scholarly journals Games with winning conditions of high Borel complexity

2006 ◽  
Vol 350 (2-3) ◽  
pp. 345-372 ◽  
Author(s):  
Olivier Serre
Keyword(s):  
Borel Complexity

scholarly journals The Borel complexity of von Neumann equivalence

2020 ◽  
pp. 102913
Author(s):  
Inessa Moroz ◽  
Asger Törnquist
Keyword(s):  
Von Neumann ◽  
Borel Complexity

scholarly journals THE CLASSIFICATION PROBLEM FOR AUTOMORPHISMS OF C*-ALGEBRAS

2015 ◽  
Vol 21 (4) ◽  
pp. 402-424 ◽  
Author(s):  
MARTINO LUPINI
Keyword(s):  
Complexity Theory ◽  
Classification Problem ◽  
C Algebras ◽  
Borel Complexity ◽  
Recent Developments

AbstractWe present an overview of the recent developments in the study of the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory.

Measure Convex and Measure Extremal Sets

2006 ◽  
Vol 49 (4) ◽  
pp. 536-548 ◽  
Author(s):  
Petr Dostál ◽  
Jaroslav Lukeš ◽  
Jiří Spurný
Keyword(s):  
Convex Sets ◽  
Borel Complexity ◽  
Extremal Sets ◽  
Positive Results

AbstractWe prove that convex sets are measure convex and extremal sets are measure extremal provided they are of low Borel complexity. We also present examples showing that the positive results cannot be strengthened.

Ideals with bases of unbounded Borel complexity

2011 ◽  
Vol 57 (6) ◽  
pp. 582-590 ◽  
Author(s):  
Piotr Borodulin-Nadzieja ◽  
Szymon Gła̧b
Keyword(s):  
Borel Complexity

scholarly journals The Weyl–von Neumann theorem and Borel complexity of unitary equivalence modulo compacts of self-adjoint operators

2015 ◽  
Vol 145 (6) ◽  
pp. 1115-1144 ◽  
Author(s):  
Hiroshi Ando ◽  
Yasumichi Matsuzawa
Keyword(s):  
Equivalence Relation ◽  
Adjoint Operator ◽  
Unitary Equivalence ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Borel Complexity ◽  
Neumann Theorem ◽  
Adjoint Operators ◽  
Von Neumann Theorem ◽  
Descriptive Set

The Weyl–von Neumann theorem asserts that two bounded self-adjoint operators A, B on a Hilbert space H are unitarily equivalent modulo compacts, i.e.uAu* + K = B for some unitary u 𝜖 u(H) and compact self-adjoint operator K, if and only if A and B have the same essential spectrum: σess (A) = σess (B). We study, using methods from descriptive set theory, the problem of whether the above Weyl–von Neumann result can be extended to unbounded operators. We show that if H is separable infinite dimensional, the relation of unitary equivalence modulo compacts for bounded self-adjoint operators is smooth, while the same equivalence relation for general self-adjoint operators contains a dense Gδ-orbit but does not admit classification by countable structures. On the other hand, the apparently related equivalence relation A ~ B ⇔ ∃u 𝜖 U(H) [u(A-i)–1u* - (B-i)–1 is compact] is shown to be smooth.

Borel complexity of isomorphism between quotient Boolean algebras

2008 ◽  
Vol 73 (4) ◽  
pp. 1328-1340
Author(s):  
Su Gao ◽  
Michael Ray Oliver
Keyword(s):  
Equivalence Relation ◽  
Boolean Algebras ◽  
Equivalence Relations ◽  
Borel Set ◽  
Borel Sets ◽  
Opposite Case ◽  
Borel Complexity ◽  
Borel Ideal ◽  
The One ◽  
Ergodic Equivalence Relations

In response to a question of Farah, “How many Boolean algebras are there?” [Far04], one of us (Oliver) proved that there are continuum-many nonisomorphic Boolean algebras of the form with I a Borel ideal on the natural numbers, and in fact that this result could be improved simultaneously in two directions:(i) “Borel ideal” may be improved to “analytic P-ideal”(ii) “continuum-many” may be improved to “E0-many”; that is, E0 is Borel reducible to the isomorphism relation on quotients by analytic P-ideals.See [Oli04].In [AdKechOO], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was to find a collection of, in some sense, strongly mutually ergodic equivalence relations, indexed by reals, and then assign to each Borel set B a sort of “direct sum” of the equivalence relations corresponding to the reals in B. Then if B1, ⊆ B2 it was easy to see that the equivalence relation thus induced by B1 was Borel reducible to the one induced by B2, whereas in the opposite case, taking x to be some element of B / B2, it was possible to show that the equivalence relation corresponding to x, which was part of the equivalence relation induced by B1, was not Borel reducible to the equivalence relation corresponding to B2.

Borel and projective sets from the point of view of compact sets

1983 ◽  
Vol 94 (3) ◽  
pp. 399-409 ◽  
Author(s):  
Kenneth Kunen ◽  
Arnold W. Miller
Keyword(s):  
Metric Spaces ◽  
Polish Space ◽  
Point Of View ◽  
Alternative Proof ◽  
Compact Sets ◽  
Borel Set ◽  
Compact Metric Spaces ◽  
Borel Complexity ◽  
Continuous Surjection

In this paper we prove several results concerning the complexity of a set relative to compact sets. We prove that for any Polish space X and Borel set B ⊆ X, if B is not , then there exists a compact zero-dimensional P ⊆ X such that p ∩ X is not . We also show that it is consistent with ZFC that, for any A ⊆ ωω, if for all compact K ⊆ ωωA ∩ K is , then A is . This generalizes to in place of assuming the consistency of some hypotheses involving determinacy. We give an alternative proof of the following theorem of Saint-Raymond. Suppose X and Y are compact metric spaces and f is a continuous surjection of X onto Y. Then, for any A ⊆ Y, A is in Y iff f−1(A) is in X. The non-trivial part of this result is to show that taking pre-images cannot reduce the Borel complexity of a set. The techniques we use are the definability of forcing and Wadge games.

Sign in / Sign up

Export Citation Format

Share Document