A combinatorial approach to monotonic independence over a C^∗-algebra

Abstract

We consider the notion of monotonic independence in a more general frame, similar to the construction of operator-valued free probability. The paper presents constructions for maps with similar properties to the H and K transforms from the literature, semi-inner-product bimodule analogues for the monotone and weakly monotone product of Hilbert spaces, an ad-hoc version of the Central Limit Theorem, an operator-valued arcsine distribution as well as a connection to operator-valued conditional freeness. IntroductionAn important notion in noncommutative probability is monotonic independence, introduced by P. Y. Lu and Naofumi Muraki. Since its beginning, the study of this notion of independence was done by constructions, techniques and developments similar to the theory of free probability. R. Speicher [1998] developed an operatorvalued analogue of free independence. The present paper addresses problems similar to ones discussed in that work, but in the context of monotonic independence.Other motivation is that while for the free Fock space over a Hilbert space there is a straightforward analogous semi-inner-product bimodule construction, as illustrated in [Pimsner 1997;Speicher 1998], there are no similar constructions for its various deformations, such as the q-Fock spaces [Effros and Popa 2003]. As shown in Section 4, the monotone and weakly monotone Fock-like spaces, which are strongly connected to monotonic independence, admit analogous semi-innerproduct bimodules.The paper is structured in six sections. Section 2 presents the definition of the monotonic independence over an algebra. In Section 3 there are constructed maps with similar properties to the maps H and K from the theory of monotonic independence, as introduced in [Muraki 2000; Bercovici 2005b]. Section 4 deals with semi-inner-product bimodule analogues of the monotone and weakly monotone products of Hilbert spaces and algebras of annihilation operators, as introduced in MSC2000: primary 46L53; secondary 46L08.

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