Mapping in normed linear spaces and characterization of orthogonality problem of best approximations in 2-norm.
Mappings of continuous functions on hyper-Stonean spaces
Mappings of finite distortion: composition operator.
Mappings of positive integers and subspaces of m
Mappings preserving zero products
Let θ : ℳ → 𝓝 be a zero-product preserving linear map between algebras. We show that under some mild conditions θ is a product of a central element and an algebra homomorphism. Our result applies to matrix algebras, standard operator algebras, C*-algebras and W*-algebras.
Maps between Banach function algebras satisfying certain norm conditions
Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let and be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ℂ{0, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB, respectively. We show that if ‖S(p)T(p′)−α‖Y=‖ρ(p)τ(p′) − α‖x for all p ∈ I and p′ ∈ I′, then there exist a real algebra isomorphism S: A → B, a clopen subset K of M B and...
Maps on idempotent operators
The set of all bounded linear idempotent operators on a Banach space X is a poset with the partial order defined by P ≤ Q if PQ = QP = P. Another natural relation on the set of idempotent operators is the orthogonality relation defined by P ⊥ Q ⇔ PQ = QP = 0. We briefly survey known theorems on maps on idempotents preserving order or orthogonality. We discuss some related results and open problems. The connections with physics, geometry, theory of automorphisms, and linear preserver problems will...
Maps preserving convergence of series
Maps preserving zero products
A linear map T from a Banach algebra A into another B preserves zero products if T(a)T(b) = 0 whenever a,b ∈ A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T: A → B that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras. Our method involves continuous bilinear maps ϕ: A × A → X (for some Banach space X) with the property...
Maps which preserve equality of distance
Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series
Let , where, for 1 ≤ r < ∞, (resp., ) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values, the condition...
Marcinkiewicz spaces, commutators and non-commutative geometry
Nigel J. Kalton was one of the most eminent guests participating in the Józef Marcinkiewicz Centenary Conference. His contribution to the scientific aspect of the meeting was very essential. Nigel was going to prepare a paper based on his plenary lecture. The editors are completely sure that the paper would be a real ornament of the Proceedings. Unfortunately, Nigel's sudden death totally destroyed editors' hopes and plans. Every mathematician knows how unique were Nigel's mathematical achievements....
Markoff property of generalized random fields
Markov chains as Evans-Hudson diffusions in Fock space
Markov Chains, Riesz Transforms and Lipschitz Maps.
Markov convexity and local rigidity of distorted metrics
It is shown that a Banach space admits an equivalent norm whose modulus of uniform convexity has power-type if and only if it is Markov -convex. Counterexamples are constructed to natural questions related to isomorphic uniform convexity of metric spaces, showing in particular that tree metrics fail to have the dichotomy property.
Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory
Markov operators on the space of vector measures; coloured fractals
We consider the family 𝓜 of measures with values in a reflexive Banach space. In 𝓜 we introduce the notion of a Markov operator and using an extension of the Fortet-Mourier norm we show some criteria of the asymptotic stability. Asymptotically stable Markov operators can be used to construct coloured fractals.
Markov traces on universal Jones algebras and subfactors of finite index.
Markovian processes on mutually commuting von Neumann algebras
The aim of this paper is to study markovianity for states on von Neumann algebras generated by the union of (not necessarily commutative) von Neumann subagebras which commute with each other. This study has been already begun in [2] using several a priori different notions of noncommutative markovianity. In this paper we assume to deal with the particular case of states which define odd stochastic couplings (as developed in [3]) for all couples of von Neumann algebras involved. In this situation...