Regularity of Villadsen algebras and characters on their central sequence algebras

TL;DR

This paper investigates the regularity properties of Villadsen algebras, establishing conditions for Jiang-Su absorption related to central sequence algebra characters, and provides a novel example of a Villadsen algebra failing the Corona Factorization Property.

Contribution

It characterizes Jiang-Su absorption in Villadsen algebras via central sequence algebra characters and presents the first example of a Villadsen algebra failing the Corona Factorization Property.

Findings

Jiang-Su absorption linked to absence of characters in central sequence algebra.

Villadsen algebra of second type with infinite stable rank fails the Corona Factorization Property.

Provides new examples of simple, nuclear C*-algebras with unique trace lacking certain regularity properties.

Abstract

We show that if A is a simple Villadsen algebra of either the first type with seed space a finite dimensional CW complex, or of the second type, then $A$ absorbs the Jiang-Su algebra tensorially if and only if the central sequence algebra of A does not admit characters. Additionally, in a joint appendix with Joan Bosa, we show that the Villadsen algebra of the second type with infinite stable rank fails the Corona Factorization Property, thus providing the first example of a unital, simple, separable and nuclear $C^\ast$-algebra with a unique tracial state which fails to have this property.