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ON LEHNER’S ‘FREE ’ NONCOMMUTATIVE ANALOGUE OF DE FINETTI’S THEOREM

By Claus K Östler

Abstract

Abstract. Inspired by Lehner’s results on exchangeability systems in [Leh06] we define ‘weak conditional freeness ’ and ‘conditional freeness ’ for stationary processes in an operator algebraic framework of noncommutative probability. We show that these two properties are equivalent and thus the process embeds into a von Neumann algebraic amalgamated free product over the fixed point algebra of the stationary process. Recently Lehner introduced ‘weak freeness ’ for exchangeability systems within a cumulant approach to *-algebraic noncommutative probability. A main result in [Leh06] is that an exchangeability system with weak freeness and certain other properties embeds into an amalgamated free product analogous to the classical De Finetti theorem. Here we investigate Lehner’s approach from an operator algebraic point of view which is motivated by recent results on a noncommutative version of De Finetti’s theorem [Kös08] and a certain ‘braided ’ extension of it [GK08]. For the classical De Finetti theorem, we refer the reader to Kallenberg’s recent monograph [Kal05] on probabilistic symmetries and invariance principles. Since tail events in probability theory lead to conditioning which goes beyond amalgamation, we define ‘conditional freeness ’ and ‘weak conditional freeness ’ as a slight generalization of Voiculescu’s ‘amalgamated freeness ’ [Voi85, VDN92] and Lehner’s ‘weak freeness ’ [Leh06], respectively. We investigate them for stationary processes in an operator algebraic setting of noncommutative probability (see [Kös08, GK08] for details). Our main results are, re-formulated in terms of infinite minimal random sequences with stationarity: ⋄ ‘Weak conditional freeness ’ and ‘conditional freeness ’ are equivalent; and the conditioning is with respect to the tail algebra of the random sequence (see Theorem 2.1). ⋄ ‘Weak freeness ’ and ‘amalgamated freeness ’ are equivalent under a certain condition; and the amalgamation is with respect to the tail algebra of the random sequence (see Theorem 3.6). ⋄ Each of these four variations of Voiculescu’s central notion of freeness implies that the random sequence canonically embeds into a certain von Neumann algebra amalgamated free product and that it enjoys exchangeabilit

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