Quasitrace
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In mathematics, especially functional analysis, a quasitrace is a not necessarily additive tracial functional on a C*-algebra. An additive quasitrace is called a trace. It is a major open problem if every quasitrace is a trace.
Definition
A quasitrace on a C*-algebra A is a map [math]\displaystyle{ \tau\colon A_+\to[0,\infty] }[/math] such that:
- [math]\displaystyle{ \tau }[/math] is homogeneous:
- [math]\displaystyle{ \tau(\lambda a)=\lambda\tau(a) }[/math] for every [math]\displaystyle{ a\in A_+ }[/math] and [math]\displaystyle{ \lambda\in[0,\infty) }[/math].
- [math]\displaystyle{ \tau }[/math] is tracial:
- [math]\displaystyle{ \tau(xx^*)=\tau(x^*x) }[/math] for every [math]\displaystyle{ x\in A }[/math].
- [math]\displaystyle{ \tau }[/math] is additive on commuting elements:
[math]\displaystyle{ \tau(a+b)=\tau(a)+\tau(b) }[/math] for every [math]\displaystyle{ a,b\in A_+ }[/math] that satisfy [math]\displaystyle{ ab=ba }[/math].
- and such that for each [math]\displaystyle{ n\geq 1 }[/math] the induced map
- [math]\displaystyle{ \tau_n\colon M_n(A)_+\to[0,\infty], (a_{j,k})_{j,k=1,...,n}\mapsto\tau(a_{11})+...\tau(a_{nn}) }[/math]
has the same properties.
A quasitrace [math]\displaystyle{ \tau }[/math] is:
- bounded if
- [math]\displaystyle{ \sup\{\tau(a):a\in A_+, \|a\|\leq 1\} \lt \infty. }[/math]
- normalized if
- [math]\displaystyle{ \sup\{\tau(a):a\in A_+, \|a\|\leq 1\} = 1. }[/math]
- lower semicontinuous if
- [math]\displaystyle{ \{a\in A_+ : \tau(a)\leq t\} }[/math] is closed for each [math]\displaystyle{ t\in[0,\infty) }[/math].
Variants
- A 1-quasitrace is a map [math]\displaystyle{ A_+\to[0,\infty] }[/math] that is just homogeneous, tracial and additive on commuting elements, but does not necessarily extend to such a map on matrix algebras over A. If a 1-quasitrace extends to the matrix algebra [math]\displaystyle{ M_n(A) }[/math], then it is called a n-quasitrace. There are examples of 1-quasitraces that are not 2-quasitraces. One can show that every 2-quasitrace is automatically a n-quasitrace for every [math]\displaystyle{ n\geq 1 }[/math]. Sometimes in the literature, a quasitrace means a 1-quasitrace and a 2-quasitrace means a quasitrace.
Properties
- A quasitrace that is additive on all elements is called a trace.
- Uffe Haagerup showed that every quasitrace on a unital, exact C*-algebra is additive and thus a trace. The article of Haagerup [1] was circulated as handwritten notes in 1991 and remained unpublished until 2014. Blanchard and Kirchberg removed the assumption of unitality in Haagerup's result.[2] As of today (August 2020) it remains an open problem if every quasitrace is additive.
- Joachim Cuntz showed that a simple, unital C*-algebra is stably finite if and only if it admits a dimension function. A simple, unital C*-algebra is stably finite if and only if it admits a normalized quasitrace. An important consequence is that every simple, unital, stably finite, exact C*-algebra admits a tracial state.
- Every quasitrace on a von Neumann algebra is a trace.
Notes
References
- Blanchard, Etienne; Kirchberg, Eberhard (February 2004). "Non-simple purely infinite C∗-algebras: the Hausdorff case". Journal of Functional Analysis 207 (2): 461–513. doi:10.1016/j.jfa.2003.06.008. https://hal.archives-ouvertes.fr/hal-00922863/file/BK04b.pdf.
- Haagerup, Uffe (2014). "Quasitraces on Exact C*-algebras are Traces". C. R. Math. Rep. Acad. Sci. Canada 36: 67–92.
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