# Quasitraces

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 $\tau\colon A_+\to[0,\infty]$ such that:

• $\tau$ is homogeneous:
$\tau(\lambda a)=\lambda\tau(a)$ for every $a\in A_+$ and $\lambda\in[0,\infty)$.
• $\tau$ is tracial:
$\tau(xx^*)=\tau(x^*x)$ for every $x\in A$.
• $\tau$ is additive on commuting elements:

$\tau(a+b)=\tau(a)+\tau(b)$ for every $a,b\in A_+$ that satisfy $ab=ba$.

• and such that for each $n\geq 1$ the induced map
$\tau_n\colon M_n(A)_+\to[0,\infty], (a_{j,k})_{j,k=1,...,n}\mapsto\tau(a_{11})+...\tau(a_{nn})$

has the same properties.

A quasitrace $\tau$ is:

• bounded if
$\sup\{\tau(a):a\in A_+, \|a\|\leq 1\} < \infty.$
• normalized if
$\sup\{\tau(a):a\in A_+, \|a\|\leq 1\} = 1.$
• lower semicontinuous if
$\{a\in A_+ : \tau(a)\leq t\}$ is closed for each $t\in[0,\infty)$.

## Variants

• A 1-quasitrace is a map $A_+\to[0,\infty]$ 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 $M_n(A)$, 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 $n\geq 1$. 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 The article of Haagerup was circulated as handwritten notes in 1991 and remained unpublished until 2014.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]

## References

1. Blanchard, Kirchberg, 2004, Remarks 2.29(i)
2. "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.
3. https://hal.archives-ouvertes.fr/hal-00922863/file/BK04b.pdf