# Purely infinite rings and C*-algebras

Von | September 23, 2020

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.

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.