K-Theory 2: 675-682, 1989 675 1989 Kluwer Academic Publishers. Printed in the Netherlands. On Super-KMS Functionals and Entire Cyclic Cohomology* ARTHUR JAFFE, ANDRZEJ LESNIEWSKI, and KONRAD OSTERWALDER** Harvard University, Cambridge, M A 02138, U.S.A. (Received: 5 December 1988) Abstract. We formulate the super-KMS condition suggestedby Connes and Kastler, in the context of entire cyclic cohomology of quantum algebras. We show that the Chern character of Jaffe, Lesniewski, and Osterwalder - associated by Kastler to a super-KMS functional - satisfies the entire growth condition. Hence, a super-KMS functional defines a cocycle for the entire cyclic cohomology of quantum algebras. Key words. Cyclic cohomology, C*-algebras, KMS condition. 1. Introduction The purpose of this note is to clarify the relation between the super-KMS property and entire cyclic cohomology of quantum algebras. Our interest in super-KMS functionals was inspired by work of Kastler [4] and by private conversations with Alain Connes. This generalization is the natural framework for entire cyclic cohomology in the case that the Laplace operator has continuous spectrum. Such situations can arise if the cohomology is based on a noncompact manifold. In intuitive terms, one would like to formulate this in terms of a noncompact, 'noncommutative manifold'. Just as in statistical mechanics where a KMS state generalizes the notion of a Gibbs state, a super-KMS functional generalizes the positive temperature supertrace functional. This allows us to deal with situations which occur in examples, such as supersymmetric field theory on a noncompact manifold: the Laplace-Beltrami operator on loop space (the Hamiltonian of such a theory) is expected to have continuum spectrum, so the heat kernel it generates will not be trace class. This is characteristic of many examples. Besides, from a conceptual point of view, it is irrelevant whether the heat kernel is trace class; this assumption can be replaced by the super-KMS property. Such functionals are not necessarily positive and, hence, are not states, but estimates can be proved by expressing co as a linear combination of states. The usual KMS property relates the cyclicity of a state co to the analytic continuation of a group a t of automorphisms. The super-KMS property also involves the super *Supported in part by the National Science Foundation. **Permanent address: ETH Zentrum, 8092 Zfirich, Switzerland.