Traces on purely infinite C*-algebras

Does there exist a purely infinite C*-algebra that admits a tracial weight taking a finite, nonzero value?

Here a C*-algebra $A$ is purely infinite if every element $a \in A_+$ is properly infinite ( $a\oplus a$ is Cuntz subequivalent to $a$ in $M_2(A)$ ). This notion was introduced and studied by Kirchberg-Rørdam in [1] and [2]. Further, a weight on a C*-algebra $A$ is a map $\varphi \colon A_+ \to [0,\infty]$ that is additive and satisfies $\varphi(\lambda a)=\lambda\varphi(a)$ for all $\lambda\in[0,\infty)$ and $a\in A_+$ . A weight $\varphi$ is tracial if $\varphi(xx^*)=\varphi(x^*x)$ for all $x\in A$ .

If $I \subseteq A$ is a (not necessarily closed) two-sided ideal that is strongly invariant (for $x\in A$ , we have $xx^* \in I$ if and only if $x^*x \in I$ ), then the map $\tau_I\colon A_+ \to [0,\infty]$ given by $\tau_I(a)=0$ if $a\in I$ and $\tau_I(a)=\infty$ otherwise, is a tracial weight. Note that these tracial weights are trivial in the sense that they only take the values $0$ and $\infty$ .

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.

Schreibe einen Kommentar Antworten abbrechen