#### Vol. 4, No. 3, 2011

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${\rm P}_1$ subalgebras of $M_n(\mathbb C)$

### Stephen Rowe, Junsheng Fang and David R. Larson

Vol. 4 (2011), No. 3, 213–250
##### Abstract

A linear subspace $B$ of $L\left(H\right)$ has the property ${P}_{1}$ if every element of its predual ${B}_{\ast }$ has the form $x+{B}_{\perp }$ with $rank\left(x\right)\le 1$. We prove that if $dimH\le 4$ and $B$ is a unital operator subalgebra of $L\left(H\right)$ which has the property ${P}_{1}$, then $dimB\le dimH$. We consider whether this is true for arbitrary $H$.

##### Keywords
property $\mathrm P_1$, 2-reflexive
##### Mathematical Subject Classification 2000
Primary: 47L05, 47L75
Secondary: 47A15