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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

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.

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

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