162,055 research outputs found
Violator Spaces: Structure and Algorithms
Sharir and Welzl introduced an abstract framework for optimization problems,
called LP-type problems or also generalized linear programming problems, which
proved useful in algorithm design. We define a new, and as we believe, simpler
and more natural framework: violator spaces, which constitute a proper
generalization of LP-type problems. We show that Clarkson's randomized
algorithms for low-dimensional linear programming work in the context of
violator spaces. For example, in this way we obtain the fastest known algorithm
for the P-matrix generalized linear complementarity problem with a constant
number of blocks. We also give two new characterizations of LP-type problems:
they are equivalent to acyclic violator spaces, as well as to concrete LP-type
problems (informally, the constraints in a concrete LP-type problem are subsets
of a linearly ordered ground set, and the value of a set of constraints is the
minimum of its intersection).Comment: 28 pages, 5 figures, extended abstract was presented at ESA 2006;
author spelling fixe
Noncommutative maximal ergodic inequality for non-tracial L1-spaces
We extend the noncommutative L1-maximal ergodic inequality for semifinite von
Neumann algebras established by Yeadon in 1977 to the framework of
noncommutative L1-spaces associated with sigma-finite von Neumann algebras.
Since the semifnite case of this result is one of the two essential parts in
the proof of noncommutative maximal ergodic inequality for tracial Lp-spaces
(1<p<infinity) by Junge-Xu in 2007, we hope our result will be helpful to
establish a complete noncommutative maximal ergodic inequality for non-tracial
Lp-spaces in the future
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
- …