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Multiplicative lattices and $$C^*$$-algebras. (English) Zbl 0676.46047
It is the main content of the Gelfand duality theorem that a commutative $${\mathbb{C}}^*$$-algebra can be recaptured from the compact completely regular locale of its closed ideals, which turns out to be the lattice of open subsets of the maximal spectrum of the $${\mathbb{C}}^*$$-algebra.
In a previous joint work with the reviewer, the author generalized this result to the case of non-commutative but postliminary $${\mathbb{C}}^*$$- algebras, using the quantale of closed right ideals as a substitute for the spectrum. This quantale is in fact a complete lattice provided with a multiplication induced by that of the $${\mathbb{C}}^*$$-algebra.
In the present paper, the author introduces a second multiplication on the same lattice of closed right ideals, which is induced by both the multiplication and the involution of the $${\mathbb{C}}^*$$-algebra. This allows him to generalise to the non commutative case various properties of compactness and complete regularity of the spectrum. It is hoped that these results will eventually allow a generalization of the characterization theorem of the previous mentioned paper.
In a very recent and not yet published work, C. J. Mulvey has taken over the ideas of the present paper, applying them in fact to the quantale of closed subspaces (instead of closed right ideals) of the $${\mathbb{C}}^*$$- algebra. This allows him to consider the involution on its own without mixing it with the multiplication.
Reviewer: F.Borceux

##### MSC:
 46L35 Classifications of $$C^*$$-algebras 06D99 Distributive lattices 18B35 Preorders, orders, domains and lattices (viewed as categories)
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##### References:
 [1] 1 C.A. Akemann , The general Stone-Weierstrass problem , J. Funct. Anal. 4 ( 1969 ), 277 - 294 . MR 251545 | Zbl 0177.17603 · Zbl 0177.17603 [2] 2 C.A. Akemann , Left ideal structures of C*-algebras , J. Funct. Anal. 6 ( 1970 ), 305 - 317 . MR 275177 | Zbl 0199.45901 · Zbl 0199.45901 [3] 3 F. Borceux & G. Van Den Bossche , Quantales and their sheaves , Order 3 ( 1986 ), 61 - 87 . MR 850399 | Zbl 0595.18003 · Zbl 0595.18003 [4] 4 F. Borceux , J. Rosický & G. Van Den Bossche , Quantales and C*-algebras , Sem. Math. Univ. Cath. Louvain , Rap. 129 ( 1988 ). MR 1053610 [5] 5 J. Dixmier , Les C*-algebras et leurs représentations , Gauthier-Villars , Paris 1964 . MR 171173 | Zbl 0152.32902 · Zbl 0152.32902 [6] 6 J. Dixmier , Les algèbres d’opérateurs dans l’espace Hilbertien (Algèbres de Von Neumann) , Gauthier-Villars , Paris 1969 . Zbl 0088.32304 · Zbl 0088.32304 [7] 7 R. Giles & H. Kummer , A non-commutative generalization of topology , Indiana Univ. Math. J. 21 ( 1971 ), 91 - 102 . MR 293408 | Zbl 0219.54003 · Zbl 0219.54003 [8] 8 P.T. Johnstone , Stone spaces , Cambridge Univ. Press , Cambridge 1982 . MR 698074 | Zbl 0499.54001 · Zbl 0499.54001 [9] 9 G. Kalmbach , Orthomodular lattices , Academic Press 1983 . MR 716496 | Zbl 0512.06011 · Zbl 0512.06011 [10] 10 M.D. MacLaren , Atomic orthocomplemented lattices , Pac. J. Math. 14 ( 1964 ), 597 - 612 . Article | MR 163860 | Zbl 0122.02201 · Zbl 0122.02201 [11] 11 C.J. Mulvey , Suppl. Rend. Circ. Mat. Palermo , Ser. II , 12 ( 1986 ), 99 - 104 . MR 853151 | Zbl 0633.46065 · Zbl 0633.46065 [12] 12 M.A. Naimark , Normed rings . Moscow 1968 . MR 355602 · Zbl 0137.31703 [13] 13 J. Paseka , Regular and normal quantales , Arch. Math. ( Brno ) 4 ( 1986 ), 203 - 210 . Article | MR 868535 | Zbl 0612.06012 · Zbl 0612.06012
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