25,520 research outputs found
A weak*-topological dichotomy with applications in operator theory
Denote by the locally compact Hausdorff space consisting of
all countable ordinals, equipped with the order topology, and let
be the Banach space of scalar-valued, continuous functions
which are defined on and vanish eventually. We show that a
weakly compact subset of the dual space of is either
uniformly Eberlein compact, or it contains a homeomorphic copy of the ordinal
interval .
Using this result, we deduce that a Banach space which is a quotient of
can either be embedded in a Hilbert-generated Banach space,
or it is isomorphic to the direct sum of and a subspace of a
Hilbert-generated Banach space. Moreover, we obtain a list of eight equivalent
conditions describing the Loy-Willis ideal, which is the unique maximal ideal
of the Banach algebra of bounded, linear operators on . As a
consequence, we find that this ideal has a bounded left approximate identity,
thus resolving a problem left open by Loy and Willis, and we give new proofs,
in some cases of stronger versions, of several known results about the Banach
space and the operators acting on it.Comment: accepted to Transactions of the London Mathematical Societ
Noncommutative Lattices and Their Continuum Limits
We consider finite approximations of a topological space by
noncommutative lattices of points. These lattices are structure spaces of
noncommutative -algebras which in turn approximate the algebra \cc(M) of
continuous functions on . We show how to recover the space and the
algebra \cc(M) from a projective system of noncommutative lattices and an
inductive system of noncommutative -algebras, respectively.Comment: 22 pages, 8 Figures included in the LaTeX Source New version, minor
modifications (typos corrected) and a correction in the list of author
Lie groups in nonequilibrium thermodynamics: Geometric structure behind viscoplasticity
Poisson brackets provide the mathematical structure required to identify the
reversible contribution to dynamic phenomena in nonequilibrium thermodynamics.
This mathematical structure is deeply linked to Lie groups and their Lie
algebras. From the characterization of all the Lie groups associated with a
given Lie algebra as quotients of a universal covering group, we obtain a
natural classification of rheological models based on the concept of discrete
reference states and, in particular, we find a clear-cut and deep distinction
between viscoplasticity and viscoelasticity. The abstract ideas are illustrated
by a naive toy model of crystal viscoplasticity, but similar kinetic models are
also used for modeling the viscoplastic behavior of glasses. We discuss some
implications for coarse graining and statistical mechanics.Comment: 11 pages, 1 figure, accepted for publication in J. Non-Newtonian
Fluid Mech. Keywords: Elastic-viscoplastic materials, Nonequilibrium
thermodynamics, GENERIC, Lie groups, Reference state
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