On the quantisation of points

Sup-lattice 2-forms and quantales

“…Remark 5.12. In [7] the points of a(n involutive) quantale Q are certain right Q-modules which are atomic as sup-lattices and whose atoms are generators, being thus everywhere principal (see also Section 7). In certain places in [7] the hypothesis that the module satisfies an additional condition known as non-triviality is assumed.…”

“…We are indebted to the work of Mulvey and Pelletier [7], which was one of the main sources of inspiration for our paper. They implicitly use parts of the theory of 2-forms, and this is reflected in the fact that we obtain, in Section 7, a much shorter proof of one of their main theorems [7, Theorem 9.1], which concerns the relation between quantales and C*-algebras.…”

“…This involutive quantale is denoted by Max A in [4,5,7,8], where it plays the role of the "noncommutative maximal spectrum" of A. If H is a Hilbert space, its norm-closed linear subspaces can be identified with the projections on H , and we denote the sup-lattice Sub C (H ) by P(H ).…”

Prime Elements of Non-integral Quantales and their Applications

“…As a remarkable result we obtain that every pure state of A can be identified with a hermitian prime element of the semi-integral regularization of the quantale Max(A) given by all closed linear subspaces of A (cf. [18]). Moreover, we construct a Q(2)-valued topology τ A on the spectrum of A such that the lattice of all closed left (right) ideals is isomorphic to the subquantale of all left-sided (right-sided) elements of τ A .…”

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] …”