Permanence of real rank zero from a centrally large subalgebra to a C*-algebra

Abstract

Let A A be an infinite dimensional simple unital C*-algebra and B B be a centrally large subalgebra of A A . Archey and Phillips [J. Operator Theory 83 (2020), pp. 353–389] showed that A A has real rank zero when B B has real rank zero and stable rank one. In this paper, we have removed the restriction of B B has stable rank one. Using matrix decomposition by some suitable projections, we have shown that A A has real rank zero if B B has real rank zero.