Weak operator limit of projections in $B(H)$

Question

Let $H$ be infinite dimensional and $\cal P$ be the set of all projections in $B(H)$. Show that $\cal P$ is weak operator dense in $(B(H))^+_{\|.\|\leq 1}$, the set of positive operators in the unit ball of $B(H)$.

Please help me. Thanks.

Answer 1

Let $T\in B(H)$ be positive, with $\|T\|\leq1$. We want to show that, for each basic wot-neighbourhood $V$ of $T$, there exists a projection $P\in V$. Such a neighbourhood $V$ is of the form $$ V=\{X\in B(H):\ |\langle(T-X)y_j,z_j\rangle|<\varepsilon,\ j=1,\ldots,m\} $$ for a fixed $\varepsilon>0$ and $y_1,\ldots,y_m,z_1,\ldots,z_m\in H$. The key idea is that, since $T$ is a positive contraction,
$$\tag{1} P=\begin{bmatrix}T&(T-T^2)^{1/2}\\ (T-T^2)^{1/2}&I-T\end{bmatrix} $$ is a projection. To implement this idea, let $K=\text{span}\,\{y_1,\ldots,y_m,z_1,\ldots,z_m\}$. As $K$ is finite-dimensional and $H$ is infinite-dimensional, $K^\perp$ is infinite-dimensional. So there exists a subspace $K_0\subset K^\perp$ with $\dim K_0=\dim K$.

Let $T_0$ be the operator that agrees with $T$ on $K$ and is zero on $K^\perp$. And now define $P$ according to (1) for the space $K\oplus K_0$, i.e. $P$ is given, for $x\in K$ and $y\in K_0$, by $$ P(x+y)=Tx+(T-T^2)^{1/2}Uy+U(T-T^2)^{1/2}x+U(I-T)Uy, $$ where $U$ is a partial isometry with initial space $K$ and range space $K_0$. We extend $P$ to all of $H$ by putting $P=0$ on $(K+K_0)^\perp$. So $P$ is a projection and, for each $j=1,\ldots,m$, $$ \langle (T-P)y_j,z_j\rangle=\langle (T-P)P_Ky_j,P_Kz_j\rangle =\langle (P_KTP_K-P_KPP_K)y_j,z_j\rangle=0 $$ (note that $P_KPP_K=P_KTP_K$ by construction).