Two isomorphic $C^*$-algebras. What is the isomorphism between corresponding Hilbert spaces?

Question

Let $H$ be a separable Hilbert space. Suppose that $\mathscr{A}$ and $\mathscr{B}$ are some unital $C^*$-algebras of operators acting on $H$, not necessary coinciding with $C^*$-algebra of all the possible operators acting on $H$. Suppose that they are $*$-isomorphic with the isomorphism $\mathfrak{n}:\mathscr{A}\to\mathscr{B}$.

What are the conditions for existing a unitary operator $\mathcal{U}:H\to H$, such that $\mathfrak{n}(\mathcal{A})=\mathcal{U}\mathcal{A}\mathcal{U}^{-1}$ for all $\mathcal{A}\in\mathscr{A}$?

Any conditions on the algebras $\mathscr{A}$ and $\mathscr{B}$, e.g. commutative algebras, UHF algebras, etc., when the statement can be true, are also welcome.

Answer 1

Indeed, we have

The $*$-isomorphism $\mathfrak{n}:\mathscr{A}\to\mathscr{B}$ is implemented by a unitary in $B(H)$ if and only if $\mathfrak{n}$ extends to a $*$-automorphism of $B(H)$.

The forward direction is trivial, and the reverse direction follows from the fact that all automorphisms of $B(H)$ are inner.