HereditaryC⁎-subalgebra lattices

Abstract

We investigate the connections between order and algebra in the hereditary C*-subalgebra lattice H(A) and *-annihilator ortholattice P(A) ⊥ . In particular, we characterize ∨-distributive elements of H(A) as ideals, answering a 25 year old question, allowing the quantale structure of H(A) to be completely determined from its lattice structure. We also show that P(A) ⊥ is separative, allowing for C*-algebra type decompositions which are completely consistent with the original von Neumann algebra type decompositions.

Gerbes over posets and twisted C*-dynamical systems

“…In particular, if X is a locally compact Hausdorff space with base Ω and A is C 0 (X)-algebra [42, §C.1], then defining A X U := C 0 (U )A, U ∈ Ω, we get the C * -precosheaf A X = (A X , ) Ω . Similar arguments apply to the lattice of hereditary C * -subalgebras of a C * -algebra [2].…”

“…The Cuntz algebra has two remarkable dynamics: the faithful action α : G → autO d , where G ⊆ U(d) is a compact Lie group, and the canonical *-endomorphism σ : [18]. Let N α O d denote the normalizer of α(G) in autO d ; a 2-group of interest is given by 2…”

“…; these gerbes are all isomorphic to the standard q -gerbe, in fact the family ν = {ν o } realizes the isomorphism ν : 2 Ǧq → 2 Ǧ .…”

“…Given a group Π, a C * -algebra B and a 2-group 2 G = (G→N ) with N ⊆ autB , in our approach a twisted C * -dynamical system is defined as a non-abelian 2 G-cocycle over Π, see §4. 2. Thus twisted C * -dynamical systems in the standard sense [43,10,27] arise when G = U (B), the unitary group of B , and N = autB .…”

Diffuse traces and Haar unitaries

“…We note that the infimum j p j in Proj(A * * ) is in general strictly larger than the open projection p ∈ O(A) : p ≤ p j for all j , which corresponds to the hereditary sub-C * -algebra A ∩ j p j A * * p j . Thus, O(A) is naturally isomorphic to the lattice of hereditary sub-C * -algebras studied in [AB15].…”