the commutant of $\mathcal{HS(H)}$

Question

Let $\mathcal{HS}(H)$ be the set of Hilber Schmidt operators on a Hilbert space,it is a $C^*$ algebra.I wonder whether we have an explicit description of the commutant of $\mathcal{HS(H)}$.Is the commutant of $\mathcal{HS(H)}$ closed in $B(H)$?

Answer 1

For any set $S\subset B(H)$, the commutant of $S$ equals the commutant of the operator norm closure of $S$. Since the operator norm closure of $\mathcal{HS}(H)$ is $K(H)$, the commutant of $\mathcal{HS}(H)$ equals the commutant of $K(H)$, which is $\mathbb C$.