arXiv subjects

Romuald Lenczewski

Publications and source records attributed to Romuald Lenczewski.

25 records · Page 2Linked to original sources

arXiv2005

Reduction of free independence to tensor independence

We show how to reduce free independence to tensor independence in the strong sense. We construct a suitable unital *-algebra of closed operators `affiliated' with a given unital *-algebra and call the associated closure `monotone'. Then we prove that monotone closed operators of the form $$ X'= \sum_{k=1}^{\infty}X(k)\bar{\otimes} p_{k}, X''=\sum_{k=1}^{\infty} p_{k}\bar{\otimes}X(k) $$ are free with respect to a tensor product state, where $X(k)$ are tensor independent copies of a random variable $X$ and $(p_{k})$ is a sequence of orthogonal projections. For unital free *-algebras, we construct a monotone closed analog of a unital *-bialgebra called a `monotone closed quantum semigroup' which implements the additive free convolution, without using the concept of dual groups.

math.QA↗

arXiv2005

Discrete interpolation between monotone probability and free probability

We construct a sequence of states called m-monotone product states which give a discrete interpolation between the monotone product of states of Muraki and the free product of states of Avitzour and Voiculescu in free probability. We derive the associated basic limit theorems and develop the combinatorics based on non-crossing ordered partitions with monotone order starting from depth m. The Hilbert space representations of the limit mixed moments in the invariance principle lead to m-monotone Gaussian operators living in m-monotone Fock spaces, which are truncations of the free Fock space over the square-integrable functions on the non-negative real line (m=1 gives the monotone Fock space). A new type of combinatorics of inner blocks leads to explicit formulas for the mixed moments of m-monotone Gaussian operators, which are new even in the case of monotone independent Gaussian operators with arcsine distributions.

math.QA↗

arXiv2001

Noncommutative extensions of the Fourier transform and its logarithm

We introduce and study noncommutative extensions of the Fourier transform and its logarithm to the algebra of functions on the free semigroup FS(2) on two generators with the convolution multiplication. These extensions are new types of moment and cumulant generating functions, respectively, the latter corresponding to the cumulants which are additive under the so-called filtered convolution on the free *-algebra on two generators. This algebra plays the role of a ``noncommutative plane'' built on the ``classical real line'' and the ``boolean real line''. The restrictions of the cumulant generating function to the commutative subsemigroups generated by single generators give the logarithm of the Fourier transform and the K-transform in the boolean case, respectively. In turn, the moment generating function is a ``semigroup interpolation'' between the Fourier transform and the Cauchy transform. Using suitable weight function $W$ on the semigroup, both generating functions become elements of the Banach algebra $l^{1}(FS(2),W)$. The main results of the paper are based on the new combinatorics developed for the cumulants, the Moebius function and the moment-cumulant formulas.

math.PR↗

arXiv2001

Unification of independence in quantum probability

For a family of unital free *-algebras with a family of states on them, we construct a sequence of noncommutative probability spaces, which are tensor product algebras with tensor product states and which approximate the free product of states in the sense of convergence of mixed moments to those of free random variables. The constructed sequence of noncommutative probability spaces is called the hierarchy of freeness with the corresponding states called m-free products of states. The first-order approximation corresponding to m=1 gives the Boolean product of states. Thus the product states for the main types of noncommutative independence can be obtained from tensor products of states. Therefore, this approach can be viewed as a unification of independence. We also show how to associate a cocommutative *-bialgebra with the m-free product of states. This approach is developed in the paper for the more general case of the conditionally free product of states and can be easily extended to families of sequences of states.

math.QA↗

arXiv2001

Limit theorems for the hierarchy of freeness

The central limit theorem, the invariance principle and the Poisson limit theorem for the hierarchy of freeness are studied. We show that for given natural m the limit laws can be expressed in terms of non-crossing partitions of depth smaller or equal to m. For the algebra of polynomials in one variable we solve the associated moment problems and find explicitly the discrete limit measures.

math.QA↗

arXiv2001

Filtered stochastic calculus

By introducing a color filtration to the multiplicity space, we extend the quantum Ito calculus on multiple symmetric Fock space to the framework of filtered adapted biprocesses. In this new notion of adaptedness,``classical'' time filtration makes the integrands similar to adapted processes, whereas ``quantum'' color filtration produces their deviations from adaptedness. An important feature of this calculus, which we call filtered stochastic calculus, is that it provides an explicit interpolation between the main types of calculi, regardless of the type of independence, including freeness, Boolean independence (more generally, m-freeness) as well as tensor independence. Moreover, it shows how boson calculus is ``deformed'' by other noncommutative notions of independence.

math.QA↗

arXiv2001

Filtered random variables, bialgebras and convolutions

We introduce the filtered *-bialgebra which is a noncommutative analog of the *-bialgebra of multivariate polynomials with the canonical coproduct and counit. We study the associated filtered convolutions, random walks and random variables. The GNS representations of the limit states lead to filtered fundamental operators which are the CCR fundamental operators on the multiple symmetric Fock space, multiplied by appropriate projections. The importance of filtered random variables and fundamental operators stems from the fact that by addition and strong limits one obtains from them the main types of noncommutative random variables and fundamental operators, respectively, regardless of the type of noncommutative independence.

math.QA↗