2,581 research outputs found
Syndetic Sets and Amenability
We prove that if an infinite, discrete semigroup has the property that every
right syndetic set is left syndetic, then the semigroup has a left invariant
mean. We prove that the weak*-closed convex hull of the two-sided translates of
every bounded function on an infinite discrete semigroup contains a constant
function. Our proofs use the algebraic properties of the Stone-Cech
compactification
A Dynamical Systems Approach to the Kadison-Singer Problem
In these notes we develop a link between the Kadison-Singer problem and
questions about certain dynamical systems. We conjecture that whether or not a
given state has a unique extension is related to certain dynamical properties
of the state. We prove that if any state corresponding to a minimal idempotent
point extends uniquely to the von Neumann algebra of the group, then every
state extends uniquely to the von Neumann algebra of the group. We prove that
if any state arising in the Kadsion-Singer problem has a unique extension, then
the injective envelope of a C*-crossed product algebra associated with the
state necessarily contains the full von Neumann algebra of the group. We prove
that this latter property holds for states arising from rare ultrafilters and
-stable ultrafilters, independent, of the group action and also for
states corresponding to non-recurrent points in the corona of the group.Comment: Typos corrected, comments and references adde
Equivariant Maps and Bimodule Projections
We construct a contractive, idempotent, MASA bimodule map on B(H), whose
range is not a ternary subalgebra of B(H). Our method uses a crossed-product to
reduce the existence of such an idempotent map to an analogous problem about
the ranges of idempotent maps that are equivariant with respect to a group
action and Hamana's theory of G-injective envelopes.Comment: 15 page
Complete positivity of the map from a basis to its dual basis
The dual of a matrix ordered space has a natural matrix ordering that makes
the dual space matrix ordered as well. The purpose of these notes is to give a
condition that describes when the linear map taking a basis of the n by n
matrices to its dual basis is a complete order isomorphism and complete
co-order isomorphism. In the case of the standard matrix units this map is a
complete order isomorphism and this is a restatement of the correspondence
between completely positive maps and the Choi matrix. However, we exhibit
natural orthonormal bases for the matrices such that this map is an order
isomorphism, but not a complete order isomorphism. Some bases yield complete
co-order isomorphisms. Included among such bases is the Pauli basis and tensor
products of the Pauli basis. Consequently, when the Pauli basis is used in
place of the the matrix unit basis, the analogue of Choi's theorem is a
characterization of completely co-positive maps
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