A $C^{*}$ algebra associated to a group [closed]

Question

Let $G$ be a compact group which act on a Hilbert space $H$. We define a linear map $T$ on the dual space $H^{*}$ with $$T(\phi)(x)=\int_{G} \phi(g.x)$$ The integration is based on the Haar measure. Since $H^{*}$ is isomorphic to $H$ we actually have a linear operator on $H$. We denote this operastor with $T$, again. So $T\in B(H)$. We consider the $C^{*}$ algebra generated by $T$. It is a subalgebra of $B(H)$. Then we consider the direct sum (over all possible (non equivalent) irreducible representation of $G$) of all $C^{*}$ algebras which we obtain in this processes.

Does this $C^{*}$ algebra have a name(and studied already)? Does it contain some information about $G$?Is it a useful $C^{*}$ algebra to study?

Answer 1

Because the group is compact one can assume the representation is isometric and the Haar measure is normalized. In this situation, $T$ is just the orthogonal projection on the space of $G$-invariant vectors. (exercice: 1) $T(\phi)$ is $G$-invariant 2) Invariant linear forms are fixed by $T$ 3) $T$ is self-adjoint)

And with this in mind, your question become trivial. (on a a non trivial irreducible representation, T is zero, and on the trivial representation T is the identity)