Continuous homomorphisms induced by isometries of almost periodic functions
by
Salvador Hernández
Universitat Jaume I, Departamento de Matemáticas, 12071-Castellón (Spain)

Let G and H be locally compact groups and consider their associate spaces of almost periodic functions AP(G) and AP(H). We investigate the continuous mappings and homomorphisms induced by isometries of AP(G) into AP(H). The following results are proved:

Theorem Let G and H be \omega-bounded maximally almost periodic locally compact groups, and let T be a basis-preserving linear isometry of AP(G) into AP(H) such that every f in AP(G) achieves its norm on G iff so does Tf on H. Then there is a closed subgroup H₀ subset or equal H, a continuous group homomorphism t of H₀ onto G and an character \gamma in [^H] such that (Tf)(h)=\gamma(h) f(t(h)) for all h in H₀ and for all f in AP(G).

Theorem Let G and H be LCA groups and H is connected. Suppose that T is a linear isometry of AP(G) into AP(H) preserving trigonometric polynomials and such that every f in AP(G) achieves its norm on G iff so does Tf on H. Then there is a closed subgroup H₀ subset or equal H, a continuous group homomorphism t of H₀ onto G, an element h₀ in H₀, a character \alpha in [^H] and an unimodular complex number a such that (Tf)(h)=a·\alpha(h) ·f(t(h-h₀))\text for all h in H₀\textand for all f in AP(G). Moreover, if T is an onto isometry then H₀=H and, as a consequence, G and H are topologically isomorphic.

Date received: June 1, 1999