arXiv subjects

Piotr Niemiec

Publications and source records attributed to Piotr Niemiec.

At least 19 recordsLinked to original sources

arXiv2025

Automorphism groups of measures on the Cantor space. Part II: Abstract homogeneous measures

The main aim of the paper is to introduce a new class of (semigroup-valued) measures that are ultrahomogeneous on the Boolean algebra of all clopen subsets of the Cantor space and to study their automorphism groups. A characterisation, in terms of the so-called trinary spectrum of a measure, of ultrahomogeneous measures such that the action of their automorphism groups is (topologically) transitive or minimal is given. Also sufficient and necessary conditions for the existence of a dense (or co-meager) conjugacy class in these groups are offered. In particular, it is shown that there are uncountably many full non-atomic probability Borel measures m on the Cantor space such that m and all its restrictions to arbitrary non-empty clopen sets have all the following properties: this measure is ultrahomogeneous and not good, the action of its automorphism group G is minimal (on a respective clopen set), and G has a dense conjugacy class. It is also shown that any minimal homeomorphism h on the Cantor space induces a homogeneous h-invariant probability measure that is universal among all h-invariant probability measures (which means that it `generates' all such measures) and this property determines this measure (in a certain sense) up to a Q-linear isomorphism.

math.DS↗

arXiv2025

Automorphism groups of measures on the Cantor space. Part I: Good measures and Rokhlin properties

We study criteria for the existence of a dense or comeager conjugacy class in the automorphism group of a given measure on the Cantor space. We concentrate on good measures, defined by Akin [\emph{Trans.\ Amer.\ Math.\ Soc.} \textbf{357} (2005), no. 7, 2681--2722], which we characterize as a particular subclass of ultrahomogeneous measures. We determine good measures with rational values on clopen sets whose automorphism group admits a comeager conjugacy class. Our approach uses the Fraïssé theory.

math.LO↗

arXiv2025

Two-point dilation-homogeneous metric spaces

The main aim of the paper is to give a full classification (up to isometry) of all metric spaces X with the following two properties: X contains a compact set with non-empty interior; and for any three distinct points a, b and c of X there exists a (bijective) dilation on X that fixes a and sends b to c. As a consequence, we obtain a new characterisation of the Euclidean spaces: these are (up to isometry) precisely all metric spaces that have the above two properties, and (in addition) contain three distinct points x, y, z that are metrically collinear (that is, for which d(x,z) = d(x,y)+d(y,z)).

math.MG↗

arXiv2024

Metric duality for Abelian groups

The main aim of the paper is to introduce the concept of metric duality in the category of topological Abelian groups that extends the classical notion of duality for normed vector spaces and behaves quite nicely for LCA groups (equipped with nice metrics). In particular, it is shown that each Polish LCA group admits a reflexive proper metric and, more generally, all LCA groups possess reflexive (proper) metric structures.

math.GR↗

arXiv2023

Extensive approach to absolute homogeneity

The main aim of the paper is to study in greater detail absolutely homogeneous structures (that is, objects with the property that each partial isomorphism extends to a global automorphism), with special emphasis on metric spaces and (possibly infinite, full) graphs with edge-coloring. Besides, a general categorical approach to this concept is presented. The main achievement of the paper is the discovery of one-to-one correspondence between absolutely homogeneous objects and certain classes (that become sets when isomorphic objects are identified) of "finite" objects that satisfy a few quite general axioms (such as amalgamation and heredity). It is also introduced and discussed in detail the concept of products for graphs with edge-coloring (that produces an absolutely homogeneous graph provided all factors are so). Among the most significant results of the paper, it is worth mentioning a full classification (up to isometry) of all absolutely homogeneous ultrametric spaces as well as of all absolutely homogeneous graphs with edge-coloring in which all triangles are isosceles or in which all triangles are (precisely) tricolored.

