TITLE: Is weak operator topology (WOT) limit of unitary operators isometry?
QUESTION [6 upvotes]: Let $(U_{\alpha})$ be net of unitary operator in $B\mathcal{(H)}$ s.t. $U_{\alpha} \xrightarrow {\text{WOT}}V$.Can we conclude that V is an isometry? If it be not true give a counter example.
Comments: I observe that if $U_{\alpha} \xrightarrow {\text{SOT}}V$ then V is necessarily isometry. But I could not able to justify the statement for weak operator topology case neither by proving nor by giving counter example.
Notations: $U_{\alpha} \xrightarrow {\text{WOT}}V$ means $\langle U_{\alpha}x,y\rangle\rightarrow\langle Vx,y\rangle$ for all x,y $\in\mathcal{H}$, $U_{\alpha} \xrightarrow {\text{SOT}}V$ means $\Vert 
U_{\alpha}x\Vert\rightarrow\Vert Vx\Vert$ for all x$\in \mathcal{H}$ and $\mathcal{H}$ is a Hilbert Space.
Any comment regarding proving the statement or giving counter example is highly appreciated.Thanks in advance.

REPLY [2 votes]: The answer is definitely no. In fact, the wot-closure of the set of unitaries is the whole unit ball. 

The proof is a generalization of the idea, with matrices, that given any $t\in\overline{\mathbb D}$, there exists a unitary  $$ \begin{bmatrix} t&*\\ *&*\end{bmatrix}.$$
Concretely, let $x\in B(H)$ be a contraction, i.e. $\|x\|\leq 1$. Neighbourhoods in the wot topology are of the form $$N=\{y:\ |\langle (x-y)\xi_j,\eta_j\rangle|\leq1:\ \xi_1,\ldots,\xi_m,\eta_1,\ldots,\eta_m\in H\}.$$
Fix one such $N$ and let $$L=\operatorname{span}\{\xi_1,\ldots,\xi_m,\eta_1,\ldots,\eta_m,x\xi_1,\ldots,x\xi_m,x\eta_1,\ldots,x\eta_m\}.$$
Let $p$ be the orthogonal projection onto $L$. Since $\dim L<\infty$, there exists a subspace $L'\subset L^\perp$ with $\dim L'=\dim L$. Let $q$ be the orthogonal projection onto $L'$, and $v$ a partial isometry $v:L'\to L$, i.e. $v^*v=q$, $vv^*=p$. Now based on the decomposition $H=L\oplus L'\oplus ( L^\perp\ominus L')$, let $$u=\begin{bmatrix} pxp & (p-(pxp)^*pxp)^{1/2}v&0\\ v^*(p-(pxp)^*pxp)^{1/2} &- v^*pxpv&0\\ 0&0&I\end{bmatrix}.
$$
It is not hard to check that $u$ is a unitary. And 
$$
\langle u\xi_j,\eta_j\rangle
=\langle up\xi_j,p\eta_j\rangle
=\langle pup\xi_j,\eta_j\rangle
=\langle pxp\xi_j,\eta_j\rangle
=\langle x\xi_j,\eta_j\rangle. 
$$
So $u\in N$. As we can do this for any neighbourhood $N$ of $x$, we can construct a net $u_N$ of unitaries with $u_n\to x$ wot. 
This argument provides a way to show that the wot and sot topologies do not agree (and are, in fact, very different).