Let $H$ be a complex Hilbert space.

Is there a compact Hausdorff space $X$ such that $C(X)$ is embeded in $B(H)$ and for every homeomorphism $\alpha$ of $X$ there exist a unitary operator $u\in B(H)$ with $u^*fu=f\circ \alpha^{-1}\quad \forall f \in C(X)\subset B(H)$?

The motivation comes from the following post:

Can every dynamical system be interpreted in terms of (unitary) conjugation in an operator algebra