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The norm of products of free random variables

Authors:	Vladislav Kargin

Institution:	(1) Courant Institute of Mathematical Sciences, 109-20 71st Road, Apt. 4A, Forest Hills, NY 11375, USA

Abstract:	Let X _i denote free identically-distributed random variables. This paper investigates how the norm of products ${\Pi_{n} = X_{1}X_{2}\cdots X_{n}}$ behaves as n approaches infinity. In addition, for positive X _i it studies the asymptotic behavior of the norm of ${Y_{n} = X_{1}\circ X_{2}\circ \cdots \circ X_{n},}$ where ${\circ}$ denotes the symmetric product of two positive operators: ${A\circ B=:A^{1/2}BA^{1/2}}$ . It is proved that if EX _i = 1, then ${\left\Vert Y_{n}\right\Vert }$ is between ${c_{1}\sqrt{n}}$ and c ₂ n for certain constant c ₁ and c ₂. For ${\left\Vert \Pi_{n}\right\Vert ,}$ it is proved that the limit of ${n^{-1}\log \left\Vert \Pi _{n}\right\Vert }$ exists and equals ${\log \sqrt{E\left( X_{i}^{\ast }X_{i}\right) }.}$ Finally, if π is a cyclic representation of the algebra generated by X _i, and if ξ is a cyclic vector, then ${n^{-1}\log \left\Vert \pi \left( \Pi _{n}\right) \xi \right\Vert = \log \sqrt{E\left( X_{i}^{\ast }X_{i}\right) }}$ for all n. These results are significantly different from analogous results for commuting random variables.

Keywords:	46L54 15A52
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