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Let $H$ be a separable Hilbert space. Let us say $s:B(H)\to B(H)$ is a simple function if $s(B(H))$, the range of $s$, is a finite set. Since (bounded) closed unit balls in $B(H)$ are all WOT-compact and metrizable then one may find a sequence of simple functions $s_n:B(H)\to B(H)$ converging pointwise-weakly to the identity mapping $I:B(H)\to B(H)$ ( where $I(x)=x$) which means that for any arbitrary operator $x$ in $B(H)$ the sequence $s_n(x)$ is WOT-convergent to $x$.

Q. Does there exists any sequence of simple functions $s_n:B(H)\to B(H)$ converging pointwise strongly to $I$?

ABB
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  • Is $H$ separable? And why doesn't this answer also answer this question? –  Sep 22 '17 at 21:32
  • @Michelle. Thanks for you pay attention. It was edited ($H$ is separable). What's your suggestion for $s_n$'s by Martin's argument? – ABB Sep 23 '17 at 04:28

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Using your previous question I think an answer could look like this:

SOT is metrizable on bounded sets, so lets look at the unit ball $B_1(0)$ first. Let $A_n$ be subsets of $B_1(0)$ that have the DS property wrt $B_1(0)$. Partition $B_1(0)$ into $|A_n|$ many disjoint blocks, let $R_n := \min_{a_n, b_n \in A_n} d(a_n,b_n)/3$ and let each $B_{R_n}(a_n)$ be contained entirely in one and only one block. Have $s_n$ send the block with $a_n$ to $a_n$.

For any $x\in B_1(0)$ you have $d(s(x),x)\to0$ which follows from the denseness of the $\{a_n\mid a_n \in A_n, n\in \Bbb N\}$.

To extend to $B(H)$ have $s_k$ be $0$ outside of $B_n(0)$ for $k<n$ and for $k≥n$ get a DS sequence for $B_k(0)-B_{k-1}(0)$ and do the same kind of construction that was done $B_1(0)$.

s.harp
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