Quasitrace

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

• ${\displaystyle \tau }$ is homogeneous:

${\displaystyle \tau (\lambda a)=\lambda \tau (a)}$ for every ${\displaystyle a\in A_{+}}$ and ${\displaystyle \lambda \in [0,\infty )}$.

• ${\displaystyle \tau }$ is tracial:

${\displaystyle \tau (xx^{*})=\tau (x^{*}x)}$ for every ${\displaystyle x\in A}$.

• ${\displaystyle \tau }$ is additive on commuting elements:

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

• and such that for each ${\displaystyle n\geq 1}$ the induced map

${\displaystyle \tau _{n}\colon M_{n}(A)_{+}\to [0,\infty ],(a_{j,k})_{j,k=1,...,n}\mapsto \tau (a_{11})+...\tau (a_{nn})}$

has the same properties.

A quasitrace ${\displaystyle \tau }$ is:

• bounded if

${\displaystyle \sup\{\tau (a):a\in A_{+},\|a\|\leq 1\}<\infty .}$

• normalized if

${\displaystyle \sup\{\tau (a):a\in A_{+},\|a\|\leq 1\}=1.}$

• lower semicontinuous if

${\displaystyle \{a\in A_{+}:\tau (a)\leq t\}}$ is closed for each ${\displaystyle t\in [0,\infty )}$.

Variants

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 The article of Haagerup ^[1] was circulated as handwritten notes in 1991 and remained unpublished until 2014.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