Continuity of functions operating on characteristic functions

Abstract

AbstractFor a locally compact abelian group G let P0(G) denote the set of all characteristic functions on G (i.e., continuous positive definite functions φ with φ(0) = 1). A complex-valued function f is said to operate on P0(G) if f(φ(·)) ∈ P0(G) whenever φ ∈ P0(G). The natural domain for functions operating on P0(G) is the set D(G) = {z ∈ C: z = φ(g), g ∈ G, φ ∈ P0(G)}. It is known that every function operating on P0(G) is continuous on the interior of D(G) for each infinite group G. On the other hand, functions discontinuous on D(G) operate on P0(G) for any discrete G. We show that, for any non-discrete group G, every function operating on P0(G) is continuous on D(G). Together with some earlier results, this statement allows us to obtain a constructive description of operating functions on the whole set D(G).