Tracial weight - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra. A weight $\tau:M_+\to[0,+\infty]$ is tracial if

\tau(x^*x)=\tau(xx^*)\qquad(x\in M).

The equality is interpreted in the extended nonnegative reals, so either side may be $+\infty$ . A tracial weight is such a weight, without any implied normality, semifiniteness, faithfulness, or finiteness. In particular, the tracial identity specifies symmetry under changing the order of the two factors, while the regularity and domain properties commonly needed in von Neumann algebra theory are independent additional hypotheses.

Invariance and finite elements

Taking $x=ua^{1/2}$ , where $u$ is unitary and $a\in M_+$ , shows that

\tau(uau^*)=\tau(a).

Thus a tracial weight is invariant under inner unitary conjugation. Its finite positive domain

\mathfrak m_\tau^+=\{a\in M_+:\tau(a)<\infty\}

is hereditary and invariant under unitary conjugation. When $\tau(1)<\infty$ , additivity and homogeneity extend $\tau$ uniquely to a bounded tracial positive linear functional on all of $M$ Takesaki, Chapter V.

Examples and distinctions

The usual operator trace on $B(H)_+$ , allowed to take $+\infty$ , is a normal semifinite faithful tracial weight. Integration on $L^\infty(X,\mu)_+$ is tracial because the algebra is commutative. By contrast, a vector state on $B(H)$ is generally not tracial: it can distinguish $x^*x$ from $xx^*$ .

A bounded trace on a $C^*$ -algebra is finite everywhere. A tracial weight may instead be unbounded and extended-valued, so the two terms should not be treated as aliases.

Conventions and scope

Some authors use “trace” to mean a normal semifinite faithful tracial weight, while others reserve “trace” for a bounded functional. Here “tracial” records only the displayed symmetry. The adjectives normal, semifinite, faithful, and finite must be stated explicitly whenever they are required.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on traces and the type decomposition of von Neumann algebras.
Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory, AMS, 1997. AMS record. Relevant: §7.2 on traces and weights.