Abstract:
Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of non-commutative integration are established. {}.

Abstract:
Hypercontractivity of a quantum dynamical semigroup has strong implications for its convergence behavior and entropy decay rate. A logarithmic Sobolev inequality and the corresponding logarithmic Sobolev constant can be inferred from the semigroup's hypercontractive norm bound. We consider completely-positive quantum mechanical semigroups described by a Lindblad master equation. To prove the norm bound, we follow an approach which has its roots in the study of classical rate equations. We use interpolation theorems for non-commutative $L_p$ spaces to obtain a general hypercontractive inequality from a particular $p \rightarrow q$-norm bound. Then, we derive a bound on the $2 \rightarrow 4$-norm from an analysis of the block diagonal structure of the semigroup's spectrum. We show that the dynamics of an $N$-qubit graph state Hamiltonian weakly coupled to a thermal environment is hypercontractive. As a consequence this allows for the efficient preparation of graph states in time ${\rm poly}(\log(N))$ by coupling at sufficiently low temperature. Furthermore, we extend our results to gapped Liouvillians arising from a weak linear coupling of a free-fermion systems.

Abstract:
We show that a four-parameter class of 3+1 dimensional NCOS theories can be obtained by dimensional reduction on a general 2-torus from OM theory. Compactifying two spatial directions of NCOS theory on a 2-torus, we study the transformation properties under the $SO(2,2;Z)$ T-duality group. We then discuss non-perturbative configurations of non-commutative super Yang-Mills theory. In particular, we calculate the tension for magnetic monopoles and (p,q) dyons and exhibit their six-dimensional origin, and construct a supergravity solution representing an instanton in the gauge theory. We also compute the potential for a monopole-antimonopole in the supergravity approximation.

Abstract:
These notes were written following lectures I had the pleasure of giving on this subject at Keio University, during November and December 2004. The first part is about new applications of Jordan algebras to the geometry of Hermitian symmetric spaces and to causal semi-simple symmetric spaces of Cayley type. The second part will present new contributions for studing (non commutative) Hardy spaces of holomorphic functions on Lie semi-groups which is a part of the so called Gelfand-Gindikin program.

Abstract:
We introduce the and vector spaces of holomorphic functions defined in the unit ball of , generalizing previous work like Ouyang et al. (1998), Stroethoff (1989), and Choa et al. (1992). Likewise, we characterize those spaces in terms of harmonic majorants as a generalization of Arellano et al. (2000). 1. Introduction 1.1. Preliminaries in One Complex Variable Let be the open unit disk in the complex plane . For , let be the M？bius transformation defined by For , we denote Green's function of with logarithmic singularity at by The Bloch space is defined as the set of analytic functions , such that For , Aulaskari and Lappan [1] introduced in 1994 the spaces as the family of analytic functions satisfying For , this definition was extended in [2]. Theorem 1 (see [2]). Let and be an analytic function. Then, if and only if With the aim of generalizing and enclosing several weighted function spaces, Zhao in [3] introduced the spaces in the next way. Let , , , and be an analytic function. We will say that belongs to if satisfies the integral condition So far, Theorem 1 is generalized for functions of by the following. Theorem 2 (see [3]). Let , , , and be an analytic function. Then, if and only if Zhao has shown that for certain intervals of , , and , makes as a Banach space. In this paper, we present the spaces of holomorphic functions in the unit ball of , that generalize the spaces introduced by Zhao in [3] for analytic functions in the unit disk. At the same time, this work generalizes, mainly, several results of Ouyang et al for -holomorphic functions appearing in [4]. However, the techniques, the methods, and the structure of our work results are completely different to the quoted reference. At the beginning of Section 2, we introduce the spaces and , that is, when we use in the integral representation, the invariant Green function, or when we use the biholomorphism . We remark that the generalization to several complex variables of requires, for and , different intervals to those used in the one-dimensional case. In Theorems 12 and 13, we draw attention to the continuity of the integral expressions defining these spaces to conclude easily the inclusions of the little classes and in and , respectively. In Section 2.2, we present what is the most natural Bloch space associated to and several characterizations that we give in Theorem 22. It is important to compare this statement with the results of [4, Proposition 3.6] and [5, Theorem 2.4]. In Section 3, we present in Corollary 30 the equivalence between and , generalizing Proposition 3.4 of [4] and

Abstract:
We prove that every reduced Stein space admits a holomorphic function without critical points. Furthermore, any closed discrete subset of such a space is the critical locus of a holomorphic function. We also show that for every complex analytic stratification with nonsingular strata on a reduced Stein space there exists a holomorphic function whose restriction to every stratum is noncritical. These result also provide some information on critical loci of holomorphic functions on desingularizations of Stein spaces. In particular, every 1-convex manifold admits a holomorphic function that is noncritical outside the exceptional variety.

Abstract:
We show that for a hypoelliptic Dirichlet form operator A on a stratified complex Lie group, if the logarithmic Sobolev inequality holds, then a holomorphic projection of A is strongly hypercontractive in the sense of Janson. This extends previous results of Gross to a setting in which the operator A is not holomorphic.

Abstract:
We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex space admitting an exhaustion function whose Levi form has at least two positive eigenvalues at every point outside a compact set, and this condition is essential. The proof involves a lifting method for the boundary of the curve and a newly developed technique of gluing holomorphic sprays over Cartan pairs in Stein manifolds whose value lie in a complex space, with control up to the boundary of the domains. (The latter technique is also exploited in the subsequent papers math.CV/0607185 and math.CV/0609706.) We also prove that any compact complex curve with C^2 boundary in a complex space admits a basis of open Stein neighborhoods. In particular, an embedded disc of class C^2 with holomorphic interior in a complex manifold admits a basis of open polydisc neighborhoods.

Abstract:
The notions of almost periodicity in the sense of Weyl and Besicovitch of the order p are extended to holomorphic functions on a strip. We prove that the spaces of holomorphic almost periodic functions in the sense of Weyl for various orders p are the same. These spaces are considerably wider than the space of holomorphic uniformly almost periodic functions and considerably narrower than the spaces of holomorphic almost periodic functions in the sense of Besicovitch. Besides we construct examples showing that the spaces of holomorphic almost periodic functions in the sense of Besicovitch for various orders p are all different.

Abstract:
We introduce the notion of numerical (strong) peak function and investigate the denseness of the norm and numerical peak functions on complex Banach spaces. Let $A_b(B_X:X)$ be the Banach space of all bounded continuous functions $f$ on the unit ball $B_X$ of a Banach space $X$ and their restrictions $f|_{B_X^\circ}$ to the open unit ball are holomorphic. In finite dimensional spaces, we show that the intersection of the set of all norm peak functions and the set of all numerical peak functions is a dense $G_\delta$ subset of $A_b(B_X:X)$. We also prove that if $X$ is a smooth Banach space with the Radon-Nikod\'ym property, then the set of all numerical strong peak functions is dense in $A_b(B_X:X)$. In particular, when $X=L_p(\mu)$ $(1