A generalisation to Birkhoff - von Neumann theorem

@article{Paunescu2015AGT,
  title={A generalisation to Birkhoff - von Neumann theorem},
  author={Liviu Paunescu and Florin Radulescu},
  journal={arXiv: Functional Analysis},
  year={2015}
}
The classic Birkhoff- von Neumann theorem states that the set of doubly stochastic matrices is the convex hull of the permutation matrices. In this paper, we study a generalisation of this theorem in the type $II_1$ setting. Namely, we replace a doubly stochastic matrix with a collection of measure preserving partial isomorphisms, of the unit interval, with similar properties. We show that a weaker version of this theorem still holds. 
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References

SHOWING 1-10 OF 17 REFERENCES
Birkhoff’s problem 111
It is well known that the doubly stochastic matrices of order n are precisely the convex combinations of permutation matrices of order n. Problem 111 of Garrett Birkhoff'sLattice theory asks for an
Doubly stochastic measures with prescribed support
SummaryDenote the set of doubly stochastic measures on the unit square X ×X that are supported on the graphs of measurable maps L,H∶X→X by ℳ(L, H). Conditions are given that imply that ℳ(L, H) is a
Coût des relations d’équivalence et des groupes
Abstract.We study a new dynamical invariant for dicrete groups: the cost. It is a real number in {1−1/n}∪[1,∞], bounded by the number of generators of the group, and it is well behaved with respect
Almost commuting permutations are near commuting permutations
Abstract We prove that the commutator is stable in permutations endowed with the Hamming distance, that is, two permutations that almost commute are near two commuting permutations. Our result
A lemma for cost attained
  • G. Hjorth
  • Mathematics, Computer Science
    Ann. Pure Appl. Log.
  • 2006
Abstract A treeable ergodic equivalence relation of integer cost is generated by a free action of the free group on the corresponding number of generators. Every countable treeable ergodic
Sofic equivalence relations
We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes' Embedding Conjecture as well as the Measurable Determinant Conjecture of Luck, Sauer and Wegner
Descriptive Set Theory
Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary
Hecke Operators and Distributing Points on S 2 . I 1
where p + 1 is the number of rotations in S . A simple example of such a set is S = { A , B , C, A-' , B-', C ' } , where A , B, C are rotations of arccos( $) about the X , Y, Z-axes, respectively.
Topics in orbit equivalence
Preface.- I. Orbit Equivalence.- II. Amenability and Hyperfiniteness.- III. Costs of Equivalence Relations and Groups.- References.- Index.
Classical descriptive set theory
Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the
...
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