173 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
Operator Algebras of Functions
We present some general theorems about operator algebras that are algebras of
functions on sets, including theories of local algebras, residually finite
dimensional operator algebras and algebras that can be represented as the
scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use
these to further develop a quantized function theory for various domains that
extends and unifies Agler's theory of commuting contractions and the
Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous
factorization theorems, prove that the algebras that we obtain are dual
operator algebras and show that for many domains, supremums over all commuting
tuples of operators satisfying certain inequalities are obtained over all
commuting tuples of matrices.Comment: 33 page
Vector spaces with an order unit
We develop a theory of ordered *-vector spaces with an order unit. We prove
fundamental results concerning positive linear functionals and states, and we
show that the order (semi)norm on the space of self-adjoint elements admits
multiple extensions to an order (semi)norm on the entire space. We single out
three of these (semi)norms for further study and discuss their significance for
operator algebras and operator systems. In addition, we introduce a functorial
method for taking an ordered space with an order unit and forming an
Archimedean ordered space. We then use this process to describe an appropriate
notion of quotients in the category of Archimedean ordered spaces.Comment: 38 pages, uses XY-pic, Version 2 comments: minor typos corrected.;
Version 3 Comments: minor typos corrected; Version 4 Comments: minor typos
corrected, hypothesis of Archimedean added to Theorem 4.22, To appear in
Indiana Univ. Math.
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