Some Permanence for Large Subalgebra

TL;DR

This paper investigates how certain properties of C*-algebras, such as real rank zero and weak comparison, can be inferred from their large or centrally large subalgebras, providing tools for property analysis.

Contribution

It establishes that properties like real rank zero and weak comparison in a C*-algebra can be deduced from those of its large or centrally large subalgebras, advancing understanding of algebraic structure.

Findings

A has real rank zero if B has real rank zero, where B is a centrally large subalgebra of A.

If A is stably finite, B has local weak comparison if A has it, and vice versa.

A has weak comparison if and only if B has weak comparison.

Abstract

In this paper, we give two properties of C*-algebra that could be deduced from the properties of its large subalgebra. Let A be an infinite dimensional simple unital C*-algebra and let B be a centrally large subalgebra of A, we prove that A has real rank zero if B has real rank zero. If A is stablely fnite in addition, B is a large subalgebra of A, we prove that B has local weak comparison if A has local weak comparison, and A has local weak comparison if M2(B) has local weak comparison. As a consequence, we show that A has weak comparison if and only if B has weak comparison. These results could be used to study some properties of C*-algebra from its large subalgebra or centrally large subalgebra.