Totally Bounded Elements in W*-probability Spaces

Abstract

We introduce the notion of a totally ($K$-) bounded element of a $W^{\ast }$-probability space $(M,\phi )$ and, borrowing ideas of Kadison, give an intrinsic characterization of the ${}^{\ast }$-subalgebra $M_{\mathrm{tb}}$ of totally bounded elements. Namely, we show that $M_{\mathrm{tb}}$ is the unique strongly dense ${}^{\ast }$-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 $\Vert \cdot {\Vert }_{\phi }^{\#}$-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). We also prove that totally $K$-bounded elements of an Ocneanu ultraproduct admit representatives with the same total bound using a careful effective estimate of the distance of a given totally bounded element to the totally 1-bounded elements. An alternative proof in the appendix uses an isometric $H^{\infty }$-lifting theorem for the Ocneanu multiplier quotient, derived from a metric $H^{\infty }$-lifting theorem for $C^{\ast }$-quotients and a $C^{\ast }$-algebraic Schur parametrization. We combine these results with Rieffel and Van Daele's bounded operator approach to modular theory to arrive at a new language and axiomatization of $W^{\ast }$-probability spaces as metric structures. Previous work of Dabrowski had axiomatized $W^{\ast }$-probability spaces using a smeared version of multiplication, but the subalgebra $M_{\mathrm{tb}}$ allows us to give an axiomatization in terms of the original algebra operations. Finally, we prove the (non-)axiomatizability of several classes of $W^{\ast }$-probability spaces.

87 pages, major revisions