Completely positive maps of order zero

We say a completely positive contractive map between two C*-algebras has order zero, if it sends orthogonal elements to orthogonal elements. We prove a structure theorem for such maps. As a consequence, order zero maps are in one-to-one correspondence with *-homomorphisms from the cone over the domain into the target algebra. Moreover, we conclude that tensor products of order zero maps are again order zero, that the composition of an order zero map with a tracial functional is again a tracial functional, and that order zero maps respect the Cuntz relation, hence induce ordered semigroup morphisms between Cuntz semigroups.Comment: 13 page

Rokhlin dimension and C*-dynamics

We develop the concept of Rokhlin dimension for integer and for finite group actions on C∗-algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of Z-stable C∗-algebras, where Z denotes the Jiang–Su algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve Z-stability. In topological dynamics our notion may be interpreted as a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. The latter result has by now been generalized by Szabó to the case of free and aperiodic Zd -actions on compact metrizable and finite dimensional spaces

Threshold phenomenon for the quintic wave equation in three dimensions

For the critical focusing wave equation $\Box u = u^5$ on $\R^{3+1}$ in the radial case, we establish the role of the "center stable" manifold $\Sigma$ constructed in \cite{KS} near the ground state $(W,0)$ as a threshold between type I blowup and scattering to zero, establishing a conjecture going back to numerical work by Bizo\'n, Chmaj, Tabor. The underlying topology is stronger than the energy norm

Global dynamics of the nonradial energy-critical wave equation above the ground state energy

In this paper we establish the existence of certain classes of solutions to the energy critical nonlinear wave equation in dimensions 3 and 5 assuming that the energy exceeds the ground state energy only by a small amount. No radial assumption is made. We find that there exist four sets in the natural energy space with nonempty interiors which correspond to all possible combinations of finite-time blowup on the one hand, and global existence and scattering to a free wave, on the other hand, as time approaches infinity. In our previous paper arxiv:1010.3799 we treated the radial case, and this paper provides the natural nonradial extension of these results. However, the present paper is self-contained and in fact develops a somewhat different formalism in order to handle the more complex nonradial situation.Comment: arXiv admin note: substantial text overlap with arXiv:1010.3799. This is the final version as it appears in Discrete and Continuous Dynamical Systems Volume 33, Issue 6, 2013 Pages 2423-2450. Some minor misprints have been correcte

Global dynamics above the ground state energy for the one-dimensional NLKG equation

In this paper we obtain a global characterization of the dynamics of even solutions to the one-dimensional nonlinear Klein-Gordon (NLKG) equation on the line with focusing nonlinearity |u|^{p-1}u, p>5, provided their energy exceeds that of the ground state only sightly. The method is the same as in the three-dimensional case arXiv:1005.4894, the major difference being in the construction of the center-stable manifold. The difficulty there lies with the weak dispersive decay of 1-dimensional NLKG. In order to address this specific issue, we establish local dispersive estimates for the perturbed linear Klein-Gordon equation, similar to those of Mizumachi arXiv:math/0605031. The essential ingredient for the latter class of estimates is the absence of a threshold resonance of the linearized operator

Slow blow-up solutions for the H^1(R^3) critical focusing semi-linear wave equation in R^3

We prove the existence of energy solutions of the energy critical focusing wave equation in R^3 which blow up exactly at x=t=0. They decompose into a bulk term plus radiation term. The bulk is a rescaled version of the stationary "soliton" type solution of the NLW. The construction depends crucially on the renormalization procedure of the "soliton" which we introduced in our companion paper on the wave map problem.Comment: 38 page

Concentration compactness for critical wave maps

By means of the concentrated compactness method of Bahouri-Gerard and Kenig-Merle, we prove global existence and regularity for wave maps with smooth data and large energy from 2+1 dimensions into the hyperbolic plane. The argument yields an apriori bound of the Coulomb gauged derivative components of our wave map relative to a suitable norm (which holds the solution) in terms of the energy alone. As a by-product of our argument, we obtain a phase-space decomposition of the gauged derivative components analogous to the one of Bahouri-Gerard.Comment: 261 pages, 7 figure