Lower semicontinuous trace

Let $A$ be a $C^*$ -algebra. A lower semicontinuous trace is a map $\tau:A_+\to[0,+\infty]$ that is additive, positively homogeneous, and satisfies

\tau(x^*x)=\tau(xx^*)\qquad(x\in A),

and whose sublevel sets $\{a\in A_+:\tau(a)\leq c\}$ are norm closed for every $c\geq0$ . Equivalently, $a_n\to a$ in norm with $a_n,a\geq0$ implies $\tau(a)\leq\liminf_n\tau(a_n)$ . Values of $+\infty$ are allowed. Neither dense definition, semifiniteness, faithfulness, nor boundedness is part of the term unless explicitly added.

Equivalent approximation formula

For $a\in A_+$ , lower semicontinuity is equivalent to

\tau(a)=\sup_{\varepsilon>0}\tau\bigl((a-\varepsilon)_+\bigr),

where $(a-\varepsilon)_+$ is obtained by continuous functional calculus. This formula is especially useful in nonunital algebras and in comparison theory. The tracial identity also implies invariance under unitary conjugation in the unitization.

Domains and extra adjectives

The finite positive domain is

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

The trace is densely defined when this cone is norm dense in $A_+$ . Semifiniteness is an order-density requirement and is a separate condition; compare semifinite weights. A bounded trace on a $C^*$ -algebra is finite everywhere and automatically lower semicontinuous, but lower semicontinuous traces need not be bounded.

Examples

The canonical operator trace on $K(H)_+$ , allowed to take $+\infty$ , is a densely defined lower semicontinuous trace: finite-rank positive operators lie in its finite domain and are norm dense in $K(H)_+$ . On $C_0(X)$ , integration

\tau(f)=\int_X f\,d\mu

against a positive Radon measure gives an extended lower semicontinuous trace. Commutativity makes the tracial identity automatic.

References

Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. Publisher record. Relevant: §§5.2 and 5.6 on lower-semicontinuous weights, traces, and their finite ideals.
Leonel Robert, “On the Comparison of Positive Elements of a $C^*$ -Algebra by Lower Semicontinuous Traces,” 2008. arXiv record. Relevant: Introduction for the extended trace convention and §2 for lower-semicontinuous trace comparison.