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Abstract. Various types of Hubert space representations for a Baer '-semigroup S are defined. The representations are characterized in terms of '-positive functions on S which possess additivity and consistency properties.1. Introduction. In this paper we characterize those Baer »-semigroups that admit various types of representations in Hubert space. These characterizations are formulated in terms of the existence of »-positive functions possessing certain properties. Such formulations have been used for general semigroups and »-semigroups [1], [10], [12], but the additional properties needed for Baer »-semigroups have not been previously considered.Representations of Baer »-semigroups are not only of interest in their own right, they can be important in the study of quantum logics [7] . This connection is not only of a purely mathematical character; it also results from a deeper understanding of the physical meaning of "operation" in quantum mechanics [15], [16]. For this reason, representations of Baer »-semigroups may be physically interpreted as representations of the set of operations for a quantum system. One then obtains a representation theory which is more fundamental than the usual representation theory for the C*-algebra of bounded observables for a quantum system [4], [9].