Find a type I group which has a type $I_\infty$ representation

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 !

• 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$_\infty$ – Adrián González-Pérez Feb 5 at 12:06
• 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! – J.F Feb 5 at 14:02