A note on the invariant subspace problem relative to a type ${\rm II}_1$ factor
| dc.creator | Fang, Junsheng | |
| dc.creator | Hadwin, Don | |
| dc.date | 2008-08-01 | |
| dc.date | 2008-11-17 | |
| dc.date.accessioned | 2026-07-07T10:18:18Z | |
| dc.date.available | 2026-07-07T10:18:18Z | |
| dc.description | Let $\M$ be a type ${\rm II}_1$ factor with a faithful normal tracial state $τ$ 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 $\{P_t\}_{0\leq t\leq 1}$ in $\M^ω$ such that $TP_t=P_tTP_t$, $P_s\leq P_t$ if $s\leq t$, and $τ_ω(P_t)=t$. 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 $ε>0$, there is an operator $S\in \mathfrak{M}$ such that $\|S\|\leq \|T\|$ and $\|S-T\|_2<ε$. We also show that $\prod_n^ωM_n(\cc)$ is not $\ast$-isomorphic to the ultrapower algebra of the hyperfinite type ${\rm II}_1$ factor. | |
| dc.description | 16 pages, minor changes based on comments from David Sherman | |
| dc.identifier | https://arxiv.org/abs/0808.0049 | |
| dc.identifier | http://arxiv.org/abs/0808.0049 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/174142 | |
| dc.subject | Operator Algebras | |
| dc.subject | 46L10, 47A15 | |
| dc.title | A note on the invariant subspace problem relative to a type ${\rm II}_1$ factor | |
| dc.type | text |