### A Banach principle for semifinite von Neumann algebras.

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Let M be a finite von Neumann algebra acting on the standard Hilbert space L²(M). We look at the space of those bounded operators on L²(M) that are compact as operators from M into L²(M). The case where M is the free group factor is particularly interesting.

In this note we study sets of normal generators of finitely presented residually $p$-finite groups. We show that if an infinite, finitely presented, residually $p$-finite group $G$ is normally generated by ${g}_{1},\cdots ,{g}_{k}$ with order ${n}_{1},\cdots ,{n}_{k}\in \{1,2,\cdots \}\cup \left\{\infty \right\}$, then $${\beta}_{1}^{\left(2\right)}\left(G\right)\le k-1-\sum _{i=1}^{k}\frac{1}{{n}_{i}}\phantom{\rule{0.166667em}{0ex}},$$ where ${\beta}_{1}^{\left(2\right)}\left(G\right)$ denotes the first ${\ell}^{2}$-Betti number of $G$. We also show that any $k$-generated group with ${\beta}_{1}^{\left(2\right)}\left(G\right)\ge k-1-\epsilon $ must have girth greater than or equal $1/\epsilon $.

We introduce a property of ergodic flows, called Property B. We prove that an ergodic hyperfinite equivalence relation of type III₀ whose associated flow has this property is not of product type. A consequence is that a properly ergodic flow with Property B is not approximately transitive. We use Property B to construct a non-AT flow which-up to conjugacy-is built under a function with the dyadic odometer as base automorphism.