Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras

A nonlinear map φ between operator algebras is said to be a numerical radius isometry if w(φ(T−S))=w(T−S) for all T, S in its domain algebra, where w(T) stands for the numerical radius of T. Let and be two atomic nests on complex Hilbert spaces H and K, respectively. Denote the nest algebra associated with and the diagonal algebra. We give a thorough classification of weakly continuous numerical radius isometries from onto and a thorough classification of numerical radius isometries from onto .     

Additive Jordan isomorphisms of nest algebras on normed spaces

Let X be a real or complex Banach space. Let and be two nest algebras on X. Suppose that φ is an additive bijective mapping from onto such that φ(A²)=φ(A)² for every . Then φ is either a ring isomorphism or a ring anti-isomorphism. Moreover, if X is a real space or an infinite dimensional complex space, then there exists a continuous (conjugate) linear bijective mapping T such that either φ(A)=TAT⁻¹ for every or φ(A)=TA∗T−1 for every .     

Some classes of non-analytic Markov semigroups

We deal with Markov semigroups T_t corresponding to second order elliptic operators Au=Δu+〈Du,F〉, where F is an unbounded locally Lipschitz vector field on . We obtain new conditions on F under which T_t is not analytic in . In particular, we prove that the one-dimensional operator Au=u″−x³u′, with domain , , is not sectorial in . Under suitable hypotheses on the growth of F, we introduce a class of non-analytic Markov semigroups in , where μ is an invariant measure for T_t.     

The Bourgain ?-index of mixed Tsirelson space

Suppose that is a sequence of regular families of finite subsets of such that contains all singletons, and (θ_n)_n=1^∞ is a nonincreasing null sequence in (0,1). The mixed Tsirelson space is the completion of c₀₀ with respect to the implicitly defined norm

     

Covering numbers of different points in Dvoretzky covering

Consider the Dvoretzky random covering on the circle T with a decreasing length sequence {?n}_n?1 such that . We study, for a given β?0, the set Fβ of points which are asymptotically covered by a number βLn of the first n randomly placed intervals where . Three typical situations arise, delimited by two “phase transitions”, according to is zero, positive-finite or infinite, where . More precisely, if ?n tends to zero rapidly enough so that then, with probability one, dimHFβ=1 for all β?0; if ?n is moderate so that then, with probability one, we have for and Fβ=∅ for where and is the interval consisting of β's such that ; eventually, if ?n is so slow that then, with probability one, F₁=T. This solves a problem raised by L. Carleson in a rather satisfactory fashion.Analogous results are obtained for the Poisson covering of the line, which is studied as a tool.     

Trace-preserving homomorphisms of semigroups

We consider those homomorphisms φ of semigroups of trace-class operators on a Hilbert space that preserve trace. If φ is a spatially induced isomorphism on a semigroup , that is φ(S)T=TS for an invertible operator T and for all S in , then φ clearly has this property. More generally, if T in the relation above is a densely defined, closed, injective operator with dense image, φ still preserves trace. We prove the converse of this statement under certain conditions. Using these results we prove simultaneous similarity theorems for semigroups of operators (on finite or infinite-dimensional spaces) whose members are individually similar to unitary or J-unitary operators.     

On the product system of a completely positive semigroup

Given a W∗-continuous semigroup φ of unital, normal, completely positive maps of B(H), we introduce its continuous tensor product system . If α is a minimal dilation E₀-semigroup of φ with Arveson product system F, then is canonically isomorphic to F. We apply this construction to a class of semigroups of arising from a modified Weyl-Moyal quantization of convolution semigroups of Borel probability measures on . This class includes the heat flow on the CCR algebra studied recently by Arveson. We prove that the minimal dilations of all such semigroups are completely spatial, and additionally, we prove that the minimal dilation of the heat flow is cocyle conjugate to the CAR/CCR flow of index two.     

Removable singularities for a Sobolev space

Let Ω⊂RN be an open set and F a relatively closed subset of Ω. We show that if the (N−1)-dimensional Hausdorff measure of F is finite, then the spaces and coincide, that is, F is a removable singularity for . Here is the closure of in H¹(Ω) and H¹(Ω) denotes the first order Sobolev space. We also give a relative capacity criterium for this removability. The space is important for defining realizations of the Laplacian with Neumann and with Robin boundary conditions. For example, if the boundary of Ω has finite (N−1)-dimensional Hausdorff measure, then our results show that we may replace Ω by the better set (which is regular in topology), i.e., Neumann boundary conditions (respectively Robin boundary conditions) on Ω and on coincide.     

A new majorization between functions, polynomials, and operator inequalities

Let P₊ be the set of all non-negative operator monotone functions defined on [0,∞), and put . Then and . For a function and a strictly increasing function h we write if is operator monotone. If and and if and , then . We will apply this result to polynomials and operator inequalities. Let and be non-increasing sequences, and put for t≧a₁ and for t≧b₁. Then v₊?u₊ if m≦n and : in particular, for a sequence of orthonormal polynomials, (p_n-1)₊?(p_n)₊. Suppose 0<r,p and s=0 or 1≦s≦1+p/r. Then 0≦A≦B implies for 0<α≦r/(p+r).     

Perturbations of subalgebras of type II1 factors

In this paper we consider two von Neumann subalgebras and of a type II₁ factor . For a map φ on , we define

     

Projective Schur groups of Henselian fields

One of the open questions that has emerged in the study of the projective Schur group of a field F is whether or not is an algebraic relative Brauer group over F, i.e. does there exist an algebraic extension L/F such that ? We show that the same question for the Schur group of a number field has a negative answer. For the projective Schur group, no counterexample is known. In this paper we prove that is an algebraic relative Brauer group for all Henselian valued fields F of equal characteristic whose residue field is a local or global field. For this, we first show how is determined by for an equicharacteristic Henselian field with arbitrary residue field k.     

Enumeration of isomorphism classes of extensions of p-adic fields

Let Ω be an algebraic closure of and let F be a finite extension of contained in Ω. Given positive integers f and e, the number of extensions K/F contained in Ω with residue degree f and ramification index e was computed by Krasner. This paper is concerned with the number of F-isomorphism classes of such extensions. We determine completely when p2?e and get partial results when p2||e. When s is large, is equal to the number of isomorphism classes of finite commutative chain rings with residue field , ramification index e, and length s.