PhD Defense Martin Søndergaard Christensen

Title: Regularity of C*-algebras and central sequence algebra

Abstract:

The main topic of this thesis is regularity properties of C*-algebras and how these regularity properties are reflected in their associated central sequence algebras.

 The primary results of the thesis are: For the class of Villadsen algebras of the first type, admitting a standard decomposition with seed space a finite CW complex and the class of Villadsen algebras of the second type, tensorial absorption of the Jiang-Su algebra is characterized by the absence of characters on the central sequence algebra. Also, for the larger class of unital, simple, separable and nuclear C*-algebras, the stronger requirement that the central sequence algebra is k-locally almost divisible suffices to conclude tensorial absorption of the Jiang-Su algebra.

 Secondary results include the proof that the Villadsen algebra of the second type with infinite stable rank fails the corona factorization property. This is the first example of a unital, simple, separable and stably finite C*-algebra with a unique tracial state failing this property. As a by-product of investigating whether the corona factorization property is equivalent to regularity, for simple and separable C*-algebras, a characterization of asymptotic regularity is given, in terms of the Cuntz semigroup associated with the C*-algebra. Finally, for a substantial class of unital, separable and Jiang-Su absorbing C*-algebras, an example of an ideal which is not sigma-ideal is provided, marking the first examples of such ideals in central sequence algebras.

Supervisor: Prof. Mikael Rørdam, Math, University of Copenhagen

Assessment committee:

Prof. Søren Eilers (Chairman), MATH, University of Copenhagen

Prof. Marius Dadarlat, Purdue University

Prof. Wilhelm Winter, Universität Münster