Purely infinite rings and C*-algebras

Von | September 23, 2020

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. [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. [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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