Sup-lattice 2-forms and quantales

Abstract

A 2-form between two sup-lattices L and R is defined to be a sup-lattice bimorphism L × R → 2. Such 2-forms are equivalent to Galois connections, and we study them and their relation to quantales, involutive quantales and quantale modules. As examples we describe applications to C*-algebras.  2004 Elsevier Inc. All rights reserved.

A Noncommutative Theory of Penrose Tilings

“…It may be noted that at present this correspondence holds only up to a conjecture, equivalent to asking [29] that for each algebraically irreducible representation of Max A the annihilator ann(x) = {a ∈ Max A | xa = 0} of at least one cyclic generator x ∈ S is a maximal right-sided element of Max A. The precise form of the conjecture in [20] is in terms of the nontriviality of the representation with respect to the mapping of pure states.…”

“…The reader is referred to [19,20,21,23,29] for background discussion of quantales of this kind. By a representation of a quantale Q on the orthocomplemented sup-lattice S is meant a homomorphism r : Q → Q(S) .…”

“…The fact that r is an involutive homomorphism can be restated equivalently by saying that S is an involutive right Q-module [29], i.e., a right Q-module satisfying the condition xa ⊥ y ⇐⇒ x ⊥ ya * for all x, y ∈ S and all a ∈ Q.…”

“…In particular, there is a functor from unital C*-algebras to the category of unital involutive quantales which is a complete invariant: to each C*-algebra A it assigns the quantale Max A of all the closed linear subspaces of A (its operator spaces [24]). This functor was proposed in [17] as a noncommutative generalisation of the classical maximal spectrum, and has been subsequently studied in various papers [13,14,15,18,20,21,22,29].…”

Remark on the Unital Quantale Q[e]

“…Following C.J. Mulvey, various types and aspects of quantales have been considered by many researchers [5][6][7][9][10][11][12][13][14][15][16]. Simple quantale, spatial quantale and idempotent quantale are very important classes of quantales and they have been studied systemically in [5,7,10,12,15,16] …”