53 research outputs found
Perturbations of nuclear C*-algebras
Kadison and Kastler introduced a natural metric on the collection of all
C*-subalgebras of the bounded operators on a separable Hilbert space. They
conjectured that sufficiently close algebras are unitarily conjugate. We
establish this conjecture when one algebra is separable and nuclear. We also
consider one-sided versions of these notions, and we obtain embeddings from
certain near inclusions involving separable nuclear C*-algebras. At the end of
the paper we demonstrate how our methods lead to improved characterisations of
some of the types of algebras that are of current interest in the
classification programme.Comment: 45 page
Relative commutants of strongly self-absorbing C*-algebras
The relative commutant of a strongly self-absorbing
algebra is indistinguishable from its ultrapower . This
applies both to the case when is the hyperfinite II factor and to the
case when it is a strongly self-absorbing C*-algebra. In the latter case we
prove analogous results for and reduced powers
corresponding to other filters on . Examples of algebras with
approximately inner flip and approximately inner half-flip are provided,
showing the optimality of our results. We also prove that strongly
self-absorbing algebras are smoothly classifiable, unlike the algebras with
approximately inner half-flip.Comment: Some minor correction
Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras
The assignment (nonstable K_0-theory), that to a ring R associates the monoid
V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices
with only finitely nonzero entries over R, extends naturally to a functor. We
prove the following lifting properties of that functor: (1) There is no functor
F, from simplicial monoids with order-unit with normalized positive
homomorphisms to exchange rings, such that VF is equivalent to the identity.
(2) There is no functor F, from simplicial monoids with order-unit with
normalized positive embeddings to C*-algebras of real rank 0 (resp., von
Neumann regular rings), such that VF is equivalent to the identity. (3) There
is a {0,1}^3-indexed commutative diagram D of simplicial monoids that can be
lifted, with respect to the functor V, by exchange rings and by C*-algebras of
real rank 1, but not by semiprimitive exchange rings, thus neither by regular
rings nor by C*-algebras of real rank 0. By using categorical tools from an
earlier paper (larders, lifters, CLL), we deduce that there exists a unital
exchange ring of cardinality aleph three (resp., an aleph three-separable
unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence
2, such that V(R) is the positive cone of a dimension group and V(R) is not
isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0
or a regular ring.Comment: 34 pages. Algebras and Representation Theory, to appea
An exact duality theory for semidefinite programming based on sums of squares
Farkas' lemma is a fundamental result from linear programming providing
linear certificates for infeasibility of systems of linear inequalities. In
semidefinite programming, such linear certificates only exist for strongly
infeasible linear matrix inequalities. We provide nonlinear algebraic
certificates for all infeasible linear matrix inequalities in the spirit of
real algebraic geometry: A linear matrix inequality is infeasible if and only
if -1 lies in the quadratic module associated to it. We also present a new
exact duality theory for semidefinite programming, motivated by the real
radical and sums of squares certificates from real algebraic geometry.Comment: arXiv admin note: substantial text overlap with arXiv:1108.593
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