Totally Bounded Elements in W*-probability Spaces

We introduce the notion of a totally ($K$-) bounded element of a W*-probability space $(M, \varphi)$ and, borrowing ideas of Kadison, give an intrinsic characterization of the $^*$-subalgebra $M_{tb}$ of totally bounded elements. Namely, we show that $M_{tb}$ is the unique strongly dense $^*$-subalgebra $M_0$ of totally bounded elements of $M$ for which the collection of totally $1$-bounded elements of $M_0$ is complete with respect to the $\|\cdot\|_\varphi^\#$-norm and for which $M_0$ is closed under all operators $h_a(\log(\Delta))$ for $a \in \mathbb{N}$, where $\Delta$ is the modular operator and $h_a(t):=1/\cosh(t-a)$ (see Theorem 4.3). As an application, we combine this characterization with Rieffel and Van Daele's bounded approach to modular theory to arrive at a new language and axiomatization of W*-probability spaces as metric structures. Previous work of Dabrowski had axiomatized W*-probability spaces using a smeared version of multiplication, but the subalgebra $M_{tb}$ allows us to give an axiomatization in terms of the original algebra operations. Finally, we prove the (non-)axiomatizability of several classes of W*-probability spaces.