Trace on a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. A bounded trace on $A$ is a positive linear functional $\tau:A\to\mathbb C$ satisfying

\tau(ab)=\tau(ba)\qquad(a,b\in A).

Equivalently, $\tau(x^*x)=\tau(xx^*)$ for every $x\in A$ . Boundedness and norm continuity are included here because $\tau$ is a positive functional on all of $A$ . A tracial state is a trace of norm one. Some sources also call an extended-valued tracial weight on $A_+$ a trace; such an object must have its domain, lower-semicontinuity, and density assumptions stated separately and is not the bounded notion defined above.

Equivalent invariance

For a bounded positive functional, the tracial identity is equivalent to invariance under inner unitary conjugation:

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

for every unitary $u$ in the unitization of $A$ . On positive elements it also implies that Murray–von Neumann equivalent projections have equal trace. These statements express that a trace records size without depending on a choice of coordinates, while positivity prevents cancellation on $A_+$ Blackadar, §II.6.

Examples and existence

The normalized matrix trace $a\mapsto n^{-1}\operatorname{Tr}(a)$ is the unique tracial state on $M_n(\mathbb C)$ . For a commutative $C^*$ -algebra, every positive linear functional is automatically tracial. In contrast, $B(H)$ for infinite-dimensional $H$ has no tracial state: its canonical operator trace is unbounded and takes $+\infty$ at the identity. Thus the existence of a tracial state is a genuine restriction on a $C^*$ -algebra.

Extended traces and scope

An extended trace is commonly formulated as a weight $\tau:A_+\to[0,+\infty]$ satisfying $\tau(x^*x)=\tau(xx^*)$ . Authors may additionally require it to be densely defined, lower semicontinuous in norm, semifinite, or normal when $A$ is a von Neumann algebra. None of those continuity and domain properties follows from the tracial identity alone. The matrix/operator trace and a trace on an abstract algebra should therefore not be identified without specifying their ambient algebra and domains.

References

Bruce Blackadar, Operator Algebras: Theory of $C^*$ -Algebras and von Neumann Algebras, Springer, 2006. DOI record. Relevant: §II.6 on traces, tracial states, and dimension functions.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: the chapters on densely defined lower-semicontinuous traces and weights.