Tracial state - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. A tracial state is a state $\tau:A\to\mathbb C$ satisfying

\tau(ab)=\tau(ba)\qquad\text{for all }a,b\in A.

Equivalently, it is a bounded trace of norm one. If $A$ is unital, normalization is $\tau(1)=1$ . For a nonunital algebra, norm one is the defining normalization. The tracial identity is additional to positivity and normalization: a general state need not be tracial. It makes the value of a two-factor product independent of cyclic interchange of its factors.

Equivalent invariance

On a unital $C^*$ -algebra, a state is tracial exactly when it is invariant under every inner unitary conjugation:

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

It then takes equal values on Murray--von Neumann equivalent projections. The set of tracial states is a convex weak-star closed subset of the state space Blackadar, treatment of traces and tracial states.

Examples and non-existence

The normalized matrix trace $\tau(a)=n^{-1}\operatorname{Tr}(a)$ is the unique tracial state on $M_n(\mathbb C)$ . Every state on a commutative $C^*$ -algebra is tracial. By contrast, $B(H)$ has no tracial state when $H$ is infinite-dimensional: two isometries with orthogonal ranges would force the state of the identity to equal twice itself.

Bounded and unbounded notions

References

Bruce Blackadar, Operator Algebras: Theory of $C^*$ -Algebras and von Neumann Algebras, Springer, 2006. DOI record. Relevant: the treatment of bounded traces, tracial states, and finite algebras.
Nathanial P. Brown and Narutaka Ozawa, $C^*$ -Algebras and Finite-Dimensional Approximations, American Mathematical Society, 2008. AMS DOI record. Relevant: the chapters using tracial states in finite and stably finite $C^*$ -algebras.