A note on the automorphisms of a WAP-compactification (Q792469)

Let S be the weak almost periodic compactification of the additive group \(R^ n (n=1,2,...)\) and let G be a group of continuous automorphisms of S. If G is provided with a topology such that the action \(S\times G\to S\), \((s,g)\to s\square g=g^{-1}(s),\) is jointly continuous, then G is discrete. Let G be the direct product \(G=R^ n\times K\), where K is a compact topological group and n is any natural number. Furthermore, let S be the weak almost periodic compactification of G and write Aut S for the group of all continuous automorphisms of S; we suppose that Aut S is provided with a topology rendering the map \(S\times Aut S\to S\), \((s,g)\to s\square g=g^{-1}(s),\) jointly continuous. Then S is the direct product \(S=S_ 1\times K\), where \(S_ 1\) is the weak almost periodic compactification of \(R^ n\) and the automorphisms \(\alpha:\quad(a,b)\to(a,\phi(a)b), \phi\) being a continuous homomorphism \(S_ 1\to K\), form an open subgroup A of Aut S. Moreover, if A is endowed with the topology of uniform convergence on S and Hom \((R{}^ n,K)\) with the compact open topology then the restriction map \(r:A\to Hom (R^ n,K), \alpha \to \alpha | R^ n\) is an isomorphism. Let S be the weak almost periodic compactification of a locally compact topological group G. Suppose N is a normal subgroup of G which (provided with the topology inherited from G) is isomorphic with \(R^ n\). Then the following assertions are equivalent: (i) The closure of N in S is isomorphic with the weak almost periodic compactification of \(R^ n\). (ii) The centralizer \(C(N)=\{g\in G| gn=ng\) for all \(n\in N\}\) of N is open in G.