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Journal of Lie Theory 19 (2009), No. 3, 613--637 Copyright Heldermann Verlag 2009 Lie Quasi-States Michael Entov Dept. of Mathematics, Technion -- Israel Inst. of Technology, Haifa 32000, Israel entov@math.technion.ac.il Leonid Polterovich School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel and: Dept. of Mathematics, University of Chicago, Chicago, IL 60637, U.S.A. polterov@runbox.com Lie quasi-states on a real Lie algebra are functionals which are linear on any abelian subalgebra. We show that on the symplectic Lie algebra of rank at least 3 there is only one continuous non-linear Lie quasi-state (up to a scalar factor, modulo linear functionals). It is related to the asymptotic Maslov index of paths of symplectic matrices. Keywords: Quasi-state, Lie algebra, Maslov index, Gleason theorem. MSC: 53D12, 17B99, 15A27, 15B99 [ Fulltext-pdf (257 KB)] for subscribers only. |