Additive conjugacy and the Bohr compactification of orthogonal representations (Q2235215)

For a (Hausdorff) topological abelian group \(A\), let \(A^\wedge\) be its dual group of continuous group homomorphisms from \(A\) to \(T=\mathbb{R}/\mathbb{Z}\), endowed with the compact-open topology. The Bohr compactification of \(A\) is defined as \((A^\wedge_d)^\wedge\), using the discrete topology on \(A^\wedge\). Every continuous homomorphism of groups \(\phi\colon A\to B\) between topological abelian groups gives rise to a continuous group homomorphism \(\phi^\wedge\colon B^\wedge\to A^\wedge\), \(\xi\mapsto \xi\circ \phi\), the dual homomorphism. Regarding \(\phi^\wedge\) as a homomorphism from \(B^\wedge_d\) to \(A^\wedge_d\), its dual homomorphism is a continuous homomorphism \(b\phi\colon bA\to bB\) which is an isomorphism of topological groups if \(\phi\) is so. For a finitely generated group \(G\), the authors study orthogonal representations on a separable real Hilbert space \(H\), i.e., homomorphisms \(\pi\colon G\to O(H)\) to the group \(O(H)\) of surjective linear isometries of \(H\). For each \(g\in G\), we get an automorphism \(b(\pi(g))\) of \(bH\), and a corresponding action \(G\times bH\to bH\), \((g,\theta)\mapsto b(\pi(g))(\theta)\). One says that \(\pi\) has almost invariant vectors if there exists a sequence of unit vectors \(v_n\) in \(H\) such that \(\pi(g)(v_n)-v_n\to 0\) in \(H\) as \(n\to\infty\), for each \(g\in G\). As the main result, the authors show that \(\pi\) has almost invariant vectors if and only if the corresponding \(G\)-action on \(bH\) has a non-zero fixed point (Theorem 1). They conclude: If \(\pi_j\colon G\to O(H_j)\) are orthogonal representations for \(j\in\{1,2\}\) and there exists a (not necessarily continuous) isomorphism \(\phi\colon (H_1,+)\to (H_2,+)\) of groups such that \(\phi\circ\pi_1(g)=\pi_2(g)\circ\phi\) for all \(g\in G\), then \(\pi_1\) has almost invariant vectors if and only if so does \(\pi_2\) (Corollary 2). They also conclude that \(G\) is amenable if and only if there exists a (not necessarily continuous) non-zero group homomorphism \(L^2(G)\to T\) which is invariant under the \(G\)-action on \(L^2(G)\) (Corollary 3). An essential tool is the identification of the Bohr compactification \(bH\) with the group \((H_d)^\wedge\) of all (not necessarily continuous) group homomorphisms \(H\to T\), endowed with the topology of pointwise convergence. In fact, it is well-known that \(E'_c\cong E^\wedge\) holds for the dual space of a topological vector space \(E\), endowed with the topology of compact convergence (see Proposition 2.3 in [\textit{W. Banaszczyk}, Additive subgroups of topological vector spaces. Berlin etc.: Springer-Verlag (1991; Zbl 0743.46002)]). If \(E\) is a Hilbert space (or reflexive Banach space), then \(E\cong E'_b\) for the topology of bounded convergence, whence \(E_d\cong E'_d\cong E^\wedge_d\) and thus \((E_d)^\wedge\cong (E^\wedge_d)^\wedge=bE\). (Reviewer's comment: Proposition 15 is not correct. If \(H\) is an infinite-dimensional separable real Hilbert space, then the weak topology on \(H\) is hemicompact, whence \((H_w)^\wedge\) is metrizable. Thus \((H_w)^\wedge\) cannot be isomorphic to \(H_w\) as a topological group, the latter not being metrizable.)