin) · Zbl 0432.46066
[7] Dilworth, R.; Ward, M., Residuated lattices, Trans. Amer. Math. Soc., 45, 335-354 (1939) · JFM 65.0084.01
[8] Dixmier, J., Les \(C^{∗>}\)-Algèbres et leurs Représentations (1964), Gauthiers-Villars: Gauthiers-Villars Paris · Zbl 0152.32902
[9] Giles, R.; Kummer, H., A noncommutative generalization of topology, Indiana Math. J., 21, 91-102 (1971) · Zbl 0219.54003
[10] Johnstone, P., Stone Spaces (1982), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0499.54001
[11] Mulvey, C. J., Banach sheaves, J. Pure Appl. Algebra, 17, 69-83 (1980) · Zbl 0475.18007
[12] Mulvey, C. J., Rend. Circ. Mat. Palermo, 12, 99-104 (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.