Let \(G\) be a group, let \(K\) be a complete non-Archimedean valued field. A function \(G\to K\) is called almost periodic if its (left) translates form a compactoid for the uniform topology. Here compactoidity is a ‘convexified’ notion of precompactness which becomes useful if \(K\) is not separable. The almost periodic functions on \(G\) form a Banach algebra \(AP(G\to K)\) over \(K\). It is also a (complete) Hopf algebra under the comultiplication \(f\mapsto \pi\circ \Delta(f)\) where \(\Delta: AP(G\to K)\to AP(G\times G\to K)\) is given by \((\Delta f)(s, t)= f(st)\) and where \(\pi\) is the canonical isomorphism \[ AP(G\times G\to K)\to AP(G\to K)\widehat\otimes AP(G\to K), \] the co-unit \(f\mapsto f(s)\) and antipode \(\eta\) given by \(\eta(f)(s)= f(s^{-1})\).
By using Hopf algebra techniques the following theorem is proved: Let \(AP(G\to K)\) have an invariant mean (i.e. a shift invariant element of \(AP(G\to K)'\) sending the constant function 1 into \(1\in K\)). Then
(i) any topologically irreducible almost periodic linear representation is finite-dimensional
(ii) if \(U\) is an almost periodic representation of \(G\) into some Banach space \(E\) then \(E\) is the topological direct sum of irreducible \(U\)- invariant subspaces.
By applying this theorem to the left regular representation of \(G\) into \(AP(G\to K)\), a Peter-Weyl-like theorem is obtained.