math.GN↗

arXiv2020

Positive Hankel operators, positive definite kernels and related topics

It is shown that a positive (bounded linear) operator on a Hilbert space with trivial kernel is unitarily equivalent to a Hankel operator that satisfies double positivity condition if and only if it is non-invertible and has simple spectrum (that is, if this operator admits a cyclic vector). More generally, for an arbitrary positive (bounded linear) operator A on a Hilbert space H with trivial kernel the collection V(A) of all linear isometries V from H into H such that AV is positive as well is investigated. In particular, operators A such that V(A) contains a pure isometry with a given deficiency index are characterized. Some applications to unbounded positive self-adjoint operators as well as to positive definite kernels are presented. In particular, positive definite matrix-type square roots of such kernels are studied and kernels that have a unique such root are characterized. The class of all positive definite kernels that have at least one such a square root is also investigated.

math.FA↗

arXiv2020

A note on functions preserving positive definiteness

The aim of the paper is to give a full characterization of functions f from I into the real line R (where I is an interval in R that satisfies certain natural conditions) such that for any I-valued positive definite kernel K defined on an arbitrary set X the kernel formed by composing f with K is positive definite as well.

math.FA↗

arXiv2018

Hyperbolic geometry for non-differential topologists

A soft presentation of hyperbolic spaces, free of differential apparatus, is offered. Fifth Euclid's postulate in such spaces is overthrown and, among other things, it is proved that spheres (equipped with great-circle distances) and hyperbolic and Euclidean spaces are the only locally compact geodesic (i.e., convex) metric spaces that are three-point homogeneous.

math.MG↗

arXiv2017

Applications of amenable semigroups in operator theory

The paper deals with continuous homomorphisms $S \ni s \mapsto T_s \in L(E)$ of amenable semigroups $S$ into the algebra $L(E)$ of all bounded linear operators on a Banach space $E$. For a closed linear subspace $F$ of $E$, sufficient conditions are given under which there exists a projection $P \in L(E)$ onto $F$ that commutes with all $T_s$. And when $E$ is a Hilbert space, sufficient conditions are given for the existence of an invertible operator $R \in L(E)$ such that all $R T_s R^{-1}$ are isometries. Also certain results on extending intertwining operators, renorming as well as on operators on hereditarily indecomposable Banach spaces are offered.

math.FA↗

arXiv2013

Models for subhomogeneous C*-algebras

A new category of topological spaces with additional structures, called m-towers, is introduced. It is shown that there is a covariant functor which establishes a one-to-one correspondences between unital (resp. arbitrary) subhomogeneous C*-algebras and proper (resp. proper pointed) m-towers of finite height, and between all *-homomorphisms between two such algebras and morphisms between m-towers corresponding to these algebras.

math.OA↗

arXiv2013

Functional calculus in finite type I von Neumann algebras

A certain class of matrix-valued Borel matrix functions is introduced and it is shown that all functions of that class naturally operate on any operator T in a finite type I von Neumann algebra M in a way such that uniformly bounded sequences f_1,f_2,... of functions that converge pointwise to 0 transform into sequences f_1[T],f_2[T],... of operators in M that converge to 0 in the *-strong operator topology. It is also demonstrated that the double *-commutant of any such operator T which acts on a separable Hilbert space coincides with the set of all operators of the form f[T] where f runs over all function from the aforementioned class. Some conclusions concerning so-called operator-spectra of such operators are drawn and a new variation of the spectral theorem for them is formulated.

math.OA↗

arXiv2013

Bounded convergence theorems

There are presented certain results on extending continuous linear operators defined on spaces of E-valued continuous functions (defined on a compact Hausdorff space X) to linear operators defined on spaces of E-valued measurable functions in a way such that uniformly bounded sequences of functions that converge pointwise in the weak (or norm) topology of E are sent to sequences that converge in the weak, norm or weak* topology of the target space. As an application, a new description of uniform closures of convex subsets of C(X,E) is given. Also new and strong results on integral representations of continuous linear operators defined on C(X,E) are presented. A new classes of vector measures are introduced and various bounded convergence theorems for them are proved.

