We develop a theory of ergodicity for unbounded functions ø: J → X, where J is a subsemigroup of a locally compact abelian group G and X is a Banach space. It is assumed that ø is continuous and dominated by a weight w defined on G. In particular, we establish total ergodicity for the orbits of an (unbounded) strongly continuous representation T: G → L(X) whose dual representation has no unitary point spectrum. Under additional conditions stability of the orbits follows. To study spectra of functions, we use Beurling algebras L1w(G) and obtain new characterizations of their maximal primary ideals, when w is non-quasianalytic, and of their minimal primary ideals, when w has polynomial growth. It follows that, relative to certain translation invariant function classes , the reduced Beurling spectrum of ø is empty if and only if ø ∈ . For the zero class, this is Wiener's tauberian theorem.

Iseki [11] defined a general notion of ergodicity suitable for functions ϕ: J → X where J is an arbitrary abelian semigroup and X is a Banach space. In this paper we develop the theory of such functions, showing in particular that it fits the general framework established by Eberlein [9] for ergodicity of semigroups of operators acting on X. Moreover, let A be a translation invariant closed subspace of the space of all bounded functions from J to X. We prove that if A contains the constant functions and ϕ is an ergodic function whose differences lie in A then ϕ ∈ A. This result has applications to spaces of sequences facilitating new proofs of theorems of Gelfand and Katznelson-Tzafriri [12]. We also obtain a decomposition for the space of ergodic vectors of a representation T: J → L(X) generalizing results known for the case J = Z+. Finally, when J is a subsemigroup of a locally compact abelian group G, we compare the Iseki integrals with the better known Cesàro integrals.