# Simple Involutive Quantales

@article{Pelletier1997SimpleIQ, title={Simple Involutive Quantales}, author={Joan Wick Pelletier and Jir{\'i} Rosick{\'y}}, journal={Journal of Algebra}, year={1997}, volume={195}, pages={367-386} }

Abstract Involutive quantales were introduced in [7] as complete lattices equipped with a multiplication and an involution. Such structures are well known from the calculus of relations: the set R el(X) of binary relations on any setXforms an involutive quantale. However, the motivating example of an involutive quantale is the spectrum Max Aof a non-commutativeC*-algebraA, where Max Ais the involutive quantale consisting of all closed linear subspaces ofA. In [8] it was indeed shown thatAcan be…

## 47 Citations

COVARIANT PRESHEAVES AND SUBALGEBRAS

- 2011

For small involutive and integral quantaloids Q it is shown that covariant presheaves on symmetric Q-categories are equivalent to certain subalgebras of a speci c monad on the category of symmetric…

Groupoid sheaves as quantale sheaves

- Mathematics
- 2012

Abstract Several notions of sheaf on various types of quantale have been proposed and studied in the last twenty five years. It is fairly standard that for an involutive quantale Q satisfying mild…

On the quantisation of points

- Mathematics
- 2001

In the study of quantales arising naturally in the context of -algebras, Gelfand quantales have emerged as providing the basic setting. In this paper, the problem of defining the concept of point of…

On the quantisation of spaces

- Mathematics
- 2002

Abstract In a previous paper (Pure Appl. Algebra 159 (2001) 231) a definition of point of a Gelfand quantale is given in terms of algebraically irreducible representations of the quantale on an…

On quantales that classify C*-algebras ∗

- Mathematics
- 2004

The functor Max of Mulvey assigns to each unital C*-algebra A the unital involutive quantale Max A of closed linear subspaces of A, and it has been remarked that it classifies unital C*-algebras up…

Tropological systems are points of quantales

- Mathematics
- 2002

We address two areas in which quantales have been used. One is of a topological nature, whereby quantales or involutive quantales are seen as generalized noncommutative spaces, and its main purpose…

A non-commutative and non-idempotent theory of quantale sets

- Mathematics, Computer ScienceFuzzy Sets Syst.
- 2011

The purpose of this paper is to combine both ideas and to present a theory of non-commutative and non-idempotent quantale sets (among other things, standard concepts like fuzzy preorders and fuzzy equivalence relations will be exhibited as special cases).

Projectives and Injectives in the Category of Quantales

- Mathematics
- 2003

The notions of multiplication-stable completely distributive lattices and weakly multiplication-stable completely distributive lattices are introduced in this paper. It is proved that the regular…

Non-Commutative Topology and Quantales

- Mathematics, Computer ScienceStud Logica
- 2000

The relationship between q-spaces (c.f. [9]) and quantum spaces (c.f. [5]) is studied, proving that both models coincide in the case of Spec A, the spectrum of a non-commutative C*-algebra A. It is…

On Quantales and Spectra of C*-Algebras

- Mathematics, Computer ScienceAppl. Categorical Struct.
- 2003

Although Max is not an equivalence of categories, therefore not providing a direct generalization of Gelfand duality to the noncommutative case, it is a faithful complete invariant of unital C*-algebras.

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