math.FA↗

arXiv2013

Direct integrals of matrices

It is shown that each linear operator on a separable Hilbert space which generates a finite type I von Neumann algebra has, up to unitary equivalence, a unique representation as a direct integral of inflations of mutually unitary inequivalent irreducible matrices. This leads to a simplification of the so-called prime (or central) decomposition and the multiplicity theory for such operators. The concept of so-called p-isomorphisms between special classes of such operators is discussed. All results are formulated in more general settings; that is, for tuples of closed densely defined operators affiliated with finite type I von Neumann algebras.

math.FA↗

arXiv2013

Algebra of operators affiliated with a finite type I von Neumann algebra

It is shown that the *-algebra of all (closed densely defined linear) operators affiliated with a finite type I von Neumann algebra admits a unique center-valued trace, which turns out to be, in a sense, normal. It is also demonstrated that for no other von Neumann algebras similar constructions can be performed.

math.OA↗

arXiv2012

Elementary approach to homogeneous C*-algebras

A C*-algebra is n-homogeneous (where n is finite) if every its nonzero irreducible representation acts on an n-dimensional Hilbert space. An elementary proof of Fell's characterization of n-homogeneous C*-algebras (by means of their spectra) is presented. A spectral theorem and a functional calculus for finite systems of elements which generate n-homogeneous C*-algebras are proposed.

math.OA↗

arXiv2012

Functional calculus for diagonalizable matrices

For an arbitrary function f:Ω\rightarrow C (where Ωis a subset of the field C) and a positive integer k let f act on all diagonalizable complex matrices whose all eigenvalues lie in Omega in the following way: f[P Diag(z1,...,zk) P-1] = P Diag(f(z1),...,f(zk)) P-1 for arbitrary numbers z1,...,zk in Ωand an invertible k \times k matrix P. The aim of the paper is to fully answer the question of when the function fop defined above is continuous for fixed k. In particular, it is shown that if Ωis open in C, then fop is continuous for fixed k > 2 iff f is holomorphic; and if Ωis an interval in R and k > 2, then fop is continuous iff f is of class Ck-2(Ω) and f(k-2) is locally Lipschitz in Ω. Also a full characterization is given when the domain of f is arbitrary as well as when fop acts on infinite-dimensional (diagonalizable) matrices.

math.FA↗

arXiv2012

Isometry groups among topological groups

It is shown that a topological group G is topologically isomorphic to the isometry group of a (complete) metric space iff G coincides with its G-delta-closure in the Rajkov completion of G (resp. if G is Rajkov-complete). It is also shown that for every Polish (resp. compact Polish; locally compact Polish) group G there is a complete (resp. proper) metric d on X inducing the topology of X such that G is isomorphic to Iso(X,d) where X = l_2 (resp. X = Q; X = Q\{point} where Q is the Hilbert cube). It is demonstrated that there are a separable Banach space E and a nonzero vector e in E such that G is isomorphic to the group of all (linear) isometries of E which leave the point e fixed. Similar results are proved for an arbitrary complete topological group.

math.GR↗

arXiv2012

Isometry groups of proper metric spaces

Given a locally compact Polish space X, a necessary and sufficient condition for a group G of homeomorphisms of X to be the full isometry group of (X,d) for some proper metric d on X is given. It is shown that every locally compact Polish group G acts freely on GxY as the full isometry group of GxY with respect to a certain proper metric on GxY, where Y is an arbitrary locally compact Polish space with (card(G),card(Y)) different from (1,2). Locally compact Polish groups which act effectively and almost transitively on complete metric spaces as full isometry groups are characterized. Locally compact Polish non-Abelian groups on which every left invariant metric is automatically right invariant are characterized and fully classified. It is demonstrated that for every locally compact Polish space X having more than two points the set of proper metrics d such that Iso(X,d) = {id} is dense in the space of all proper metrics on X.

math.GR↗