Non injectivity of the q-deformed von Neumann algebra

By Alexandre Nou. 2004

77 citations recorded by the source corpus.

Abstract.In this paper we prove that the von Neumann algebra generated by q-gaussians is not injective as soon as the dimension of the underlying Hilbert space is greater than 1. Our approach is based on a suitable vector valued Khintchine type inequality for Wick products. The same proof also works for the more general setting of a Yang-Baxter deformation. Our techniques can also be extended to the so called q-Araki-Woods von Neumann algebras recently introduced by Hiai. In this latter case, we obtain the non injectivity under some asssumption on the spectral set of the positive operator associated with the deformation.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8adde2f54aea66bff25b904117cae0a82115c37a

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Citations in the corpus

A free stochastic partial differential equation (2010, background): “A short proof of the first part can be found in [28] (basically a variant of [5] without writing the computations), the argument obviously giving the second part too.”
Some estimates for non-microstates free entropy dimension with applications to q-semicircular families (2003, background): “, Xn is represented on a deformed Fock space and generates a finite von Neumann algebra, which is non-hyperfinite for n ≥ 2 and q ∈ (−1, 1) [7].”
Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors (2010, unknown): “This formula expresses W ( e 1 ⊗ · · · ⊗ e n ) as an element of T ( H ) and has many consequences such as Khintchine type inequalities in [5, 25, 4] for instance.”
Rosenthal Type inequalities for free chaos. (2005, background): “A. Nou, Non injectivity of the q-deformed von Neumann algebra.”