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

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 $$B(H)$$?

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 $$\mathbb C$$.