Some Permanence for Large Subalgebra

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.