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.