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 . As noted in Problem 8.6 in , 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 . 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 . (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 . (The definition in  only considers positive elements in , but it equivalent to the definition given here.)
Background: By Proposition 3.17 in , 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 ), 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 , if is purely infinite as a ring, then is purely infinite. Thus, in this particular instance, Question 1 has a positive answer.
- 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.
- 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.