Approximation by elements of finite spectra for C* Algebras of higher real rank

TL;DR

The paper investigates extending density of finite-spectrum approximants from real rank zero C*-algebras to higher real rank by focusing on $1$-diagonal C*-algebras and the diagonal of $A^2_{sa}$. It proves that for a $1$-diagonal C*-algebra $A$, the set $A^2_{finite}$ is dense in $diag(A^2)_{sa}$ using a constructive approach that mirrors projection-based approximations of continuous functions via the continuous functional calculus. The method is validated in commutative and matrix-valued settings (e.g., $C([0,1])$ and $C([0,1],M_n(\, ext{C}))$, where rr$=1$), via spectral partitioning and projections $p_i=\chi_{F_i}(y^n)$ to produce finite-spectrum approximants. The work connects to Thiel's generator rank program, raising open questions about how the generator rank behaves under direct sums for rr$=1$ and higher (notably Questions 2.3 and 2.4), and discusses an erroneous lemma that motivates further investigation.

Abstract

In this article, we extend a well known result about real rank zero C* Algebras to higher real rank C* Algebras. The main technique used here is similar to the method in which we approximate continuous functions using projections. What we reach at the end, is similar to the fact that the self-adjoint elements of a real rank zero C* Algebra can be approximated by elements of finite spectrum. We achieve the result for the diagonal of the self-adjoint elements of A^2, where A is a real rank one C* Algebra.