A characterization of locally compact amenable groups by means of tensor products
There is a variety of characterizations of amenability of a locally compact group G. One, which is due to Hulanicki and Reiter and proved very important in representation theory, says that G is amenable if and only if the trivial l-dimensional representation \(l_ G\) is weakly contained in the left regular representation \(\lambda_ G\) \((l_ G\prec \lambda_ G)\). For unitary representations \(\pi\) and \(\rho\) of G, denote by \(\pi\) \(\otimes \rho\) their tensor product and by \({\bar \pi}\) the conjugate representation of \(\pi\). Observing that \(\lambda_ G\otimes {\bar \lambda}_ G\) is a multiple of \(\lambda_ G\), the Hulanicki- Reiter theorem implies that G is amenable provided that \(l_ G\prec \pi \otimes {\bar \pi}\) for some representation \(\pi\) such that \(\pi \prec \lambda_ G\). The author proves the following nice and important converse result: If G is amenable, then \(l_ G\prec \pi \otimes {\bar \pi}\) for any unitary representation \(\pi\) of G. This had previously been shown in a number of particular cases. A main tool in the proof is that \(\pi\) \(\otimes {\bar \pi}\) can be realized on the space of Hilbert- Schmidt operators in the Hilbert space of \(\pi\).
scientific article; zbMATH DE number 4099569