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I am looking for an example of a locally compact group $G$ that is type $I$ and has a unitary representation $(\pi, H_\pi)$ of $G$ such that $\pi(G)\cong B(H)$, as C*-algebras, where $H$ is an infinite dimensional (separable) Hilbert space. By $\pi(G)$, I really mean the (extension to $C^*_{\rm{max}}(G)$) of the integrated representation of $\pi$.

I know that I have to look for non-second countable examples, otherwise $\pi(G)$ would be separable as a C*-algebra (while $B(H)$ is not). I also have to avoid virtually abelian groups and compact groups because of theorem such as Peter-Weyl.

Thank you for your help !

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  • $\begingroup$ I think that the terminology of type I$_\infty$ representation is misleading, since representations with $\pi[C_\max^*(G)] = K(H)$ are referred as type I$_\infty$ $\endgroup$ – Adrián González-Pérez Feb 5 at 12:06
  • $\begingroup$ Oh, I was not aware of that terminology, my bad! I invite you to change the title to make it more appropriate, if you have a better suggestion! $\endgroup$ – J.F Feb 5 at 14:02

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