Quantales and C∗-Algebras

HereditaryC⁎-subalgebra lattices

“…By Theorem 4, the ideals in H(A) can be identified purely from the order structure of H(A), so the lattice H(A) completely determines the quantale H(A). As any postliminary A is completely determined by the quantale H(A), by the theorem at the end of[BRVdB89], 8 we also get the following strengthening of [BRVdB89] Proposition 5.…”

“…We then show in §6 that the superficially similar notion of a ∧-pseudocomplement turns out to have a quite different algebraic characterization in H(A), namely as an annihilator ideal. Next, in §7, we show that arbitrary ideals in H(A) can be characterized as the ∨-distributive elements, answering a long-standing question from [BRVdB89]. Quantales, as introduced in [Mul86], have often been considered the appropriate non-commutative analogs of locales, and this characterization shows that the natural quantale structure on H(A) is, in fact, completely determined by its lattice structure.…”

“…satisfying aIb ⊆ I ⇒ a ∈ I or b ∈ I, for all a, b ∈ A (see [Ped79] 3.13.7). We also call I ∈ I(A) primitive if I is the largest element of I(A) contained in some coatom B ∈ H(A), which is equivalent saying I is the kernel of some π ∈Â (see [BRVdB89]). Coatoms in I(A) are usually just called maximal, and we have the following relationships between these concepts in…”

Value semigroups, value quantales, and positivity domains

“…The term quantale, a portemanteau on quantum locale, was introduced by Mulvey in [21]. In [21], as well as in other texts (e.g., and this is far from an exshaustive list, [22,23], which are also general references for quantales, [12,24,25,26,27] for quantales in the context of topology, [28,29,30] for quantales in the context of categorical logic and computer science, and [31,32] for quantales in the context of C * -algebras) the notion of a (unital) quantale is defined as a monoid object in the category of complete join semilattices. Consequently, if L is a value quantale, then L op , the opposite of L, is a quantale in the sense of, e.g., [22], which is in fact an integral quantale.…”

Quantales

“…Then p: R(A) ---> L(H) is a strong homomorphism of quantales which is surjective on dual atoms.Proof. Following[4], pis a strong homomorphism of quantales. Due to Lemma 2.1, it suffices to prove that p. is injective on dual atoms.…”