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

TL;DR

This paper extends approximation results for self-adjoint elements in real rank zero C* algebras to those of higher real rank, specifically for real rank one algebras, using techniques similar to function approximation with projections.

Contribution

It generalizes the approximation of self-adjoint elements by finite spectrum elements from real rank zero to higher real rank C* algebras, focusing on the diagonal of the self-adjoint part.

Findings

Approximation of self-adjoint elements by finite spectrum elements in higher real rank C* algebras.

Extension of known results from real rank zero to real rank one C* algebras.

Methodology similar to continuous function approximation using projections.

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.