Abstract:We prove a noncommutative $p$-Poincaré inequality for GNS-detailed-balance quantum Markov semigroups (QMSs) on non-tracial $\sigma$-finite von Neumann algebras, assuming only the existence of a spectral gap. Extending the semi-commutative results of Huang and Tropp, we first establish the inequality for tracial von Neumann algebras. We use Markov dilations to obtain chain-rule estimates for Dirichlet forms and amalgamated free products to define an appropriate noncommutative derivation. We then extend the result to QMSs satisfying GNS detailed balance on non-tracial $\sigma$-finite von Neumann algebras, using Haagerup's reduction and Kosaki's interpolation theorem. As applications, we recover sub-exponential concentration inequalities and estimate the Lipschitz and completely bounded Lipschitz diameters of the QMSs.
From: Jia Wang [view email]
[v1]
Fri, 9 Jan 2026 18:41:12 UTC (65 KB)
[v2]
Mon, 31 Aug 2026 18:10:55 UTC (68 KB)
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