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Maysam Maysami Sadr

Publications and source records attributed to Maysam Maysami Sadr.

At least 19 recordsLinked to original sources
arXiv2024

Dynamical entropy of probability measures on infinite product spaces

The aim of this note is to introduce a notion of dynamical entropy, which we call infinite-product entropy, for probability measures on (countable) infinite cartesian product of any measurable space with itself. The idea behind the definition is that any infinite product space may be considered as a type of dynamical object. We have considered in a previous note a similar idea in topological dynamics to define a notion of dynamical entropy for arbitrary subsets of infinite products of compact topological spaces. We consider some basic properties of infinite-product entropy, e.g. shift invariance, convexity, subadditivity with respect to product of probability measures, the behavior with respect to dilation and restriction. We show that for a translation invariant probability measure the infinite-product entropy coincides with the usual entropy of a shift transformation. We consider some basic examples and computations. We also consider a variational inequality related to infinite-product entropy and topological entropy of subsets of infinite product spaces.

math.PR↗
arXiv2023

On the Differential Geometry of Some Classes of Infinite Dimensional Manifolds

Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space $Γ_X$ of any manifold $X$. The name comes from the fact that various elements of the geometry of $Γ_X$ are constructed via lifting of the corresponding elements of the geometry of $X$. In this note, we construct a general algebraic framework for lifted geometry which can be applied to various ``infinite dimensional spaces'' associated to $X$. In order to define a lifted geometry for a ``space'', one dose not need any topology or local coordinate system on the space. As example and application, lifted geometry for spaces of Radon measures on $X$, mappings into $X$, embedded submanifolds of $X$, and tilings on $X$, are considered. The gradient operator in the lifted geometry of Radon measures is considered. Also, the construction of a natural Dirichlet form associated to a Random measure is discussed. It is shown that Stokes' Theorem appears as ``differentiability'' of ``boundary operator'' in the lifted geometry of spaces of submanifolds. It is shown that (generalized) action functionals associated with Lagrangian densities on $X$ form the algebra of smooth functions in a specific lifted geometry for the path-space of $X$.

math.DG↗
arXiv2023

A modification of Hardy-Littlewood maximal-function on Lie groups

For a real-valued function $f$ on a metric measure space $(X,d,μ)$ the Hardy-Littlewood maximal-function of $f$ is given by the following `supremum-norm': $$Mf(x):=\sup_{r>0}\frac{1}{μ(\mathcal{B}_{x,r})}\int_{\mathcal{B}_{x,r}}|f|dμ.$$ In this note, we replace the supremum-norm on parameters $r$ by $\mathcal{L}_p$-norm with weight $w$ on parameters $r$ and define Hardy-Littlewood integral-function $I_{p,w}f$. It is shown that $I_{p,w}f$ converges pointwise to $Mf$ as $p\to\infty$. Boundedness of the sublinear operator $I_{p,w}$ and continuity of the function $I_{p,w}f$ in case that $X$ is a Lie group, $d$ is a left-invariant metric, and $μ$ is a left Haar-measure (resp. right Haar-measure) are studied.

math.FA↗
arXiv2022

Random Quantum Maps and Their Associated Quantum Markov Chains

The notion of `quantum family of maps' (QFM) has been defined by Piotr Soltan as a noncommutative analogue of `parameterized family of continuous maps' between locally compact spaces. A QFM between C*-algebras $B,A$, is given by a pair $(C,ϕ)$ where $C$ is a C*-algebra and $ϕ:B\rightarrow A\check{\otimes}C$ is a $*$-morphism. The main goal of this note, is to introduce the notion of `random quantum map' (RQM), which is a noncommutative analogue of `random continuous map' between compact spaces. We define a RQM between $B,A$, to be given by a triple $(C,ϕ,ν)$ where $(C,ϕ)$ is a QFM and $ν$ a state (normalized positive linear functional) on $C$. Our first application of RQMs takes place in theory of completely positive maps (CPM): RQMs give rise canonically to a class of CPMs which we call implemented CPMs. We consider some partial results about the natural and important problem of characterization of implemented CPMs. For instance, using Stinespring's Theorem, we show that any CPM from $B$ to $A$ is implemented if $A$ is finite-dimensional. Our second application of RQMs takes place in theory of quantum stochastic processes: We show that iterations of any RQM with $B=A$, gives rise to a quantum Markov chain in a sense introduced by Luigi Accardi.

math.OA↗
arXiv2022

A Remark on Contractible Banach Algebras of Operators

For a Banach algebra $A$, we say that an element $M$ in $A\otimes^γA$ is a hyper-commutator if $(a\otimes 1)M=M(1\otimes a)$ for every $a\in A$. A diagonal for a Banach algebra is a hyper-commutator which its image under diagonal mapping is $1$. It is well-known that a Banach algebra is contractible iff it has a diagonal. The main aim of this note is to show that for any Banach subalgebra $A\subseteq\mathcal{L}(X)$ of bounded linear operators on infinite-dimensional Banach space $X$, which contains the ideal of finite-rank operators, the image of any hyper-commutator of $A$ under the canonical algebra-morphism $\mathcal{L}(X)\otimes^γ\mathcal{L}(X)\to\mathcal{L}(X\otimes^γX)$, vanishes.

