Question 1:If is a *-subalgebra of bounded linear operators on a separable Hilbert space such that is purely infinite as a ring, is the norm-closure 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 is a unital, simple, purely infinite ring. In the converse direction, it seems natural to ask:

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

**Definitions: **The relation on a ring is defined by setting if there exist such that ; see Definition 2.1 in [1]. A ring is *purely infinite* if no quotient of is a division ring, and if any satisfy if (and only if) ; see Definition 3.1 in [1]. (Here, denotes the two-sided ideal generated by , that is, .)

Given a C*-algebra , the Cuntz subequivalence relation is defined by setting if there exist sequences and in such that . A C*-algebra is purely infinite if it admits no nonzero one-dimensional representations and if any satisfy if and only if belongs to , the closed, two-sided ideal generated by ; see Definition 4.1 in [2]. (The definition in [2] only considers positive elements in , 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 , let denote its Pedersen ideal (the minimal dense ideal in ). By Proposition 8.5 in [1], if is purely infinite as a ring, then 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.