Purely infinite rings and C*-algebras

Question 1: If $A_0\subseteq\mathcal{B}(H)$ is a *-subalgebra of bounded linear operators on a separable Hilbert space $H$ such that $A_0$ is purely infinite as a ring, is the norm-closure $\overline{A_0}$ purely infinite as a C*-algebra?

This is Problem 8.4 in [1]. As noted in Problem 8.6 in [1], this is even unclear if $A_0$ is a unital, simple, purely infinite ring. In the converse direction, it seems natural to ask:

Question 2: Given a purely infinite C*-algebra $A$ , is there a dense *-subalgebra $A_0\subseteq A$ that is purely infinite as a ring?

Definitions: The relation $\precsim$ on a ring $R$ is defined by setting $x\precsim y$ if there exist $a,b\in R$ such that $x=ayb$ ; see Definition 2.1 in [1]. A ring $R$ is purely infinite if no quotient of $R$ is a division ring, and if any $x,y\in R$ satisfy $x\precsim y$ if (and only if) $x\in RyR$ ; see Definition 3.1 in [1]. (Here, $RyR$ denotes the two-sided ideal generated by $y$ , that is, $RyR=\{a_1yb_1+\ldots+a_nyb_n : n\geq 1, a_i,b_i\in R\}$ .)

Given a C*-algebra $A$ , the Cuntz subequivalence relation $\precsim$ is defined by setting $x\precsim y$ if there exist sequences $(a_n)_n$ and $(b_n)_n$ in $A$ such that $\lim_{n\to\infty} \| x - a_nyb_n \| = 0$ . A C*-algebra $A$ is purely infinite if it admits no nonzero one-dimensional representations and if any $x,y\in A$ satisfy $x\precsim y$ if and only if $x$ belongs to $\overline{\mathrm{span}}AyA$ , the closed, two-sided ideal generated by $y$ ; see Definition 4.1 in [2]. (The definition in [2] only considers positive elements in $A$ , but it equivalent to the definition given here.)

Background: By Proposition 3.17 in [1], if a C*-algebra is purely infinite as a ring, then it is purely infinite as a C*-algebra. The converse does probably not hold (Remark 3.18 in [1]), which is why we ask Question 2 above. A unital, simple C*-algebra is purely infinite as a C*-algebra if and only if it is purely infinite as a ring. Thus, Question 2 has a positive answer in this case.

Given a C*-algebra $A$ , let $\mathrm{Ped}(A)\subseteq A$ denote its Pedersen ideal (the minimal dense ideal in $A$ ). By Proposition 8.5 in [1], if $\mathrm{Ped}(A)$ is purely infinite as a ring, then $A$ is purely infinite. Thus, in this particular instance, Question 1 has a positive answer.

[1]
G. Aranda Pino, K.R. Goodearl, F. Perera, M. Siles Molina, Non-simple purely infinite rings, American Journal of Mathematics. (2010) 563–610. https://doi.org/10.1353/ajm.0.0119.
[2]
E. Kirchberg, M. Rordam, Non-simple purely infinite C*-algebras, American Journal of Mathematics. (2000) 637–666. https://doi.org/10.1353/ajm.2000.0021.

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