When central sequence C*-algebras have characters

TL;DR

This paper explores the properties of C*-algebras with characterless central sequence algebras, linking their structure to the Jiang-Su algebra and regularity properties, and presents new results on their divisibility and absorption characteristics.

Contribution

It establishes a connection between the absence of characters in the central sequence algebra and the strong Corona Factorization Property, and investigates conditions under which the Jiang-Su algebra embeds.

Findings

Absence of characters implies the strong Corona Factorization Property.

Some simple nuclear C*-algebras have central sequence algebras with characters.

Stronger divisibility in the central sequence algebra leads to enhanced regularity in the C*-algebra.

Abstract

We investigate C*-algebras whose central sequence algebra has no characters, and we raise the question if such C*-algebras necessarily must absorb the Jiang-Su algebra (provided that they also are separable). We relate this question to a question of Dadarlat and Toms if the Jiang-Su algebra always embeds into the infinite tensor power of any unital C*-algebra without characters. We show that absence of characters of the central sequence algebra implies that the C*-algebra has the so-called strong Corona Factorization Property, and we use this result to exhibit simple nuclear separable unital C*-algebras whose central sequence algebra does admit a character. We show how stronger divisibility properties on the central sequence algebra imply stronger regularity properties of the underlying C*-algebra.