Abstract
Let G be a countable discrete group with an orthogonal representation α on a real Hilbert space H. We prove Lp Poincaré inequalities for the group measure space L∞(ΩH,γ)⋊G, where both the group action and the Gaussian measure space (ΩH,γ) are associated with the representation α. The idea of proof comes from Pisierʼs method on the boundedness of Riesz transform and Lust-Piquardʼs work on spin systems. Then we deduce a transportation type inequality from the Lp Poincaré inequalities in the general noncommutative setting. This inequality is sharp up to a constant (in the Gaussian setting). Several applications are given, including Wiener/Rademacher chaos estimation and new examples of Rieffelʼs compact quantum metric spaces.
Published Version
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