Cyclic Tracial Operator Algebras and Their Traces

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation.

1. (Cyclic tracial operator algebras)¶ A cyclic tracial operator algebra is a triple $(H,\mathcal{A},\Omega)$ consisting of a complex Hilbert space $H$, a set $\mathcal{A}\subseteq\mathcal{L}(H)$ and a vector $\Omega\in H$, called its cyclic vector, with the following three properties.

¶(a) $I\in\mathcal{A}$, and $S+T$, $cS$, $ST$ and $S^{*}$ belong to $\mathcal{A}$ for all $S,T\in\mathcal{A}$ and $c\in\mathbb{C}$.

¶(b) $\lVert\Omega\rVert=1$, and the set $\mathcal{A}\Omega=\{S\Omega:\ S\in\mathcal{A}\}$ is dense in $H$.

¶(c) $\langle\Omega,ST\Omega\rangle=\langle\Omega,TS\Omega\rangle$ for all $S,T\in\mathcal{A}$.

2. (Trace)¶ The trace of $(H,\mathcal{A},\Omega)$ is the map $\tau_{\mathcal{A}}:\mathcal{A}\to\mathbb{C}$, $\tau_{\mathcal{A}}(S)=\langle\Omega,S\Omega\rangle$.