Essential commutative Cartan subalgebras of $C^*$-algebras

Abstract

We define essential commutative Cartan pairs of C * -algebras generalising the definition of Renault [12] and show that such pairs are given by essential twisted groupoid C * -algebras as defined by Kwaśniewski and Meyer [7]. We show that the underlying twisted groupoid is effective, and is unique up to isomorphism among twists over effective groupoids giving rise to the essential commutative Cartan pair. We also show that for twists over effective groupoids giving rise to such pairs, the automorphism group of the twist is isomorphic to the automorphism group of the induced essential Cartan pair via explicit constructions. The Weyl groupoid and Weyl twistLet G be an étale groupoid. A twist over G is a central extension Σ of G by G (0) × T; the unit space times the circle groupBy definition, Σ carries a central action of T. From this one may construct a canonical line bundle associated to the twist (G, Σ) as L := Σ×C T , where the quotient is by the action z(σ, λ) = (zσ, zλ) for z ∈ T, σ ∈ Σ, and λ ∈ C. Sections of this bundle are then equivalent to functions Σ → C satisfying f (zσ) = zf (σ), and we may often identify the two.Renault defines the Weyl groupoid and Weyl twist in [12] for commutative and nondegenerate inclusions of C * -algebras. In particular, we need not alter the definition to accommodate for our more general definition of essential Cartan pairs. Definition 2.1. Let A = C 0 (X) be a commutative C * -subalgebra of B containing an approximate unit for B. For each n ∈ N (A, B) we have n * n, nn * ∈ A by [12, Lemma 4.5]. Define dom(n) := {x ∈ X : n * n(x) > 0} and ran(n) := {x ∈ X : nn * (x) > 0}. Kumjian [5] described how normalisers of an inclusion A ⊆ B of a commutative C *algebra act as partial homeomorphisms on the Gelfand dual X of A = C 0 (X).