Unital affine semigroups

Affine semigroups are convex sets on which there exists an associative binary operation which is affine separately in either variable. They were introduced by Cohen and Collins in 1959. We look at examples of affine semigroups which are of interest to matrix and operator theory and we prove some new results on the extreme points and the absorbing elements of certain types of affine semigroups. Most notably we improve a result of Wendel that every invertible element in a compact affine semigroup is extreme by extending this result to linearly bounded affine semigroups.     

Representations of product systems over semigroups and dilations of commuting CP maps

We prove that every pair of commuting CP maps on a von Neumann algebra M can be dilated to a commuting pair of endomorphisms (on a larger von Neumann algebra). To achieve this, we first prove that every completely contractive representation of a product system of C^∗-correspondences over the semigroup N² can be dilated to an isometric (or Toeplitz) representation.     

A continuity theorem for Stinespring's dilation

We show a continuity theorem for Stinespring's dilation: two completely positive maps between arbitrary C^∗-algebras are close in cb-norm if and only if we can find corresponding dilations that are close in operator norm. The proof establishes the equivalence of the cb-norm distance and the Bures distance for completely positive maps. We briefly discuss applications to quantum information theory.     

Fourier algebras on homogeneous spaces

Spectral synthesis and operator synthesis on a homogeneous space G/K, where K is a compact subgroup of a locally compact group G, are studied. Injection theorem for sets of spectral synthesis for A(G/K) is proved, extending the classical result of Reiter and more recent results of Kaniuth–Lau, Parthasarathy–Prakash and others. A simple direct image theorem for spectral synthesis is proved and an extension of the subgroup theorem and an alternate proof of the injection theorem are obtained as consequences. The relation between synthesis in the Fourier algebra A(G/K) and an appropriate Varopoulos algebra is obtained, subsuming earlier results of Varopoulos, Spronk–Turowska and Parthasarathy–Prakash. Study of relations between spectral synthesis and operator synthesis pioneered by Arveson and carried forward recently by Shulman–Turowska, Parthasarathy–Prakash and Ludwig–Turowska is undertaken on homogeneous spaces. Operator space methods are needed for this study, and more specifically, a characterisation of completely bounded multipliers on A(G/K) as the invariant part of a suitable weak^? Haagerup tensor product (or the space of Schur multipliers) is given and is used for this study.     

Semigroups of Herz–Schur multipliers

In order to investigate the relationship between weak amenability and the Haagerup property for groups, we introduce the weak Haagerup property, and we prove that having this approximation property is equivalent to the existence of a semigroup of Herz–Schur multipliers generated by a proper function (see Theorem 1.2). It is then shown that a (not necessarily proper) generator of a semigroup of Herz–Schur multipliers splits into a positive definite kernel and a conditionally negative definite kernel. We also show that the generator has a particularly pleasant form if and only if the group is amenable. In the second half of the paper we study semigroups of radial Herz–Schur multipliers on free groups. We prove that a generator of such a semigroup is linearly bounded by the word length function (see Theorem 1.6).     

On Representations Associated with Completely n-Positive Linear Maps on Pro-C*-Algebras

It is shown that an n × n matrix of continuous linear maps from a pro-C*-algebra A to L(H), which verifies the condition of complete positivity, is of the formlinear operator from H to K, and [Tij]ni,j=1 is a positive element in the C*-algebra of all Suen in Proc. Amer. Math. Soc., 112(3), 1991, 709-712. Also, a covariant version of this construction is given.     

矩阵的Schur补的性态

本文讨论了一些常见矩阵如正定阵、完全主正阵等的Ｓｃｈｕｒ补的性质。     

Completely p-summing maps on the operator Hilbert space OH

We prove the completely p-summing ideals of OH are all equal as sets for 1?p<2. A phase transition then occurs at p=2 as we also show for p?2, the completely p-summing ideals of OH turn out as sets to be Schatten ideal classes with the limiting case being the Schatten 4-class ideal S₄ when p→∞.     

Factorization Index for Completely Positive Graphs

In this paper, we prove that the set of all factorization indices of a completely positive graph has no gaps. In other words, we give an affirmative answer to a question raised by N. Kogan and A. Berman [8] in the case of completely positive graphs. Received December 9, 1997, Accepted May 16, 2002     

A simplified completely positive reformulation for binary quadratic programs

In this paper, we propose a simplified completely positive programming reformulation for binary quadratic programs. The linear equality constraints associated with the binary constraints in the original problem can be aggregated into a single linear equality constraint without changing the feasible set of the classic completely positive reformulation proposed in the literature. We also show that the dual of the proposed simplified formulation is strictly feasible under a mild assumption.     

复正定矩阵的充要条件

本文研究Hermite正定矩阵、实正定矩阵的推广概念复正定矩阵,给出了复方阵复正定性的一些充分必要条件。     

The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let M be a von Neumann algebra equipped with a normal faithful semifinite trace τ, and let E be an r.i. space on (0,∞). Let E(M) be the associated symmetric space of measurable operators. Then to any bounded linear map T from E(M) into a Hilbert space H corresponds a positive norm one functional f∈E(2)^∗(M) such that

     

Kolmogorov’s existence theorem for Markov processes inC* algebras

Given a family of transition probability functions between measure spaces and an initial distribution Kolmogorov’s existence theorem associates a unique Markov process on the product space. Here a canonical non-commutative analogue of this result is established for families of completely positive maps betweenC* algebras satisfying the Chapman-Kolmogorov equations. This could be the starting point for a theory of quantum Markov processes. Dedicated to the memory of Professor K G Ramanathan