11,241 research outputs found
The Isomorphism Relation Between Tree-Automatic Structures
An -tree-automatic structure is a relational structure whose domain
and relations are accepted by Muller or Rabin tree automata. We investigate in
this paper the isomorphism problem for -tree-automatic structures. We
prove first that the isomorphism relation for -tree-automatic boolean
algebras (respectively, partial orders, rings, commutative rings, non
commutative rings, non commutative groups, nilpotent groups of class n >1) is
not determined by the axiomatic system ZFC. Then we prove that the isomorphism
problem for -tree-automatic boolean algebras (respectively, partial
orders, rings, commutative rings, non commutative rings, non commutative
groups, nilpotent groups of class n >1) is neither a -set nor a
-set
Automatic closure of invariant linear manifolds for operator algebras
Kadison's transitivity theorem implies that, for irreducible representations
of C*-algebras, every invariant linear manifold is closed. It is known that CSL
algebras have this propery if, and only if, the lattice is hyperatomic (every
projection is generated by a finite number of atoms). We show several other
conditions are equivalent, including the conditon that every invariant linear
manifold is singly generated.
We show that two families of norm closed operator algebras have this
property. First, let L be a CSL and suppose A is a norm closed algebra which is
weakly dense in Alg L and is a bimodule over the (not necessarily closed)
algebra generated by the atoms of L. If L is hyperatomic and the compression of
A to each atom of L is a C*-algebra, then every linear manifold invariant under
A is closed. Secondly, if A is the image of a strongly maximal triangular AF
algebra under a multiplicity free nest representation, where the nest has order
type -N, then every linear manifold invariant under A is closed and is singly
generated.Comment: AMS-LaTeX, 15 pages, minor revision
Continuous fields of C*-algebras over finite dimensional spaces
Let be a finite dimensional compact metrizable space. We study a
technique which employs semiprojectivity as a tool to produce approximations of
-algebras by -subalgebras with controlled complexity. The following
applications are given. All unital separable continuous fields of C*-algebras
over with fibers isomorphic to a fixed Cuntz algebra ,
are locally trivial. They are trivial if or
. For finite, such a field is trivial if and only if
in , where is the C*-algebra of continuous sections
of the field.
We give a complete list of the Kirchberg algebras satisfying the UCT and
having finitely generated K-theory groups for which every unital separable
continuous field over with fibers isomorphic to is automatically
(locally) trivial.
In a more general context, we show that a separable unital continuous field
over with fibers isomorphic to a -semiprojective is trivial if and only
if it satisfies a K-theoretical Fell type condition.Comment: 31 page
Some Automatic Continuity Theorems for Operator Algebras and Centralizers of Pedersen's Ideal
We prove automatic continuity theorems for "decomposable" or "local" linear
transformations between certain natural subspaces of operator algebras. The
transformations involved are not algebra homomorphisms but often are module
homomorphisms. We show that all left (respectively quasi-) centralizers of the
Pedersen ideal of a C*-algebra A are locally bounded if and only if A has no
infinite dimensional elementary direct summand. It has previously been shown by
Lazar and Taylor and Phillips that double centralizers of Pedersen's ideal are
always locally bounded.Comment: A slightly revised version of this paper is being published by
Integral Equations and Operator Theory (published online on October 11 2016
Dualizability of automatic algebras
We make a start on one of George McNulty's Dozen Easy Problems: "Which finite
automatic algebras are dualizable?" We give some necessary and some sufficient
conditions for dualizability. For example, we prove that a finite automatic
algebra is dualizable if its letters act as an abelian group of permutations on
its states. To illustrate the potential difficulty of the general problem, we
exhibit an infinite ascending chain of finite automatic algebras that are alternately dualizable and
non-dualizable
Measure continuous derivations on von Neumann algebras and applications to L^2-cohomology
We prove that norm continuous derivations from a von Neumann algebra into the
algebra of operators affiliated with its tensor square are automatically
continuous for both the strong operator topology and the measure topology.
Furthermore, we prove that the first continuous L^2-Betti number scales
quadratically when passing to corner algebras and derive an upper bound given
by Shen's generator invariant. This, in turn, yields vanishing of the first
continuous L^2-Betti number for II_1 factors with property (T), for finitely
generated factors with non-trivial fundamental group and for factors with
property Gamma.Comment: 17 page
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