The pure infiniteness transfer problem

Abstract

Problem 8.4 of Pino, Goodearl, Perera and Molina in [arxiv.org/abs/0806.4156] asks whether a dense subalgebra $A_{0}$ of a $C^*$-algebra $A$ that is purely infinite as a ring forces $A$ to be purely infinite as a $C^*$-algebra. We call this the $\textit{pure infiniteness transfer problem}$, the transfer being from the ring to its $C^*$-completion. The problem is open, even when $A_{0}$ is unital and simple. We settle it under two hypotheses: $A$ has real rank zero, and $A_{0}$ is closed under holomorphic functional calculus. Under these hypotheses $A_{0}$ is purely infinite simple as a ring if and only if $A$ is purely infinite simple as a $C^*$-algebra. We also prove an obstruction: a unital $C^*$-algebra with a nonzero finite projection has no dense hfc-closed purely infinite simple unital subring. Finally, the hypotheses hold for proper subalgebras, for the gauge action of $\mathbb{T}$ on a Cuntz algebra $\mathcal{O}_n$ , the smooth subalgebra $\mathcal{O}_n^\infty$ is a proper dense hfc-closed subalgebra that is purely infinite simple as a ring.