353 research outputs found
On a question of Abraham Robinson's
In this note we give a negative answer to Abraham Robinson's question whether
a finitely generated extension of an undecidable field is always undecidable.
We construct 'natural' undecidable fields of transcendence degree 1 over Q all
of whose proper finite extensions are decidable. We also construct undecidable
algebraic extensions of Q that allow decidable finite extensions
Two New Settings for Examples of von Neumann Dimension
Let , let be a lattice in , and let
be an irreducible unitary representation of with
square-integrable matrix coefficients. A theorem in [Goodman, de la Harpe,
Jones 1989] states that the von Neumann dimension of as a
-module is equal to the formal dimension of the discrete series
representation times the covolume of , calculated with
respect to the same Haar measure. We prove two results inspired by this
theorem. First, we show there is a representation of on a subspace
of cuspidal automorphic functions in , where
and are lattices in ; and this representation is
unitarily equivalent to one of the representations in [Goodman, de la Harpe,
Jones 1989]. Next, we calculate von Neumann dimensions when is ,
for a local non-archimedean field of characteristic with residue field
of order not divisible by 2; is a torsion-free lattice in ,
which, by a theorem of Ihara, is a free group; and is the
Steinberg representation, or a depth-zero supercuspidal representation, each
yielding a different dimension.Comment: This is the author's Ph.D. thesi
Shapiroʼs Theorem for subspaces
AbstractIn the previous paper (Almira and Oikhberg, 2010 [4]), the authors investigated the existence of an element x of a quasi-Banach space X whose errors of best approximation by a given approximation scheme (An) (defined by E(x,An)=infa∈An‖x−an‖) decay arbitrarily slowly. In this work, we consider the question of whether x witnessing the slowness rate of approximation can be selected in a prescribed subspace of X. In many particular cases, the answer turns out to be positive
Kadison -Singer algebras with applications to von Neumann algebras
I develop the theory of Kadison-Singer algebras, introduced recently by Ge and Yuan. I prove basic structure theorems, construct several new examples and explore connections to other areas of operator algebras. In chapter 1, I survey those aspects of the theory of non-selfadjoint algebras that are relevant to this work. In chapter 2, I define Kadison-Singer algebras and give different proofs of results of Ge-Yuan, which will be further extended in the last chapter. In chapter 3, I analyse in detail a class of elementary Kadison-Singer algebras that contain Hinfinity and describe their lattices of projections. In chapter 4, I use ideas from free probability theory to construct Kadison-Singer algebras with core the free group factors L(Fr) for r \u3c 2. I then introduce two constructions that yield new Kadison-Singer algebras - The maximal join and the minimal join. In chapter 5, I analyse tensor products of Kadison-Singer algebras, showing that they are never Kadison-Singer. I then show how under certain conditions, one may construct a Kadison-Singer algebra with core the tensor product of the core of two given Kadison-Singer algebras
Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture
Let be a real algebraic group acting equivariantly with finitely many
orbits on a real algebraic manifold and a real algebraic bundle
on . Let be the Lie algebra of . Let
be the space of Schwartz sections of
. We prove that is a
closed subspace of of finite codimension.
We give an application of this result in the case when is a real
spherical subgroup of a real reductive group . We deduce an equivalence of
two old conjectures due to Casselman: the automatic continuity and the
comparison conjecture for zero homology. Namely, let be a
Casselman-Wallach representation of and be the corresponding
Harish-Chandra module. Then the natural morphism of coinvariants
is an isomorphism if and only if any
linear -invariant functional on is continuous in the topology
induced from . The latter statement is known to hold in two important
special cases: if includes a symmetric subgroup, and if includes the
nilradical of a minimal parabolic subgroup of .Comment: v4: version appearing in Math. Z + erratum added in the en
The classification of bifurcations with hidden symmetries
We set up a singularity-theoretic framework for classifying one-parameter steady-state bifurcations with hidden symmetries. This framework also permits a non-trivial linearization at the bifurcation point. Many problems can be reduced to this situation; for instance, the bifurcation of steady or periodic solutions to certain elliptic partial differential equations with Neumann or Dirichlet boundary conditions. We formulate an appropriate equivalence relation with its associated tangent spaces, so that the usual methods of singularity theory become applicable. We also present an alternative method for computing those matrix-valued germs that appear in the equivalence relations employed in the classification of equivariant bifurcation problems. This result is motivated by hidden symmetries appearing in a class of partial differential equations defined on an N-dimensional rectangle under Neumann boundary conditions
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