Does there exist a purely infinite C*-algebra that admits a tracial weight taking a finite, nonzero value?
Here a C*-algebra is purely infinite if every element
is properly infinite (
is Cuntz subequivalent to
in
). This notion was introduced and studied by Kirchberg-Rørdam in [1] and [2]. Further, a weight on a C*-algebra
is a map
that is additive and satisfies
for all
and
. A weight
is tracial if
for all
.
If is a (not necessarily closed) two-sided ideal that is strongly invariant (for
, we have
if and only if
), then the map
given by
if
and
otherwise, is a tracial weight. Note that these tracial weights are trivial in the sense that they only take the values
and
.
The question is if there exists a nontrivial tracial weight on a purely infinite C*-algebra.
It is well-known that every lower-semicontinuous tracial weight on a purely infinite C*-algebra is trivial. In particular, purely infinite C*-algebras do not admit tracial states. The question is about tracial weights that are not lower-semicontinuous. Sometimes, such tracial weights are called singular traces.
- [1]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.
- [2]E. Kirchberg, M. Rørdam, Infinite Non-simple C*-Algebras: Absorbing the Cuntz Algebra O∞, Advances in Mathematics. (2002) 195–264. https://doi.org/10.1006/aima.2001.2041.