Let $H$ and $K$ be infinite dimensional (separable) Hilbert spaces and $X=B(H,K)$ denote the space of bounded linear operators. For $T_1, T_2$ in $X$, we define $D_{T_1,T_2}:X \to X$ as $D_{T_1,T_2}(T)=TT_1^*T_2$. Finally define $V^0=\operatorname{span}\{D_{T_1,T_2}, T_1, T_2 \in X\}$. One can check that $V^0$ is a pre-$C^*$-algebra with involution $D_{T_1,T_2}^*=D_{T_2,T_1}$. Let $V$ denotes the closure of $V^0$ inside $B(X)$.

Is $V$ isomorphic to some well known $C^*$-algebra?

P.S: This question was first posted on Math Stackexchange here. Also, this question is particular case of a more general construction given at Trying to understand construction of $C^*$-algebra corresponding to a ternary $C^*$-ring from a paper.

`$V^0 = \operatorname{span} \{…\}$`

, not $V^0 =$span$\{…\}$`$V^0 =$span$\{…\}$`

. I have edited accordingly. $\endgroup$