math.FA↗
arXiv2022

Path-Connected Components of Affine Schemes and Algebraic K-Theory

We introduce a functor $\mathfrak{M}:\mathbf{Alg}\times\mathbf{Alg}^\mathrm{op}\rightarrow\mathrm{pro}\text{-}\mathbf{Alg}$ constructed from representations of $\mathrm{Hom}_\mathbf{Alg}(A,B\otimes ? )$. As applications, the following items are introduced and studied: (i) Analogue of the functor $π_0$ for algebras and affine schemes. (ii) Cotype of Weibel's concept of strict homotopization. (iii) A homotopy invariant intrinsic singular cohomology theory for affine schemes with cup product. (iv) Some extensions of $\mathbf{Alg}$ that are enriched over idempotent semigroups. (v) Classifying homotopy pro-algebras for Cortiñas-Thom's KK-groups and Weibel's homotopy K-groups.

math.KT↗
arXiv2020

Topological Entropy for Arbitrary Subsets of Infinite Product Spaces

In this note a notion of generalized topological entropy for arbitrary subsets of the space of all sequences in a compact topological space is introduced. It is shown that for a continuous map on a compact space the generalized topological entropy of the set of all orbits of the map coincides with the classical topological entropy of the map. Some basic properties of this new notion of entropy are considered; among them are: the behavior of the entropy with respect to disjoint union, cartesian product, component restriction and dilation, shift mapping, and some continuity properties with respect to Vietoris topology. As an example, it is shown that any self-similar structure of a fractal given by a finite family of contractions gives rise to a notion of intrinsic topological entropy for subsets of the fractal. A generalized notion of Bowen's entropy associated to any increasing sequence of compatible semimetrics on a topological space is introduced and some of its basic properties are considered. As a special case for $1\leq p\leq\infty$ the Bowen $p$-entropy of sets of sequences of any metric space is introduced. It is shown that the notions of generalized topological entropy and Bowen $\infty$-entropy for compact metric spaces coincide.

math.DS↗
arXiv2020

Abstract Hardy-Littlewood Maximal Inequality

In this note besides two abstract versions of the Vitali Covering Lemma an abstract Hardy-Littlewood Maximal Inequality, generalizing weak type (1,1) maximal function inequality, associated to any outer measure and a family of subsets on a set is introduced. The inequality is (effectively) satisfied if and only if a special numerical constant called Hardy-Littelwood maximal constant is finite. Two general sufficient conditions for the finiteness of this constant are given and upper bounds for this constant associated to the family of (centered) balls in homogeneous spaces, family of dyadic cubes in Euclidean spaces, family of admissible trapezoids in homogeneous trees, and family of Calderón-Zygmund sets in (ax+b)-group, are considered. Also a very simple application to find some nontrivial estimates about mass density in Classical Mechanics is given.

math.FA↗
arXiv2019

Morita Equivalence of Brandt Semigroup Algebras

We prove that for every group $G$ and any two sets $I,J$, the Brandt semigroup algebras $\ell(B(I,G))$ and $\ell(B(J,G))$ are Morita equivalent with respect to the Morita theory of self-induced Banach algebras introduced by Gronbaek. As applications, we show that if $G$ is an amenable group, then for a wide class of Banach $\ell(B(I,G))$-bimodules $E$, and every $n>0$, the bounded Hochschild cohomology groups $H^n(\ell(B(I,G)),E^*)$ are trivial, and also, the notion of approximate amenability is not Morita invariant.

math.FA↗
arXiv2019

Continuous Bounded Cohomology of Topological Semigroups

In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach space. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.

math.FA↗
arXiv2019

Banach Algebras Associated to Metric Operator Fields

Motivated by noncommutative geometry and quantum physics, the concept of `metric operator field' is introduced. Roughly speaking, a metric operator field is a vector field on a set with values in self tensor product of a bundle of C*-algebras, satisfying properties similar to an ordinary metric (distance function). It is proved that to any such object there naturally correspond a Banach *-algebra that we call Lipschitz algebra, a class of probabilistic metrics, and (under some conditions) a (nontrivial) continuous field of C*-algebras in the sense of Dixmier. It is proved that for metric operator fields with values in von Neumann algebras the associated Lipschitz algebras are dual Banach spaces, and under some conditions, they are not amenable Banach algebras. Some examples and constructions are considered. We also discuss very briefly a possible application to quantum gravity.

math.OA↗
arXiv2019

Quantum metric spaces of quantum maps

We show that any quantum family of maps from a non commutative space to a compact quantum metric space has a canonical quantum semi metric structure.

math.OA↗
arXiv2019

On the structure of noncommutative mapping schemes

The following three types of objects are considered in a dual functorial formalism: (i) ind-scheme of mappings between two schemes, (ii) for a quantum group G, ind-scheme of G-mappings between two G-schemes, and (iii) ind-scheme of group homomorphisms between two quantum group. By schemes and quantum groups here we mean objects which are respectively dual to unital associative algebras and Hopf algebras.

math.AG↗
arXiv2019

A Note on the Frankl Conjecture

The Frankl conjecture (called also union-closed sets conjecture) is one of the famous unsolved conjectures in combinatorics of finite sets. In this short note, we introduce and to some extent justify some variants of the Frankl conjecture.

math.CO↗
arXiv2019

A note on the best approximation in spaces of affine functions

The proximinality of certain subspaces of spaces of bounded affine functions is proved. The results presented here are some linear versions of an old result due to Mazur. For the proofs we use some sandwich theorems of Fenchel's duality theory.

math.FA↗
arXiv2019

Some Inequalities for Continuous Algebra-Multiplications on a Banach Space

In this short note, we first consider some inequalities for comparison of some algebraic properties of two continuous algebra-multiplications on an arbitrary Banach space and then, as an application, we consider some very basic observations on the space of all continuous algebra-multiplications on a Banach space.

math.FA↗