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

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 $H$.