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  and . 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.
- 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.
- 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.