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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

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    $\begingroup$ Yes, the one point space. I assume you actually wanted to ask something more specific? $\endgroup$ – Yemon Choi Aug 14 at 1:41
  • $\begingroup$ @YemonChoi Yes I mean non degenerate space. $\endgroup$ – Ali Taghavi Aug 14 at 1:50
  • $\begingroup$ @YemonChoi a space with at least 3 point. However infinite space or even connected space would be ideal examples. $\endgroup$ – Ali Taghavi Aug 14 at 1:54
  • $\begingroup$ @YemonChoi One can generalize the question in a more algebraic manner as follows: For a goven $C^*$ algebra $A$, one can think to all subalgebras $B$ such that every automorphism of $B$ is in the form of a unitary conjugation. $\endgroup$ – Ali Taghavi Aug 14 at 2:50
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    $\begingroup$ Can't you just take any faithful representation of the $C^*$-algebra $C(X) \rtimes {\rm Homeo}(X)$? $\endgroup$ – Jesse Peterson Aug 14 at 15:35

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