Multiplicative lattices and \(C^*\)-algebras.

*(English)*Zbl 0676.46047It 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.

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) |

##### Keywords:

Gelfand duality theorem; commutative \({\mathbb{C}}^*\)-algebra; compact completely regular locale of its closed ideals; lattice of open subsets of the maximal spectrum; non-commutative but postliminary \({\mathbb{C}}^*\)- algebras; quantale of closed right ideals; lattice of closed right ideals
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\textit{J. Rosický}, Cah. Topologie Géom. Différ. Catégoriques 30, No. 2, 95--110 (1989; Zbl 0676.46047)

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