2025/01/24 by Arulseelan, Jananan, Goldbring, Isaac, Hart, Bradd +1 · 3 citations
#03C66 (Secondary) #46L10 (Primary) 46L53 #FOS: Mathematics #Logic (math.LO) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2501.14153
We introduce the notion of a totally (K-) bounded element of a W*-probability space (M, φ) and, borrowing ideas of Kadison, give an intrinsic characterization of the ^*-subalgebra Mtb of totally bounded elements. Namely, we show that Mtb is the unique strongly dense ^*-subalgebra M0 of totally bounded elements of M for which the collection of totally 1-bounded elements of M0 is complete with respect to the ‖⋅‖φ^#-norm and for which M0 is closed under all operators ha(log(Δ)) for a ∈ ℕ, where Δ is the modular operator and ha(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 Mtb 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.