The linear span of projections in AH algebras and for inclusions of C*-algebras

TL;DR

This paper characterizes when AH algebras and certain inclusions of C*-algebras have the LP property, linking it to spectral functions and Rokhlin actions, with implications for fixed point and crossed product algebras.

Contribution

It provides a characterization of the LP property in AH algebras via spectral functions and explores how it is preserved under Rokhlin actions and inclusions.

Findings

AH algebra has LP property iff spectral functions are in the closure of projections

Diagonal AH-algebras with small eigenvalue variation have LP property

Rokhlin property actions preserve LP property in fixed point and crossed product algebras

Abstract

A $C^*$-algebra is said to have the LP property if the linear span of projections is dense in a given algebra. In the first part of this paper, we show that an AH algebra $A = \underrightarrow{\lim}(A_i,\phi_i)$ has the LP property if and only if every real-valued continuous function on the spectrum of $A_i$ (as an element of $A_i$ via the non-unital embedding) belongs to the closure of the linear span of projections in $A$. As a consequence, a diagonal AH-algebra has the LP property if it has small eigenvalue variation. The second contribution of this paper is that for an inclusion of unital $C^*$-algebras $P \subset A$ with a finite Watatani Index, if a faithful conditional expectation $E\colon A \rightarrow P$ has the Rokhlin property in the sense of Osaka and Teruya, then $P$ has the LP property under the condition $A$ has the LP property. As an application, let $A$ be a simple unital $C^*$-algebra with the LP property, $G$ a finite group and $\alpha$ an action of $G$ onto $\mathrm{Aut}(A)$. If $\alpha$ has the Rokhlin property in the sense of Izumi, then the fixed point algebra $A^G$ and the crossed product algebra $A \rtimes_\alpha G$ have the LP property. We also point out that there is a symmetry on CAR algebra, which is constructed by Elliott, such that its fixed point algebra does not have the LP property.