A note on the invariant subspace problem relative to a type ${\mathrm{II}}_1$ factor

Abstract

Let $\M$ be a type ${\mathrm{II}}_1$ factor with a faithful normal tracial state $\tau$ and let $\M^ω$ be the ultrapower algebra of $\M$. In this paper, we prove that for every operator $T\in \M^ω$, there is a family of projections $\{P_t{\}}_{0\le t\le 1}$ in $\M^ω$ such that $T{P}_t=P_{t}T{P}_t$, $P_s\le P_t$ if $s\le t$, and ${\tau }_{\omega }(P_t)=t$. Let $\mathfrak{M}=\{Z \in \M: \text{there is a family of projections} \{P_t\}_{0\leq t\leq 1} \text{in} \M \text{such that} ZP_t=P_tZP_t, P_s\leq P_t \text{if} s\leq t, \text{and} τ(P_t)=t\}$. As an application we show that for every operator $T\in \M$ and $\epsilon >0$, there is an operator $S\in \mathfrak{M}$ such that $\Vert S\Vert \le \Vert T\Vert$ and $\Vert S-T{\Vert }_2<\epsilon$. We also show that $\prod_n^ωM_n(\cc)$ is not $\ast$-isomorphic to the ultrapower algebra of the hyperfinite type ${\mathrm{II}}_1$ factor.

16 pages, minor changes based on comments from David Sherman