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20
The complexity of classifying separable Banach spaces up to isomorphism
, 2006
"... 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 mathematica ..."
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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.
Turbulence, orbit equivalence, and the classification of nuclear C ∗ algebras. Journal für die reine und angewandte Mathematik
, 2011
"... Abstract. We bound the Borel cardinality of the isomorphism relation for nuclear simple separable C ∗algebras: It is turbulent, yet Borel reducible to the action of the automorphism group of the Cuntz algebra O2 on its closed subsets. The same bounds are obtained for affine homeomorphism of metriza ..."
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Abstract. We bound the Borel cardinality of the isomorphism relation for nuclear simple separable C ∗algebras: It is turbulent, yet Borel reducible to the action of the automorphism group of the Cuntz algebra O2 on its closed subsets. The same bounds are obtained for affine homeomorphism of metrizable Choquet simplexes. As a byproduct we recover a result of Kechris and Solecki, namely, that homeomorphism of compacta in the Hilbert cube is Borel reducible to a Polish group action. These results depend intimately on the classification theory of nuclear simple C ∗algebras by Ktheory and traces. Both of necessity and in order to lay the groundwork for further study on the Borel complexity of C ∗algebras, we prove that many standard C ∗algebra constructions and relations are Borel, and we prove Borel versions of Kirchberg’s O2stability and embedding theorems. We also find a C ∗algebraic witness for a Kσ hard equivalence relation. The authors dedicate this article to the memory of Greg Hjorth.
Completeness of the isomorphism problem for separable C*algebras
, 2013
"... We prove that the isomorphism problem for separable nuclear C*algebras is complete in the class of orbit equivalence relations. In fact, already the isomorphism of simple, separable AI C*algebras is a complete orbit equivalence relation. This means that any isomorphism problem arsing from a cont ..."
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We prove that the isomorphism problem for separable nuclear C*algebras is complete in the class of orbit equivalence relations. In fact, already the isomorphism of simple, separable AI C*algebras is a complete orbit equivalence relation. This means that any isomorphism problem arsing from a continuous action of a separable completely metrizable group can be reduced to the isomorphism of simple, separable AI C*algebras. This sheds new light on the classification problem for separable C*algebras, which dates back to the 1960’s. In particular, this answers questions posed by Elliott, Farah, Hjorth, Paulsen, Rosendal, Toms and Törnquist.
ON THE COMPLEXITY OF THE QUASIISOMETRY AND VIRTUAL ISOMORPHISM PROBLEMS FOR FINITELY GENERATED GROUPS
"... Abstract. We study the Borel complexity of the quasiisometry and virtual isomorphism problems for the class of finitely generated groups. 1. ..."
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Abstract. We study the Borel complexity of the quasiisometry and virtual isomorphism problems for the class of finitely generated groups. 1.
Analytic equivalence relations and biembeddability
 J. Symbolic Logic
"... Abstract. Louveau and Rosendal [5] have shown that the relation of biembeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphi ..."
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Abstract. Louveau and Rosendal [5] have shown that the relation of biembeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of Lω1ω) is far from complete (see [5, 2]). In this article we strengthen the results of [5] by showing that not only does biembeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and biembeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of Lω1ω, but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids. 1.
Isomorphism Relations on Computable Structures
, 2011
"... We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of F Freducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all Σ 1 1 equivalence relations on hyperarithmetical subsets of ..."
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Cited by 4 (0 self)
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We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of F Freducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all Σ 1 1 equivalence relations on hyperarithmetical subsets of ω.
Isomorphism and BiEmbeddability Relations on Computable Structures
, 2010
"... We study the complexity of natural equivalence relations on classes of computable structures such as isomorphism and biembeddability. We use the notion of tcreducibility to show completeness of the isomorphism relation on many familiar classes in the context of all Σ 1 1 equivalence relations on h ..."
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Cited by 4 (2 self)
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We study the complexity of natural equivalence relations on classes of computable structures such as isomorphism and biembeddability. We use the notion of tcreducibility to show completeness of the isomorphism relation on many familiar classes in the context of all Σ 1 1 equivalence relations on hyperarithmetical subsets of ω. We also show that the biembeddability relation on an appropriate hyperarithmetical class of computable structures may have the same complexity as any given Σ 1 1 equivalence relation on ω.
SELECTED APPLICATIONS OF LOGIC TO CLASSIFICATION PROBLEM FOR C*ALGEBRAS
"... 1.1. Subspaces and subalgebras of B(H) 4 ..."
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On the complexity of the relations of isomorphism and biembeddability
"... Abstract. Given an Lω1ωelementary class C, that is the collection of the countable models of some Lω1ωsentence, denote by ∼=C and ≡C the analytic equivalence relations of, respectively, isomorphism and biembeddability on C. Generalizing some questions of Louveau and Rosendal [9], in [3] it was pr ..."
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Abstract. Given an Lω1ωelementary class C, that is the collection of the countable models of some Lω1ωsentence, denote by ∼=C and ≡C the analytic equivalence relations of, respectively, isomorphism and biembeddability on C. Generalizing some questions of Louveau and Rosendal [9], in [3] it was proposed the problem of determining which pairs of analytic equivalence relations (E,F) can be realized (up to Borelequivalence) as pairs of the form (∼=C,≡C), C some Lω1ωelementary class (together with a partial answer for some specific cases). Here we will provide an almost complete solution to such problem: under very mild conditions on E and F, it is always possible to find such an Lω1ωelementary class C. 1.
Complexity and homogeneity in Banach spaces
, 2007
"... We provide an overview of a number of results concerning the complexity of isomorphism between separable Banach spaces. We also include some new results on the lattice structure of the set of spreading models of a Banach space. ..."
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Cited by 3 (2 self)
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We provide an overview of a number of results concerning the complexity of isomorphism between separable Banach spaces. We also include some new results on the lattice structure of the set of spreading models of a Banach space.