#### Date of Award

1993

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Raymond Fabec

#### Abstract

It is shown that the C* algebra of a groupoid with Haar system has a natural split abelian extension. For a split abelian extension of a C* algebra it is shown that all representations of the original algebra extend to the split abelian extension. Under a reasonable assumption it is shown that states extend to a split abelian extension. Definitions for quasi-invariant and ergodic measures are given for split abelian extension of C* algebras, and it is shown when the split abelian extension is the natural extension of the C* algebra of a groupoid with Haar system that these definitions are equivalent to the groupoid definitions of quasi-invariant and ergodic. Irreducible representations that live over orbits on principal groupoids with Haar system are shown to be determined by the orbit. And in the case of an r-discrete, principal groupoid, it is shown how to reconstruct the Borel equivalence relation from the states of the natural extension of the C* algebra of the groupoid.

#### Recommended Citation

Curole, Mark Andrew, "Split Abelian Extensions of Calgebras." (1993). *LSU Historical Dissertations and Theses*. 5563.

http://digitalcommons.lsu.edu/gradschool_disstheses/5563

#### Pages

65