arXiv subjects

Robert R. Kallman

Publications and source records attributed to Robert R. Kallman.

4 recordsLinked to original sources

arXiv2023

An Application of Descriptive Set Theory to Complex Analysis

The purpose of this paper is to prove a new general result about rings of complex analytic functions. Let $Ω$ be an arbitrary nonempty open subset of the complex plane $\mathbb C$, $\mathcal{A}(Ω)$ be the set of holomorphic functions on $Ω$ viewed as a Polish ring (not a Polish algebra over $\mathbb C$) in the usual compact open topology, let $R$ be a Polish ring and let $φ: R \to \mathcal{A}(Ω)$ be an abstract algebraic isomorphism. The main goal of this paper is to prove Theorem 36 that $φ$ is a topological isomorphism. A special result of Bers is an easy corollary. Two additional items supplement these results, viz., that $B(\mathbb{D})$, the abstract ring of bounded analytic functions on the unit disk, cannot be made into a Polish ring and that $\mathcal{M}(Ω)$, the abstract field of meromorphic functions on $Ω$, cannot be made into a Polish field.

math.CV↗

arXiv2017

An Extension of the Baire Property

The purpose of this paper is to define for every Polish space $X$ a class of sets, the $EBP(X)$-sets or the extended Baire property sets, to work out many properties of the $EBP(X)$-sets and to show their usefulness in analysis. For example, a proper generalization of the Pettis Theorem is proved in this context that furnishes a new automatic continuity result for Polish groups. The name extended Baire property sets is reasonable since $EBP(X)$ contains the Baire property sets $BP(X)$ and it is consistent with ZFC that the containment is proper.

math.LO↗

arXiv2014

$\R^{n} \rtimes G(n)$ is Algebraically Determined

Let $G$ be a Polish (i.e., complete separable metric topological) group. Define $G$ to be an algebraically determined Polish group if for any Polish group $L$ and algebraic isomorphism $φ: L \mapsto G$, we have that $φ$ is a topological isomorphism. Let $M(n,\R)$ be the set of $n \times n$ matrices with real coefficients and let the group $G$ in the above definition be the natural semidirect product $\R^{n} \rtimes G(n)$, where $n \ge 2$ and $G(n)$ is one of the following groups: either the general linear group $GL(n,\R) = \left\{ A \in M(n,\R) \ | \ \det(A) \ne 0 \right\}$, or the special linear group $SL(n,\R) = \left\{ A \in GL(n,\R) \ | \ \det(A) = 1 \right\}$, or $|SL(n,\R)| = \left\{ A \in GL(n,\R) \ | \ |\det(A)| = 1 \right\}$ or $GL^{+}(n,\R) = \left\{ A \in GL(n,\R) \ | \ \det(A) > 0 \right\}$. These groups are of fundamental importance for linear algebra and geometry. The purpose of this paper is to prove that the natural semidirect product $\R^{n} \rtimes G(n)$ is an algebraically determined Polish group. Such a result is not true for $\complexes^{n} \rtimes GL(n,\complexes)$ nor even for $\R^{3} \rtimes SO(3,\R)$. The proof of this result is done in a sequence of steps designed to verify the hypotheses of the road map Theorem 2. A key intermediate result is that $φ^{-1}(SO(n,\R))$ is an analytic subgroup of $L$ for every $n \ge 2$.

math.GN↗

arXiv2014

$\mathop{\rm PL}_+(I)$ is not a Polish group

The group $\mathop{\rm PL}_+(I)$ of increasing piecewise linear self-homeomorphisms of the interval $I=[0,1]$ may not be assigned a topology in such a way that it becomes a Polish group. The same statement holds for the groups $\mathop{\rm Homeo}_+^{Lip}(I)$ of bi-Lipschitz homeomorphisms of $I$, and $\mathop{\rm Diff}_+^{1+ε}(I)$ of diffeomorphisms of $I$ whose derivatives are Hölder continuous with exponent $ε$.

math.GR↗