26,995 research outputs found
Optimal domain of -concave operators and vector measure representation of -concave Banach lattices
Given a Banach space valued -concave linear operator defined on a
-order continuous quasi-Banach function space, we provide a description
of the optimal domain of preserving -concavity, that is, the largest
-order continuous quasi-Banach function space to which can be
extended as a -concave operator. We show in this way the existence of
maximal extensions for -concave operators. As an application, we show a
representation theorem for -concave Banach lattices through spaces of
integrable functions with respect to a vector measure. This result culminates a
series of representation theorems for Banach lattices using vector measures
that have been obtained in the last twenty years
Noncommmutative theorems: Gelfand Duality, Spectral, Invariant Subspace, and Pontryagin Duality
We extend the Gelfand-Naimark duality of commutative C*-algebras, "A
COMMUTATIVE C*-ALGEBRA -- A LOCALLY COMPACT HAUSDORFF SPACE" to "A
C*-ALGEBRA--A QUOTIENT OF A LOCALLY COMPACT HAUSDORFF SPACE". Thus, a
C*-algebra is isomorphic to the convolution algebra of continuous regular Borel
measures on the topological equivalence relation given by the above mentioned
quotient. In commutative case this reduces to Gelfand-Naimark theorem.
Applications: 1) A simultaneous extension, to arbitrary Hilbert space
operators, of Jordan Canonical Form and Spectral Theorem of normal operators 2)
A functional calculus for arbitrary operators. 3) Affirmative solution of
Invariant Subspace Problem. 4) Extension of Pontryagin duality to nonabelian
groups, and inevitably to groups whose underlying topological space is
noncommutative.Comment: 10 page
Quantum statistical mechanics over function fields
In this paper we construct a noncommutative space of ``pointed Drinfeld
modules'' that generalizes to the case of function fields the noncommutative
spaces of commensurability classes of Q-lattices. It extends the usual moduli
spaces of Drinfeld modules to possibly degenerate level structures. In the
second part of the paper we develop some notions of quantum statistical
mechanics in positive characteristic and we show that, in the case of Drinfeld
modules of rank one, there is a natural time evolution on the associated
noncommutative space, which is closely related to the positive characteristic
L-functions introduced by Goss. The points of the usual moduli space of
Drinfeld modules define KMS functionals for this time evolution. We also show
that the scaling action on the dual system is induced by a Frobenius action, up
to a Wick rotation to imaginary time.Comment: 28 pages, LaTeX; v2: last section expande
The mass of unimodular lattices
The purpose of this paper is to show how to obtain the mass of a unimodular
lattice from the point of view of the Bruhat-Tits theory. This is achieved by
relating the local stabilizer of the lattice to a maximal parahoric subgroup of
the special orthogonal group, and appealing to an explicit mass formula for
parahoric subgroups developed by Gan, Hanke and Yu.
Of course, the exact mass formula for positive defined unimodular lattices is
well-known. Moreover, the exact formula for lattices of signature (1,n) (which
give rise to hyperbolic orbifolds) was obtained by Ratcliffe and Tschantz,
starting from the fundamental work of Siegel. Our approach works uniformly for
the lattices of arbitrary signature (r,s) and hopefully gives a more conceptual
way of deriving the above known results.Comment: 15 pages, to appear in J. Number Theor
A quantum logical and geometrical approach to the study of improper mixtures
We study improper mixtures from a quantum logical and geometrical point of
view. Taking into account the fact that improper mixtures do not admit an
ignorance interpretation and must be considered as states in their own right,
we do not follow the standard approach which considers improper mixtures as
measures over the algebra of projections. Instead of it, we use the convex set
of states in order to construct a new lattice whose atoms are all physical
states: pure states and improper mixtures. This is done in order to overcome
one of the problems which appear in the standard quantum logical formalism,
namely, that for a subsystem of a larger system in an entangled state, the
conjunction of all actual properties of the subsystem does not yield its actual
state. In fact, its state is an improper mixture and cannot be represented in
the von Neumann lattice as a minimal property which determines all other
properties as is the case for pure states or classical systems. The new lattice
also contains all propositions of the von Neumann lattice. We argue that this
extension expresses in an algebraic form the fact that -alike the classical
case- quantum interactions produce non trivial correlations between the
systems. Finally, we study the maps which can be defined between the extended
lattice of a compound system and the lattices of its subsystems.Comment: submitted to the Journal of Mathematical Physic
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