471 research outputs found
Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis
We study two classes of extension problems, and their interconnections: (i)
Extension of positive definite (p.d.) continuous functions defined on subsets
in locally compact groups ; (ii) In case of Lie groups, representations of
the associated Lie algebras by unbounded skew-Hermitian
operators acting in a reproducing kernel Hilbert space (RKHS)
.
Why extensions? In science, experimentalists frequently gather spectral data
in cases when the observed data is limited, for example limited by the
precision of instruments; or on account of a variety of other limiting external
factors. Given this fact of life, it is both an art and a science to still
produce solid conclusions from restricted or limited data. In a general sense,
our monograph deals with the mathematics of extending some such given partial
data-sets obtained from experiments. More specifically, we are concerned with
the problems of extending available partial information, obtained, for example,
from sampling. In our case, the limited information is a restriction, and the
extension in turn is the full positive definite function (in a dual variable);
so an extension if available will be an everywhere defined generating function
for the exact probability distribution which reflects the data; if it were
fully available. Such extensions of local information (in the form of positive
definite functions) will in turn furnish us with spectral information. In this
form, the problem becomes an operator extension problem, referring to operators
in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we
have stressed hands-on-examples. Extensions are almost never unique, and so we
deal with both the question of existence, and if there are extensions, how they
relate back to the initial completion problem.Comment: 235 pages, 42 figures, 7 tables. arXiv admin note: substantial text
overlap with arXiv:1401.478
Holomorphic geometric models for representations of -algebras
Representations of -algebras are realized on section spaces of
holomorphic homogeneous vector bundles. The corresponding section spaces are
investigated by means of a new notion of reproducing kernel, suitable for
dealing with involutive diffeomorphisms defined on the base spaces of the
bundles. Applications of this technique to dilation theory of completely
positive maps are explored and the critical role of complexified homogeneous
spaces in connection with the Stinespring dilations is pointed out. The general
results are further illustrated by a discussion of several specific topics,
including similarity orbits of representations of amenable Banach algebras,
similarity orbits of conditional expectations, geometric models of
representations of Cuntz algebras, the relationship to endomorphisms of
, and non-commutative stochastic analysis.Comment: 45 page
Hidden Symmetries of Stochastic Models
In the matrix product states approach to species diffusion processes the
stationary probability distribution is expressed as a matrix product state with
respect to a quadratic algebra determined by the dynamics of the process. The
quadratic algebra defines a noncommutative space with a quantum group
action as its symmetry. Boundary processes amount to the appearance of
parameter dependent linear terms in the algebraic relations and lead to a
reduction of the symmetry. We argue that the boundary operators of
the asymmetric simple exclusion process generate a tridiagonal algebra whose
irriducible representations are expressed in terms of the Askey-Wilson
polynomials. The Askey-Wilson algebra arises as a symmetry of the boundary
problem and allows to solve the model exactly.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium
on Non-Perturbative and Symmetry Methods in Field Theory (June 2006,
Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Symmetries of L\'evy processes on compact quantum groups, their Markov semigroups and potential theory
Strongly continuous semigroups of unital completely positive maps (i.e.
quantum Markov semigroups or quantum dynamical semigroups) on compact quantum
groups are studied. We show that quantum Markov semigroups on the universal or
reduced C-algebra of a compact quantum group which are translation
invariant (w.r.t. to the coproduct) are in one-to-one correspondence with
L\'evy processes on its -Hopf algebra. We use the theory of L\'evy processes
on involutive bialgebras to characterize symmetry properties of the associated
quantum Markov semigroup. It turns out that the quantum Markov semigroup is
GNS-symmetric (resp. KMS-symmetric) if and only if the generating functional of
the L\'evy process is invariant under the antipode (resp. the unitary
antipode). Furthermore, we study L\'evy processes whose marginal states are
invariant under the adjoint action. In particular, we give a complete
description of generating functionals on the free orthogonal quantum group
that are invariant under the adjoint action. Finally, some aspects of
the potential theory are investigated. We describe how the Dirichlet form and a
derivation can be recovered from a quantum Markov semigroup and its L\'evy
process and we show how, under the assumption of GNS-symmetry and using the
associated Sch\"urmann triple, this gives rise to spectral triples. We discuss
in details how the above results apply to compact groups, group C-algebras
of countable discrete groups, free orthogonal quantum groups and the
twisted quantum group.Comment: 54 pages, thoroughly revised, to appear in the Journal of Functional
Analysi
L\'evy Processes on Quantum Permutation Groups
We describe basic motivations behind quantum or noncommutative probability,
introduce quantum L\'evy processes on compact quantum groups, and discuss
several aspects of the study of the latter in the example of quantum
permutation groups. The first half of this paper is a survey on quantum
probability, compact quantum groups, and L\'evy processes on compact quantum
groups. In the second half the theory is applied to quantum permutations
groups. Explicit examples are constructed and certain classes of such L\'evy
processes are classified.Comment: 60 page
Infinite Dimensional Lie Theory
The workshop focussed on recent developments in infinite-dimensional Lie theory. The talks covered a broad range of topics, such as structure and classification theory of infinite-dimensional Lie algebras, geometry of infinite-dimensional Lie groups and homogeneous spaces and representations theory of infinite-dimensional Lie groups, Lie algebras and Lie-superalgebras
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