Let $G$ be a locally compact Hausdorff group.
It is known that $G$ can be topologically embedded in $W^{\ast}(G)$ , its universal $W^{\ast}$-algebra (with the $\sigma$-weak topology). An element $T \in W^{\ast}(G)$ is a function assigning to each representation $\pi$ a bounded operator $T(\pi) \in B(H\_{\pi})$. This $T$ must be compatible with interwiners and $T(\pi)$ must be uniformly bounded.


This was done (in a slightly different language) by J. Ernest in [A new group algebra for locally compact groups](https://doi.org/10.2307/2373020).


Now define $G\_{\otimes}= \{ T \in W^\*(G)\_{\neq 0} / T(\pi\_1 \otimes \pi\_2) = T(\pi\_1) \otimes T(\pi\_2) \}$


It's not hard to see that elements in $G\_{\otimes}$ are unitaries. This is briefly proven in [Yuhjtman - Some considerations regarding the universal $W^\*$ algebra of a topological group](http://cms.dm.uba.ar/Members/sergioyuhjtman/WG2.pdf) (proposition 4.2).


Now Tatsuuma's duality theorem applies ([Tatsuuma - A duality theorem for locally compact groups, I](https://doi.org/10.3792/pja/1195522200), proposition 2) so $G=G\_\otimes$. But $G\_\otimes$ is closed and inside the unit ball, so it is compact (always $\sigma$-weak topolgy). Therefore $G$ is compact.



Universal $W^*$-algebras of locally compact groups: where is the error in this argument?
A link to the literature: I think of $C^\*(G)^\*$ as being $B(G)$, the Fourier-Stieltjes algebra, realised as a (non-closed) algebra of continuous functions on $G$. Any member of $C^\*(G)^\*$ can be realised as the composition of a representation $\pi$ on $H$ with a vector functional $\omega\_{\xi,\eta}$ on $H$. The resulting function in $B(G)$ is $g\mapsto (\pi(g)\xi|\eta)$.


Then $W^\*(G)$ is $B(G)^\*$. As the tensor product of representations corresponds to the product in $B(G)$, it follows that $G\_{\otimes}$ is actually just the collection of *characters* on $B(G)$, namely algebra homomorphisms $B(G)\rightarrow\mathbb C$. Such things were explored by Walter in his paper ["On the structure of the Fourier–Stieltjes algebra"](http://projecteuclid.org/euclid.pjm/1102905857).


It's shown that $G\_{\otimes}$ is not a group, and that it contains proper partial isometries and projections; it is a semigroup though.


