1,720,977 research outputs found
Invariant subspaces for H2 spaces of σ-finite algebras
No full textWe show that a Beurling type theory of invariant subspaces of noncommutative H
2
spaces
holds true in the setting of subdiagonal subalgebras of σ-finite von Neumann algebras. This
extends earlier work by Blecher and Labuschagne on finite algebras, and complements more
recent contributions in this regard by Bekjan and Chen, Hadwin and Shen in the finite setting,
and Sager in the semifinite setting.
We also introduce the notion of an analytically conditioned algebra, and go on to show that
in the class of analytically conditioned algebras this Beurling type theory is part of a list of
properties which all turn out to be equivalent to the maximal subdiagonality of the given algebra.
This list includes a Gleason–Whitney type theorem, as well the pairing of the unique normal
state extension property and an L
2
density conditio
A crossed product approach to Orlicz spaces
No full textWe show how the known theory of non-commutative Orlicz spaces for semifinite von Neumann
algebras equipped with an faithful normal semifinite trace may be recovered using crossed
product techniques. Then using this as a template, we construct analogues of such spaces for type
III algebras. The constructed spaces naturally dovetail with and closely mimic the behaviour of
Haagerup Lp-spaces.We then define a modified K-method of interpolation which seems to better
fit the present context, and give a formal prescription for using this method to define what may
be regarded as type III Riesz–Fischer spacesNational Research Foundatio
Multipliers on noncommutative Orlicz spaces
No full textWe establish very general criteria for the existence of multiplication
operators between noncommutative Orlicz spaces L 0 (fM) and L 1 (fM). We then
show that these criteria contain existing results, before going on to briefly look at
the extent to which the theory of multipliers on Orlicz spaces differs from that of
Lp-spaces. In closing we describe the compactness properties of such operator
Maximal ergodic inequalities for Banach function spaces
No full textWe analyse the Transfer Principle, which is used to generate weak type maximal inequalities for ergodic operators, and extend it to the general case of σσ-compact locally compact Hausdorff groups acting measure-preservingly on σσ-finite measure spaces. We show how the techniques developed here generate various weak type maximal inequalities on different Banach function spaces, and how the properties of these function spaces influence the weak type inequalities that can be obtained. Next we demonstrate how the techniques developed imply almost sure pointwise convergence of a wide class of ergodic averages. In closing we briefly indicate the utility of these results for Statistical Physic
A Helson-Szegö theorem for subdiagonal subalgebras with applications to Toeplitz operators
No full textWe formulate and establish a noncommutative version of the well known Helson–Szegö theorem about
the angle between past and future for subdiagonal subalgebras. We then proceed to use this theorem to
characterise the symbols of invertible Toeplitz operators on the noncommutative Hardy spaces associated
to subdiagonal subalgebras
Outers for noncommutative Hp revisted
No full textWe continue our study of outer elements of the noncommutative Hp spaces associated with Arveson's subdiagonal algebras. We extend our generalized inner-outer factorization theorem, and our characterization of outer elements, to include the case of elements with zero determinant. In addition, we make several further contributions to the theory of outers. For example, we generalize the classical fact that outers in Hp actually satisfy the stronger condition that there exist an∈A with han∈Ball(A) and han→1 in p-normNational Research Foundatio
On vector-valued characters for noncommutative function algebras
No full textLet A be a closed subalgebra of a C∗-algebra, that is a norm-closed algebra of Hilbert space operators. We generalize to such operator algebras several key theorems and concepts from the theory of classical function algebras. In particular we consider several problems that arise when generalizing classical function algebra results involving characters (nontrivial homomorphisms from the algebra into the scalars). For example, the Jensen inequality, the related Bishop–Ito–Schreiber theorem, and the theory of Gleason parts. Inspired by Arveson’s work on noncommutative Hardy spaces, we replace characters (classical function algebra case) by D-characters; certain completely contractive homomorphisms Φ:A→D, where D is a C∗-subalgebra of A. Using Brown’s measure and a potential theoretic balayage argument we prove a partial noncommutative Jensen inequality appropriate for C∗-algebras with a tracial state. We also show that this Jensen inequality characterizes D-characters among the module maps. Other advances include a theory of noncommutative Gleason parts appropriate for D-characters, which uses Harris’ noncommutative hyperbolic metric and Schwarz–Pick inequality, and other ingredients. As an application of Gleason parts we show that in the antisymmetric case, one is guaranteed the existence of a ‘quantum’ Wermer embedding function, and also of non-trivial compact Hankel operators, whenever the Gleason part of the canonical trace is rich in tracial stat
Ueda’s peak set theorem for general von Neumann algebras
No full textWe extend Ueda’s peak set theorem for subdiagonal subalgebras
of tracial finite von Neumann algebras to σ-finite von Neumann algebras (that
is, von Neumann algebras with a faithful state, which includes those on a
separable Hilbert space or with separable predual). To achieve this extension,
completely new strategies had to be invented at certain key points, ultimately
resulting in a more operator algebraic proof of the result. Ueda showed in
the case of finite von Neumann algebras that his peak set theorem is the
fountainhead of many other very elegant results, like the uniqueness of the
predual of such subalgebras, a highly refined F & M Riesz type theorem, and
a Gleason-Whitney theorem. The same is true in our more general setting,
and indeed we obtain a quite strong variant of the last mentioned theorem.
We also show that set theoretic issues dash hopes for extending the theorem to
some other large general classes of von Neumann algebras, for example finite
or semi-finite ones. Indeed certain cases of Ueda’s peak set theorem for a von
Neumann algebra M may be seen as ‘set theoretic statements’ about M that
require the sets to not be ‘too large
Why Are Orlicz Spaces Useful for Statistical Physics?
No full textWe review a new formalism based on Orlicz spaces for the description of large regular statistical systems. Our presentation includes both classical and quantum systems. This approach has the advantage that statistical mechanics is much better settle
On Applications of Orlicz Spaces to Statistical Physics
Get PDFWe present a new rigorous approach based on Orlicz spaces for
the description of the statistics of large regular statistical systems, both
classical and quantum. The pair of Orlicz spaces we explicitly use are,
respectively, built on the exponential function (for the description of regular
observables) and on an entropic type function (for the corresponding
states). They form a dual pair (both for classical and quantum systems).
This pair has the advantage of being general enough to encompass regular
observables, and specific enough for the latter Orlicz space to select
states with a well-defined entropy function. Moreover for small quantum
systems, this pair is shown to agree with the classical pairing of bounded
linear operators on a Hilbert space, and the trace-class operators.Grant Number N N202 208238; Foundation
for Polish Science TEAM project cofinanced by the EU European Regional
Development Fund for W.A. Majewski; National Research
Foundation for L.E. Labuschagn
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