arXiv subjects

Jacob Mashburn

Publications and source records attributed to Jacob Mashburn.

2 recordsLinked to original sources

arXiv2023

Fock representation of free convolution powers

Let $B$ be a star-algebra with a state $ϕ$, and $t > 0$. Through a Fock space construction, we define two states $Φ_t$ and $Ψ_t$ on the tensor algebra $T(B, ϕ)$ such that under the natural map $(B, ϕ) \rightarrow (T(B, ϕ), Φ_t, Ψ_t)$, free independence of arguments leads to free independence, while Boolean independence of centered arguments leads to conditionally free independence. The construction gives a new operator realization of the $(1+t)$'th free convolution power of any joint (star) distribution. We also compute several von Neumann algebras which arise.

math.OA↗

arXiv2022

Some Fock spaces with depth two action

The subject of this paper are operators represented on a Fock space which act only on the two leading components of the tensor. We unify the constructions from arXiv:math/0702158, arXiv:0709.4334, arXiv:0812.0895, and arXiv:1003.2998, and extend a number of results from these papers to our more general setting. The results include the quadratic relation satisfied by (the kernel of) the free cumulant generating function, the resolvent form of the generating function for the Wick polynomials, and classification results for the case when the vacuum state on the operator algebra is tracial. We handle the generating functions in infinitely many variables by considering their matrix-valued versions.

math.OA↗