TY - BOOK

T1 - Regularity of C*-algebras and central sequence algebras

AU - Christensen, Martin S.

PY - 2017

Y1 - 2017

N2 - The main topic of this thesis is regularity properties of C*-algebras and how these regularityproperties are reected in their associated central sequence algebras. The thesis consists of anintroduction followed by four papers [A], [B], [C], [D].In [A], we show that for the class of simple Villadsen algebra of either the rst type withseed space a nite dimensional CW complex, or the second type, tensorial absorption of theJiang-Su algebra is characterized by the absence of characters on the central sequence algebra.Additionally, in a joint appendix with Joan Bosa, we show that the Villadsen algebra of thesecond type with innite stable rank fails the corona factorization property.In [B], we consider the class of separable C*-algebras which do not admit characters on theircentral sequence algebra, and show that it has nice permanence properties. We also introducea new divisibility property, that we call local divisibility, and relate Jiang-Su stability of unital,separable C*-algebras to the local divisibility property for central sequence algebras. In particular,we show that a unital, simple, separable, nuclear C*-algebra absorbs the Jiang-Su algebra if,and only if, there exists k ≥ 1 such that the central sequence algebra is k-locally almost divisible.In [C], we show that for a substantial class of unital, separable and Ζ-stable C*-algebras, thereexists a closed 2-sided ideal in the central sequence algebra which is not a σ-ideal.In [D], we give a characterization of asymptotic regularity in terms of the Cuntz semigroup forsimple, separable C*-algebras, and show that any simple, separable C*-algebra which is neitherstably nite nor purely innite is not asymptotically regular either.

AB - The main topic of this thesis is regularity properties of C*-algebras and how these regularityproperties are reected in their associated central sequence algebras. The thesis consists of anintroduction followed by four papers [A], [B], [C], [D].In [A], we show that for the class of simple Villadsen algebra of either the rst type withseed space a nite dimensional CW complex, or the second type, tensorial absorption of theJiang-Su algebra is characterized by the absence of characters on the central sequence algebra.Additionally, in a joint appendix with Joan Bosa, we show that the Villadsen algebra of thesecond type with innite stable rank fails the corona factorization property.In [B], we consider the class of separable C*-algebras which do not admit characters on theircentral sequence algebra, and show that it has nice permanence properties. We also introducea new divisibility property, that we call local divisibility, and relate Jiang-Su stability of unital,separable C*-algebras to the local divisibility property for central sequence algebras. In particular,we show that a unital, simple, separable, nuclear C*-algebra absorbs the Jiang-Su algebra if,and only if, there exists k ≥ 1 such that the central sequence algebra is k-locally almost divisible.In [C], we show that for a substantial class of unital, separable and Ζ-stable C*-algebras, thereexists a closed 2-sided ideal in the central sequence algebra which is not a σ-ideal.In [D], we give a characterization of asymptotic regularity in terms of the Cuntz semigroup forsimple, separable C*-algebras, and show that any simple, separable C*-algebra which is neitherstably nite nor purely innite is not asymptotically regular either.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/fbp0ps/alma99122895064305763

M3 - Ph.D. thesis

BT - Regularity of C*-algebras and central sequence algebras

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -