# Quasitraces

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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>\tau\colon A_+\to[0,\infty]</math> such that:

- <math>\tau</math> is
**homogeneous**:

- <math>\tau(\lambda a)=\lambda\tau(a)</math> for every <math>a\in A_+</math> and <math>\lambda\in[0,\infty)</math>.

- <math>\tau</math> is
**tracial**:

- <math>\tau(xx^*)=\tau(x^*x)</math> for every <math>x\in A</math>.

- <math>\tau</math> is
**additive**on commuting elements:

<math>\tau(a+b)=\tau(a)+\tau(b)</math> for every <math>a,b\in A_+</math> that satisfy <math>ab=ba</math>.

- and such that for each <math>n\geq 1</math> the induced map

- <math>\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>\tau</math> is:

**bounded**if

- <math>\sup\{\tau(a):a\in A_+, \|a\|\leq 1\} < \infty.</math>

**normalized**if

- <math>\sup\{\tau(a):a\in A_+, \|a\|\leq 1\} = 1.</math>

**lower semicontinuous**if

- <math>\{a\in A_+ : \tau(a)\leq t\}</math> is closed for each <math>t\in[0,\infty)</math>.

## Variants

- A
**1-quasitrace**is a map <math>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>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>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 The article of Haagerup was circulated as handwritten notes in 1991 and remained unpublished until 2014. Blanchard and Kirchberg removed the assumption of unitality in Haagerup's result.
^{[1]}As of today (August 2020) it remains an open problem if every quasitrace is additive.^{[2]}

- 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.
^{[3]}

## In the media

## References

- ↑ Blanchard, Kirchberg, 2004, Remarks 2.29(i)
- ↑ "Non-simple purely infinite C∗-algebras: the Hausdorff case".
*Journal of Functional Analysis*.**207**(2): 461–513. 2004-02-15. doi:10.1016/j.jfa.2003.06.008. ISSN 0022-1236. - ↑ https://hal.archives-ouvertes.fr/hal-00922863/file/BK04b.pdf

## External links

- Quasitraces on exact C*-algebras are traces
- Edwards' condition for quasitraces on C*-algebras
- QUASITRACES AND AW ∗-ALGEBRAS Contents 1
- Quasitraces on exact C*-algebras are traces | Request PDF
- Quasitraces are Traces: A Short Proof of the Finite-Nuclear
- Quasitraces on Exact C*-algebras are Traces - Mathematical
- Edwards' condition for quasitraces on C*-algebras

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