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… 
COVARIANT PRESHEAVES AND SUBALGEBRAS
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
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
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
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 ∗
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
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
TLDR
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
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
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
TLDR
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.
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 12 REFERENCES
Representation of modular lattices and of relation algebras
Introduction. This paper gives an axiomatic characterization of the class of all those (modular) lattices which are isomorphic to lattices of commuting equivalence relations. As might be expected,
Von Neumann Algebras and Hilbert Quantales
  • J. Pelletier
  • Mathematics, Computer Science
    Appl. Categorical Struct.
  • 1997
TLDR
It is proved that Maxw A is a von Neumann quantale for all vonNeumann algebras A, and its restriction to right-sided elements is an isomorphism and this leads to a new characterization of Von Neumann factors.
Handbook of Categorical Algebra
The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of
The equational theory of union-free algebras of relations
We solve a problem of Jónsson [12] by showing that the class ℛ of (isomorphs of) algebras of binary relations, under the operations of relative product, conversion, and intersection, and with the
On Orthomodular Lattices
The paper On complemented lattices was the third paper in the new theory of orthomodular lattices which started in 1936 with Birkhoff and von Neumann’s idea of developing a new many-valued logic for
Characterizing spatial quantales
We will prove that an idempotent, right-sided quantale is spatial iff it is a subquantale of a product of simple quantales.
Bredikhin, The equational theory of union-free algebras of ́ Ž
  • Algebra Unï ersalis
  • 1995
A quantisation of the calculus of relations, in ‘‘Canad
  • Math. Soc. Conf. Proc.,’’
  • 1992
...
1
2
...