1,561 research outputs found
Strongly ergodic actions have local spectral gap
We show that an ergodic measure preserving action of a discrete group on a -finite measure space
satisfies the local spectral gap property (introduced by Boutonnet,
Ioana and Salehi Golsefidy) if and only if it is strongly ergodic. In fact, we
prove a more general local spectral gap criterion in arbitrary von Neumann
algebras. Using this criterion, we also obtain a short and elementary proof of
Connes' spectral gap theorem for full factors as well as its
recent generalization to full type factors.Comment: 6 page
Fullness of crossed products of factors by discrete groups
Let be an arbitrary factor and an
action of a discrete group. In this paper, we study the fullness of the crossed
product . When is amenable, we obtain a
complete characterization: the crossed product factor
is full if and only if is full and the quotient map has finite kernel and discrete image. This
answers a question of Jones from 1981. When is full and is
arbitrary, we give a sufficient condition for to be
full which generalizes both Jones' criterion and Choda's criterion. In
particular, we show that if is any full factor (possibly of type
) and is a non-inner amenable group, then the crossed
product is full.Comment: 9 pages. arXiv admin note: text overlap with arXiv:1611.0791
Solidity of type III Bernoulli crossed products
We generalize a theorem of Chifan and Ioana by proving that for any, possibly
type III, amenable von Neumann algebra , any faithful normal state
and any discrete group , the associated Bernoulli crossed
product von Neumann algebra
is solid
relatively to . In particular, if is
solid then is solid and if is non-amenable and then is a full prime factor. This gives many new examples of
solid or prime type factors. Following Chifan and Ioana, we also
obtain the first examples of solid non-amenable type equivalence
relations.Comment: 18 page
Stability of products of equivalence relations
An ergodic p.m.p. equivalence relation is said to be stable if
where is
the unique hyperfinite ergodic type equivalence relation. We
prove that a direct product of two ergodic
p.m.p. equivalence relations is stable if and only if one of the two components
or is stable. This result is deduced from a new
local characterization of stable equivalence relations. The similar question on
McDuff factors is also discussed and some partial results are
given.Comment: 14 page
Spectral gap characterization of full type III factors
We give a spectral gap characterization of fullness for type
factors which is the analog of a theorem of Connes in the tracial case. Using
this criterion, we generalize a theorem of Jones by proving that if is a
full factor and is an outer action of
a discrete group whose image in is discrete then the
crossed product von Neumann algebra is also a full factor.
We apply this result to prove the following conjecture of Tomatsu-Ueda: the
continuous core of a type factor is full if and only if
is full and its invariant is the usual topology on .Comment: 13 page
Full factors, bicentralizer flow and approximately inner automorphisms
We show that a factor is full if and only if the -algebra generated
by its left and right regular representations contains the compact operators.
We prove that the bicentralizer flow of a type factor is
always ergodic. As a consequence, for any type factor and
any , there exists an irreducible AFD type
subfactor with expectation in . Moreover, any type
factor which satisfies for
some has trivial bicentralizer. Finally, we give a
counter-example to the characterization of approximately inner automorphisms
conjectured by Connes and we prove a weaker version of this conjecture. In
particular, we obtain a new proof of Kawahigashi-Sutherland-Takesaki's result
that every automorphism of the AFD type factor is
approximately inner.Comment: 16 page
Tensor product decompositions and rigidity of full factors
We obtain several rigidity results regarding tensor product decompositions of
factors. First, we show that any full factor with separable predual has at most
countably many tensor product decompositions up to stable unitary conjugacy. We
use this to show that the class of separable full factors with countable
fundamental group is stable under tensor products. Next, we obtain new
primeness and unique prime factorization results for crossed products coming
from compact actions of higher rank lattices (e.g.\ ) and noncommutative Bernoulli shifts with arbitrary base (not
necessarily amenable). Finally, we provide examples of full factors without any
prime factorization.Comment: 30 page
Full factors and co-amenable inclusions
We show that if is a full factor and is a co-amenable
subfactor with expectation, then is also full. This answers a question of
Popa from 1986. We also generalize a theorem of Tomatsu by showing that if
is a full factor and is an outer action of
a compact group , then is automatically minimal and is a full
factor which has w-spectral gap in . Finally, in the appendix, we give a
proof of the fact that several natural notions of co-amenability for an
inclusion of von Neumann algebras are equivalent, thus closing the
cycle of implications given in Anantharaman-Delaroche's paper in 1995.Comment: 15 pages, Remark 3.6 is adde
Strongly ergodic equivalence relations: spectral gap and type III invariants
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 -cocycles with values into locally compact abelian groups are
strongly ergodic. By analogy with the work of Connes on full factors, we
introduce the Sd and invariants for type strongly ergodic
equivalence relations. As a corollary to our main results, we show that for any
type ergodic equivalence relation , the Maharam
extension is strongly ergodic if and only if
is strongly ergodic and the invariant is the
usual topology on . 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 invariants of the orbit equivalence relation and of the
associated group measure space von Neumann factor coincide.Comment: 28 pages. To appear in Ergodic Theory Dynam. System
Conformal Fourth-Rank Gravity
We consider the consequences of describing the metric properties of space-
time through a quartic line element . The associated "metric" is a fourth-rank tensor
. We construct a theory for the gravitational field
based on the fourth-rank metric which is conformally
invariant in four dimensions. In the absence of matter the fourth-rank metric
becomes of the form
therefore we recover a Riemannian behaviour of the geometry; furthermore, the
theory coincides with General Relativity. In the presence of matter we can keep
Riemannianicity, but now gravitation couples in a different way to matter as
compared to General Relativity. We develop a simple cosmological model based on
a FRW metric with matter described by a perfect fluid. Our field equations
predict that the entropy is an increasing function of time. For the
field equations predict , where ; for
we obtain . can be
estimated from the mean random velocity of typical galaxies to be
. For the early universe there is no violation of
causality for .Comment: 39 pages, plain TE
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