A Non-Commutative Generalization of Topology

A Gelfand representation theory forC∗-algebras

“…Identify A and zAczM and call M the pure state g-space of A. (The terminology is lifted from [11].) We have already defined g-open projections in M, and their complements (in M) are called g-closed.…”

“…Since p is also g-closed in M by Lemma II.4, the theorem follows from [3, 11.10] M. If A were an abelian C*-algebra of functions containing the constants and separating the points of the topological space Ω, then A consists of all continuous functions on Ω if and only if Ω is compact. Following [11] we define a g-space to be an atomic TF*-algebra. In [7] Dixmier introduces the ideal center of a C*-algebra which is a C *-subalgebra of ikf (A) containing A. Dixmier constructs it in A** but Lemma III.5 assures us the idea carries over to M as well.…”

Coupled Quantales and a Non-Commutative Approach to Bitopological Spaces

“…Mulvey [7] proposed the term quantale as a non-commutative extension of the concept of frame ( a complete lattice satisfying the first infinite distributive law of finite meets over arbitrary sups). The purpose was to develop the concept of non-commutative topology, introduced by R. Giles and H. Kummer [5], while providing constructive foundations for the theory of quantum mechanics and non-commutative logic [11]. Nowadays, the notion of quantale can boast many areas of application, e. g., in the field of non-commutative topology [8,9,3].…”

Open projections do not form a right residuated lattice

“…In [4] and [1] the authors considered the lattice of closed right ideals (or equivalently so called open projections) as a spectrum of non-commutative C*-algebra. The embedding of a C*-algebra to an enveloping W*-algebra provides a representation of the lattice to an orthomodular * Supported by the project "Algebraic Methods in Quantum Logic" by ESF, No.…”

“…Recall from [4] that a projection p ∈ A * * (here A * * is the enveloping W*-algebra of C*-algebra A) is called open if it is a support of some a ∈ A, i.e. the smallest projection such that ap = a.…”