Amenable unitary representations of locally compact groups
The concept of amenability of a locally compact group (more generally, of a homogeneous space) plays an important role in various parts of mathematics, in particular in harmonic analysis and group representation theory. In the paper under review the notion of an amenable unitary representation is introduced and successfully studied. Thus, let G be a locally compact group and \(\pi\) a unitary representation of G in the Hilbert space \(H_{\pi}\). Then \(\pi\) is said to be amenable if there exists a state M on the algebra \(B(H_{\pi})\) of bounded linear operators in \(H_{\pi}\) satisfying the equation \(M(\pi (x)T\pi (x^{- 1}))=M(T)\) for all \(T\in B(H_{\pi})\), \(x\in G\). The author proves a variety of impressive results about amenability of group representations. A homogeneous space G/H is amenable if and only if the quasi-regular representation of G on \(L^ 2(G/H)\) is amenable. In particular, G is amenable if and only if its left regular representation is so, and this latter condition is equivalent to that all the (irreducible) unitary representations of G are amenable. Moreover, G is inner-amenable exactly when the conjugation representation is amenable. Amenability of \(\pi\) also turns out to be equivalent to certain of Reiter's properties and Følner's condition. The most interesting characterization, maybe, of amenability of a unitary representation \(\pi\) is by means of tensor products and weak containment. Indeed, \(\pi\) is shown to be amenable if and only the tensor product \(\pi\otimes {\bar \pi}\) of \(\pi\) with its conjugate \({\bar \pi}\) weakly contains the trivial representation. The paper also contains several illustrating examples and certainly represents a significant contribution to unitary group representation theory. An interesting open problem is to relate amenability of \(\pi\) to amenability of some Banach algebra associated to \(\pi\).
scientific article; zbMATH DE number 4150532
scientific article