Let $H$ be a Hilbert space, let $\mathcal{B}(H)$ be the algebra of bounded operators on $H$, and let $A\subseteq\mathcal{B}(H)$ be a closed subalgebra (but not necessarily a $*$-algebra). Let $B$ be an $C^*$-algebra and let $\theta:A\rightarrow B$ be a (bounded) algebra isomorphism (again, we ignore the involution). Can we say anything about how $A$ and $H$ relate?
For example, I could start with $B\subseteq \mathcal{B}(H)$ being a closed $*$-subalgebra. Let $T\in\mathcal{B}(H)$ be invertible, and let $A = \{ TxT^{-1} : x\in B \}$. If $T$ is not unitary, then in general $A$ will not be a $*$-algebra, but it is isomorphic to $B$.
- Do all examples arise from some variation of this construction?
- If I require that $\theta$ be an isometry, can we say more?