Abstract
In 1973, in the paper [GGV1] and then in [GGV2] for the group SO(n, 1), SU(n, 1) the so-called canonical states were defined. The purpose was to construct an irreducible representation out of the current groups with values in Lie groups. The general scheme of the construction of such representations in Fock space (due to H. Araki) requires for the group of the coefficients the existence of nontrivial (vector-valued) cocycles. It was proved in these papers that for the above group such cocycles exist. Later in [Kar-V], it was proved that one cocycle could exist in the irreducible representation which is “glued” to unity (which means that the trivial representation is “close” to it in the Fell topology); so we need groups without the Kazdan property T, as are the above semisimple Lie groups. The cocycle can be described in a different way, directly as a map from the group to the representation space, or to the canonical state, which is nothing more than the exponent of the norm of the cocycle. In the last terms, a canonical state is an infinite-divisible positive definite function (so we have a one-parameter semigroup of such functions) generated by a conditionally positive definite and unbounded function.
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© 1996 Birkhäuser Boston
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Kuhn, G., Vershik, A. (1996). Canonical States on the Group of Automorphisms of a Homogeneous Tree. In: Gelfand, I.M., Lepowsky, J., Smirnov, M.M. (eds) The Gelfand Mathematical Seminars, 1993–1995. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4082-2_10
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DOI: https://doi.org/10.1007/978-1-4612-4082-2_10
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