# Journal of Operator Theory

Volume 92, Issue 1, Summer 2024 pp. 77-99.

Fock representation of free convolution powers

**Authors**:
Michael Anshelevich (1), Jacob Mashburn (2)

**Author institution:**(1) Department of Mathematics, Texas A and M University, College Station, TX 77843-3368, U.S.A.

(2) Department of Mathematics, Texas A and M University, College Station, TX 77843-3368, U.S.A.

**Summary: ** Let $\mathcal{B}$ be a $*$-algebra with a state $\phi$, and $t\!\! >\! \!0$. Through a Fock space construction, we define two states $\Phi_t$ and $\Psi_t$ on the tensor algebra $\mathcal{T}(\mathcal{B}, \phi)$ such that under the natural map $(\mathcal{B}, \phi) \rightarrow (\mathcal{T}(\mathcal{B}, \phi), \Phi_t, \Psi_t)$, 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.

**DOI: **http://dx.doi.org/10.7900/jot.2022aug08.2422

**Keywords: ** Fock space, free independence, Boolean independence, conditionally free independence

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