838 research outputs found
Noncommutative Choquet theory
We introduce a new and extensive theory of noncommutative convexity along
with a corresponding theory of noncommutative functions. We establish
noncommutative analogues of the fundamental results from classical convexity
theory, and apply these ideas to develop a noncommutative Choquet theory that
generalizes much of classical Choquet theory.
The central objects of interest in noncommutative convexity are
noncommutative convex sets. The category of compact noncommutative sets is dual
to the category of operator systems, and there is a robust notion of extreme
point for a noncommutative convex set that is dual to Arveson's notion of
boundary representation for an operator system.
We identify the C*-algebra of continuous noncommutative functions on a
compact noncommutative convex set as the maximal C*-algebra of the operator
system of continuous noncommutative affine functions on the set. In the
noncommutative setting, unital completely positive maps on this C*-algebra play
the role of representing measures in the classical setting.
The continuous convex noncommutative functions determine an order on the set
of unital completely positive maps that is analogous to the classical Choquet
order on probability measures. We characterize this order in terms of the
extensions and dilations of the maps, providing a powerful new perspective on
the structure of completely positive maps on operator systems.
Finally, we establish a noncommutative generalization of the
Choquet-Bishop-de Leeuw theorem asserting that every point in a compact
noncommutative convex set has a representing map that is supported on the
extreme boundary. In the separable case, we obtain a corresponding integral
representation theorem.Comment: 81 pages; minor change
Max-stable random sup-measures with comonotonic tail dependence
Several objects in the Extremes literature are special instances of
max-stable random sup-measures. This perspective opens connections to the
theory of random sets and the theory of risk measures and makes it possible to
extend corresponding notions and results from the literature with streamlined
proofs. In particular, it clarifies the role of Choquet random sup-measures and
their stochastic dominance property. Key tools are the LePage representation of
a max-stable random sup-measure and the dual representation of its tail
dependence functional. Properties such as complete randomness, continuity,
separability, coupling, continuous choice, invariance and transformations are
also analysed.Comment: 28 pages, 1 figur
The noncommutative Choquet boundary
Let S be an operator system -- a self-adjoint linear subspace of a unital
C*-algebra A such that contains 1 and A=C*(S) is generated by S. A boundary
representation for S is an irreducible representation \pi of C*(S) on a Hilbert
space with the property that has a unique completely
positive extension to C*(S). The set of all (unitary equivalence
classes of) boundary representations is the noncommutative counterpart of the
Choquet boundary of a function system that separates points
of X.
It is known that the closure of the Choquet boundary of a function system S
is the Silov boundary of X relative to S. The corresponding noncommutative
problem of whether every operator system has "sufficiently many" boundary
representations was formulated in 1969, but has remained unsolved despite
progress on related issues. In particular, it was unknown if is
nonempty for generic S. In this paper we show that every separable operator
system has sufficiently many boundary representations. Our methods use
separability in an essential way.Comment: 22 pages. A significant revision, including a new section and many
clarifications. No change in the basic mathematic
Multifunctions determined by integrable functions
Integral properties of multifunctions determined by vector valued functions are presented. Such multifunctions quite often serve as examples and counterexamples. In particular it can be observed that the properties of being integrable in the sense of Bochner, McShane or Birkhoff can be transferred to the generated multifunction while Henstock integrability does not guarantee i
Baire classes of affine vector-valued functions
We investigate Baire classes of strongly affine mappings with values in
Fr\'echet spaces. We show, in particular, that the validity of the
vector-valued Mokobodzki's result on affine functions of the first Baire class
is related to the approximation property of the range space. We further extend
several results known for scalar functions on Choquet simplices or on dual
balls of -preduals to the vector-valued case. This concerns, in
particular, affine classes of strongly affine Baire mappings, the abstract
Dirichlet problem and the weak Dirichlet problem for Baire mappings. Some of
these results have weaker conclusions than their scalar versions. We also
establish an affine version of the Jayne-Rogers selection theorem.Comment: 43 pages; we added some explanations and references, corrected some
misprints and simplified the proof of one lemm
Finitely additive extensions of distribution functions and moment sequences: The coherent lower prevision approach
We study the information that a distribution function provides about the finitely additive probability measure inducing it. We show that in general there is an infinite number of finitely additive probabilities associated with the same distribution function. Secondly, we investigate the relationship between a distribution function and its given sequence of moments. We provide formulae for the sets of distribution functions, and finitely additive probabilities, associated with some moment sequence, and determine under which conditions the moments determine the distribution function uniquely. We show that all these problems can be addressed efficiently using the theory of coherent lower previsions
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