14 research outputs found
On Borel equivalence relations related to self-adjoint operators
In a recent work, the authors studied various Borel equivalence relations
defined on the Polish space of all (not necessarily bounded)
self-adjoint operators on a separable infinite-dimensional Hilbert space .
In this paper we study the domain equivalence relation
given by
and
determine its exact Borel complexity: is an
(but not ) equivalence relation which is continuously
bireducible with the orbit equivalence relation
of the standard Borel group
on
. This, by Rosendal's Theorem, shows that
is universal for equivalence
relations. Moreover, we show that generic self-adjoint operators have purely
singular continuous spectrum equal to .Comment: 10 pages, added more detail of the proof of Proposition 3.8 after the
referee's suggestio
The complexity of classifying separable Banach spaces up to isomorphism
It is proved that the relation of isomorphism between separable Banach spaces
is a complete analytic equivalence relation, i.e., that any analytic
equivalence relation Borel reduces to it. Thus, separable Banach spaces up to
isomorphism provide complete invariants for a great number of mathematical
structures up to their corresponding notion of isomorphism. The same is shown
to hold for (1) complete separable metric spaces up to uniform homeomorphism,
(2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to
(complemented) biembeddability, (4) Polish groups up to topological
isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the
constructions rely on methods recently developed by S. Argyros and P. Dodos
On the number of permutatively inequivalent basic sequences in a Banach space
AbstractLet X be a Banach space with a Schauder basis (en)n∈N. The relation E0 is Borel reducible to permutative equivalence between normalized block-sequences of (en)n∈N or X is c0 or ℓp saturated for some 1⩽p<+∞. If (en)n∈N is shrinking unconditional then either it is equivalent to the canonical basis of c0 or ℓp, 1<p<+∞, or the relation E0 is Borel reducible to permutative equivalence between sequences of normalized disjoint blocks of X or of X∗. If (en)n∈N is unconditional, then either X is isomorphic to ℓ2, or X contains 2ω subspaces or 2ω quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases
Relative Primeness and Borel Partition Properties for Equivalence Relations
We introduce a notion of relative primeness for equivalence relations, strengthening the notion of non-reducibility, and show for many standard benchmark equivalence relations that non-reducibility may be strengthened to relative primeness. We introduce several analogues of cardinal properties for Borel equivalence relations, including the notion of a prime equivalence relation and Borel partition properties on quotient spaces. In particular, we introduce a notion of Borel weak compactness, and characterize partition properties for the equivalence relations 2 and 1. We also discuss dichotomies related to primeness, and see that many natural questions related to Borel reducibility of equivalence relations may be viewed in the framework of relative primeness and Borel partition properties