65 research outputs found
Towards Strong Banach property (T) for SL(3,R)
We prove that SL(3,R) has Strong Banach property (T) in Lafforgue's sense
with respect to the Banach spaces that are interpolation spaces (for
the complex interpolation method) between an arbitrary Banach space and a
Banach space with sufficiently good type and cotype. As a consequence, every
action of SL(3,R) or its lattices by affine isometries on such a Banach space X
has a fixed point, and the expanders contructed from SL(3,Z) do not admit a
coarse embedding into X. We also prove a quantitative decay of matrix
coefficients (Howe-Moore property) for representations with small exponential
growth of SL(3,R) on X.
This class of Banach spaces contains many superreflexive spaces and some
nonreflexive spaces as well. We see no obstruction for this class to be equal
to all spaces with nontrivial type.Comment: 29 pages, 3 figures. Final version, to appear in Israel journal of
math. v3: introduction shortened and small changes according to referee's
suggestions. Also, I found a gap in the proof of Lemma 3.5 of v2. This Lemma
was not used in the paper and was therefore removed from v3. But this Lemma
is true, and the interested reader can find a correct proof due to Pisier in
arXiv:1403.641
A local characterization of Kazhdan projections and applications
We give a local characterization of the existence of Kazhdan projections for
arbitary families of Banach space representations of a compactly generated
locally compact group . We also define and study a natural generalization of
the Fell topology to arbitrary Banach space representations of a locally
compact group. We give several applications in terms of stability of rigidity
under perturbations. Among them, we show a Banach-space version of the
Delorme--Guichardet theorem stating that property (T) and (FH) are equivalent
for -compact locally compact groups. Another kind of applications is
that many forms of Banach strong property (T) are open in the space of marked
groups, and more generally every group with such a property is a quotient of a
compactly presented group with the same property. We also investigate the
notions of central and non central Kazhdan projections, and present examples of
non central Kazhdan projections coming from hyperbolic groups.Comment: 31 pages. v2: small changes to the introduction. Added a discussion
on the speed of convergence, and on a notion of positivity for Kazhdan
constants (p 14). This version was submitted to a journal v3: small changes
to the presentation, background details added on Banach space geometry.
Accepted for publication to Commentarii Mathematici Helvetic
On norms taking integer values on the integer lattice
We present a new proof of Thurston's theorem that the unit ball of a seminorm
on taking integer values on is a polyhedra
defined by finitely many inequalities with integer coefficients.Comment: Bilingual french-english note; 3 pages, 1 figur
Non commutative Lp spaces without the completely bounded approximation property
For any 1\leq p \leq \infty different from 2, we give examples of
non-commutative Lp spaces without the completely bounded approximation
property. Let F be a non-archimedian local field. If p>4 or p<4/3 and r\geq 3
these examples are the non-commutative Lp-spaces of the von Neumann algebra of
lattices in SL_r(F) or in SL_r(\R). For other values of p the examples are the
non-commutative Lp-spaces of the von Neumann algebra of lattices in SL_r(F) for
r large enough depending on p.
We also prove that if r \geq 3 lattices in SL_r(F) or SL_r(\R) do not have
the Approximation Property of Haagerup and Kraus. This provides examples of
exact C^*-algebras without the operator space approximation property.Comment: v3; Minor corrections according to the referee
Complete boundedness of the Heat Semigroups on the von Neumann Algebra of hyperbolic groups
We prove that defines a
completely bounded semigroup of multipliers on the von Neuman algebra of
hyperbolic groups for all real number . One ingredient in the proof is the
observation that a construction of Ozawa allows to characterize the radial
multipliers that are bounded on every hyperbolic graph, partially generalizing
results of Haagerup--Steenstrup--Szwarc and Wysocza\'nski. Another ingredient
is an upper estimate of trace class norms for Hankel matrices, which is based
on Peller's characterization of such norms.Comment: v2: 28 pages, with new examples, new results, motivations and
hopefully a better presentatio
Invariant means for the wobbling group
Given a metric space , the wobbling group of is the group of
bijections satisfying . We study algebraic and analytic properties of in
relation with the metric space structure of , such as amenability of the
action of the lamplighter group on and property (T).Comment: 8 pages. v3: final version, with new presentation; to appear in the
Bulletin of the BM
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