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Fixed-point spectrum for group actions b...
Lavy, Omer...
Fixed-point spectrum for group actions by affine isometries on Lp-spaces by Lavy, Omer ( Author )
N.A
01-01-2014
The fixed-point spectrum of a locally compact second countable group G on lp is defined to be the set of real numbers p such that every action by affine isometries of G on lp admits a fixed-point. We show that this set is either empty, or is equal to a set of one of the following forms : [1,\pc[, [1,\pc[\{2} for some \pc<\infty or \pc=\infty, or [1,\pc], [1,\pc]\{2} for some pc<infty. This answers a question closely related to a conjecture of C. Drutu which asserts that the fixed-point spectrum is connected for isometric actions on Lp(0,1). We also study the topological properties of the fixed-point spectrum on Lp(X,\mu) for general measure spaces (X,\mu), and show partial results toward the conjecture for actions on Lp(0,1). In particular, we prove that the spectrum F_{L^{\infty}(X,\mu)(G,\pi) of actions with linear part \pi is either empty, or an interval of the form [1,\pc] or [1,\infty[, whenever \pi is an orthogonal representation associated to a measure-preserving ergodic action on a finite measure space (X,\mu).
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Article
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English
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MYR 0.01
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http://arxiv.org/abs/1410.0227
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