Strongly ergodic equivalence relations: spectral gap and type III invariants – CORRIGENDUM

2020 ◽  
pp. 1-2
Author(s):  
CYRIL HOUDAYER ◽  
AMINE MARRAKCHI ◽  
PETER VERRAEDT
Keyword(s):  
Spectral Gap ◽  
Equivalence Relations ◽  
Type Iii ◽  
Ergodic Equivalence Relations

Strongly ergodic equivalence relations: spectral gap and type III invariants

2017 ◽  
Vol 39 (7) ◽  
pp. 1904-1935 ◽  
Author(s):  
CYRIL HOUDAYER ◽  
AMINE MARRAKCHI ◽  
PETER VERRAEDT
Keyword(s):  
Equivalence Relation ◽  
Spectral Gap ◽  
Structure Theorem ◽  
Equivalence Relations ◽  
Almost Periodic ◽  
Von Neumann ◽  
Measure Spaces ◽  
Compact Abelian Groups ◽  
Usual Topology ◽  
Ergodic Equivalence Relations

We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product equivalence relations arising from measurable $1$-cocycles with values in locally compact abelian groups are strongly ergodic. By analogy with the work of Connes on full factors, we introduce the Sd and $\unicode[STIX]{x1D70F}$ invariants for type $\text{III}$ strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type $\text{III}_{1}$ ergodic equivalence relation ${\mathcal{R}}$, the Maharam extension $\text{c}({\mathcal{R}})$ is strongly ergodic if and only if ${\mathcal{R}}$ is strongly ergodic and the invariant $\unicode[STIX]{x1D70F}({\mathcal{R}})$ is the usual topology on $\mathbb{R}$. We also obtain a structure theorem for almost periodic strongly ergodic equivalence relations analogous to Connes’ structure theorem for almost periodic full factors. Finally, we prove that for arbitrary strongly ergodic free actions of bi-exact groups (e.g. hyperbolic groups), the Sd and $\unicode[STIX]{x1D70F}$ invariants of the orbit equivalence relation and of the associated group measure space von Neumann factor coincide.

Orbital factor map

1993 ◽  
Vol 13 (3) ◽  
pp. 515-532 ◽  
Author(s):  
Toshihiro Hamachi ◽  
Hideki Kosaki
Keyword(s):  
Index Theory ◽  
The Other ◽  
Equivalence Relations ◽  
Index Formula ◽  
Type Ii ◽  
Type Iii ◽  
Factor Map ◽  
Ergodic Equivalence Relations ◽  
Basic Extension

AbstractFor a certain pair of discrete measured ergodic equivalence relations, we specify how one is imbedded into the other and obtain a factor and its subfactor. For this pair of factors index theory is developed. We obtain an index formula, an ‘extended’ relation corresponding to the basic extension, and so on. Our formulas as well as construction are very explicit and are given in terms of ‘measure theoretic’ data. We also present examples showing the contrast between type III0 index theory and type II1 (or IIIλ) index theory.

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.

On full groups of non-ergodic probability-measure-preserving equivalence relations

2015 ◽  
Vol 36 (7) ◽  
pp. 2218-2245 ◽  
Author(s):  
FRANÇOIS LE MAÎTRE
Keyword(s):  
Probability Measure ◽  
Operator Algebras ◽  
Baire Category ◽  
Equivalence Relations ◽  
Continuity Property ◽  
Algebraic Characterization ◽  
Full Group ◽  
Automatic Continuity ◽  
Ergodic Equivalence Relations ◽  
The Cost

This article generalizes our previous results [Le Maître. The number of topological generators for full groups of ergodic equivalence relations. Invent. Math. 198 (2014), 261–268] to the non-ergodic case by giving a formula relating the topological rank of the full group of an aperiodic probability-measure-preserving (pmp) equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups that have a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to provide an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing a result of Giordano and Pestov to the non-ergodic case [Giordano and Pestov. Some extremely amenable groups related to operator algebras and ergodic theory. J. Inst. Math. Jussieu6(2) (2007), 279–315, Theorem 5.7].

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