771 research outputs found
Duality for convex monoids
Every C*-algebra gives rise to an effect module and a convex space of states,
which are connected via Kadison duality. We explore this duality in several
examples, where the C*-algebra is equipped with the structure of a
finite-dimensional Hopf algebra. When the Hopf algebra is the function algebra
or group algebra of a finite group, the resulting state spaces form convex
monoids. We will prove that both these convex monoids can be obtained from the
other one by taking a coproduct of density matrices on the irreducible
representations. We will also show that the same holds for a tensor product of
a group and a function algebra.Comment: 13 page
Relating Operator Spaces via Adjunctions
This chapter uses categorical techniques to describe relations between
various sets of operators on a Hilbert space, such as self-adjoint, positive,
density, effect and projection operators. These relations, including various
Hilbert-Schmidt isomorphisms of the form tr(A-), are expressed in terms of dual
adjunctions, and maps between them. Of particular interest is the connection
with quantum structures, via a dual adjunction between convex sets and effect
modules. The approach systematically uses categories of modules, via their
description as Eilenberg-Moore algebras of a monad
Toroidal crossings and logarithmic structures
We generalize Friedman's notion of d-semistability, which is a necessary
condition for spaces with normal crossings to admit smoothings with regular
total space. Our generalization deals with spaces that locally look like the
boundary divisor in Gorenstein toroidal embeddings. In this situation, we
replace d-semistability by the existence of global log structures for a given
gerbe of local log structures. This leads to cohomological descriptions for the
obstructions, existence, and automorphisms of log structures. We also apply
toroidal crossings to mirror symmetry, by giving a duality construction
involving toroidal crossing varieties whose irreducible components are toric
varieties. This duality reproduces a version of Batyrev's construction of
mirror pairs for hypersurfaces in toric varieties, but it applies to a larger
class, including degenerate abelian varieties.Comment: 34 pages, 1 figure, notational changes, to appear in Adv. Mat
The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations
This survey article begins with a discussion of the original form of the
Strominger-Yau-Zaslow conjecture, surveys the state of knowledge concering this
conjecture, and explains how thinking about this conjecture naturally leads to
the program initiated by the author and Bernd Siebert to study mirror symmetry
via degenerations of Calabi-Yau manifolds and log structures.Comment: 44 pages, to appear in the Proceedings of the 2005 AMS Symposium on
Algebraic Geometry, Seattl
On seminormal monoid rings
Given a seminormal affine monoid M we consider several monoid properties of M
and their connections to ring properties of the associated affine monoid ring
K[M] over a field K. We characterize when K[M] satisfies Serre's condition
(S_2) and analyze the local cohomology of K[M]. As an application we present
criteria which imply that K[M] is Cohen--Macaulay and we give lower bounds for
the depth of K[M]. Finally, the seminormality of an arbitrary affine monoid M
is studied with characteristic p methods.Comment: 23 page
A note on Herglotz's theorem for time series on function spaces
In this article, we prove Herglotz's theorem for Hilbert-valued time series.
This requires the notion of an operator-valued measure, which we shall make
precise for our setting. Herglotz's theorem for functional time series allows
to generalize existing results that are central to frequency domain analysis on
the function space. In particular, we use this result to prove the existence of
a functional Cram{\'e}r representation of a large class of processes, including
those with jumps in the spectral distribution and long-memory processes. We
furthermore obtain an optimal finite dimensional reduction of the time series
under weaker assumptions than available in the literature. The results of this
paper therefore enable Fourier analysis for processes of which the spectral
density operator does not necessarily exist
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