Strongly closed $*$-subalgebra of $B(H)$ which does not contain the identity element of $B(H)$

Question

Let $H$ be a Hilbert space and $B(H)$ is the set of all bounded operators on $H$. Can you give me an example of a strongly closed $*$-subalgebra $A$ of $B(H)$, which does not contain the identity element of $B(H)$, that is, $\text{id}_H \notin A$.
Somehow I thought about $K(H)$, the set of all compact operators on $H$, but then I realize that $K(H)$ is not strongly closed.
Thank you for you time and effort. Please help me to find one.

Answer 1

The most trivial example is $\{0\}$. Alternatively you may split $H=H_1\oplus H_2$ and consider $A=B(H_1)\oplus \{0\}$.

The most general example among the strongly closed $^*$-algebras is $B\oplus \{0\}$, where $B$ is a unital, strongly closed $^*$-subalgebra of $B(H_1)$.