Contractivity properties of Ornstein–Uhlenbeck semigroup for mixed q-Araki–Woods von Neumann algebras

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We study certain non-tracial von Neumann algebras generated by some self-adjoint operators satisfying mixed q-commutation relations. Such algebras are discussed in the work of Bikram et al. [“Mixed q-deformed Araki-Woods von Neumann algebras,” (submitted)]. We prove the analog of Nelson’s hypercontractivity inequality for the mixed q-Ornstein–Uhlenbeck semigroup. We also show that the mixed q-Ornstein–Uhlenbeck semigroup is ultracontractive.