On C*-Diagonals

Abstract

Preface. The impetus for this study arose from the belief that the structure of a C*-algebra is illuminated by an understanding of the manner in which abelian subalgebras embed in it. Posed in its full generality, the question concerning abelian subalgebras would seem impossible to answer. A notion of diagonal subalgebra is, however, proposed which has the virtue that one can associate a “topological”; invariant to the pair consisting of the diagonal and the ambient algebra, from which these algebras may be retrieved.In the setting of von Neumann algebras, the analogous question was addressed in the seminal work of Feldman and Moore [13]. Their definition of Cartan subalgebra permits the abstraction of a complete invariant consisting of a Borel equivalence relation together with a certain cohomology class on the relation from which the Cartan pair may be recovered. Our development parallels theirs in spirit; differences in substance derive from the topological flavor of C*-theory.

Representations of triangular subalgebras of groupoid C*-algebras

“…Throughout the paper B will always denote a nuclear C* -algebra having a diagonal D in the sense of Kumjian [10]. A normalizer of D is simply an element b e B such that b*db and bdb* are in D whenever d € D. Such a normalizer is called free if b 2 = 0.…”

“…More precisely, Kumjian's representation theorem [10] asserts that there is a T-groupoid E over an r-discrete, locally compact, principal groupoid G, whose unit space G <0) may be identified with the maximal ideal space of D, such that B is isomorphic to C* ed (G, E). We shall now explain what all this means.…”

“…Let B be a nuclear C* -algebra that has a diagonal subalgebra D in the sense of Kumjian [10]; that is, we assume D is a masa in B with certain properties that allow B to be represented (essentially) in terms of matrices indexed by an equivalence relation (= a principal groupoid) on the maximal ideal [3] Representations of triangular subalgebras 291 space of D. From the spectral theorem for bimodules [12] we may assert that any (norm closed) subalgebra A of B containing D must consist of all matrices supported on a transitive, reflexive subrelation of the equivalence relation indexing B. The algebras we consider here will also be assumed to satisfy: AD A* = D (that is, they will be triangular) and A + A* is dense in B.…”

Fell bundles over groupoids

“…Then C * r (E) has a diagonal subalgebra (see [Ku,§2]) isomorphic to C 0 (G 0 ); the twist invariant for the diagonal pair (C * r (E), C 0 (G 0 )) is the inverse of [Σ]. iv.…”

“…Assume that Γ is an r-discrete groupoid (see [Re1]). Let p : E → Γ be a Fell bundle; we construct the analog of the reduced C*-algebra C * r (E) as in [Ku,§2] (in [Yg] the full C*-algebra is constructed). Given f, g ∈ C c (E), define multiplication and involution by means of the formulas…”