Traces on purely infinite C*-algebras

Von | Oktober 26, 2022

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. [1]
    E. Kirchberg, M. Rordam, Non-simple purely infinite C*-algebras, American Journal of Mathematics. (2000) 637–666.
  2. [2]
    E. Kirchberg, M. Rørdam, Infinite Non-simple C*-Algebras: Absorbing the Cuntz Algebra O∞, Advances in Mathematics. (2002) 195–264.

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