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Raymond Mortini

Publications and source records attributed to Raymond Mortini.

At least 19 recordsLinked to original sources
arXiv2025

Real and Complex Analysis: Solutions to Problems in Amer. Math. Monthly, Math. Magazine, College Math. J., Elemente der Math., Crux Math., EMS Newsletter, Math. Gazette

In this arxiv-post I present my solutions (published or not) to Problems that appeared in Amer. Math. Monthly, Math. Magazine, Elemente der Mathematik and CRUX, that were mostly done in collaboration with Rudolf Rupp. Some of them (including a few own proposals which were published) were also done in cooperation with Rainer Br\"uck, Bikash Chakraborty, Pamela Gorkin, Gerd Herzog, J\'er\^ome No\"el, Peter Pflug and Amol Sasane.

math.HO↗
arXiv2018

One-component inner functions II

We continue our study of the set $\mathfrak I_c$ of inner functions $u$ in $H^\infty$ with the property that there is $η\in ]0,1[$ such that the level set $Ω_u(η):=\{z\in\mathbb D: |u(z)|<η\}$ is connected. These functions are called one-component inner functions. They play an important role in function theory and operator theory. Here we show that the composition of two one-component inner functions is again in $\mathfrak I_c$. We also give conditions under which a factor of one-component inner function belongs to $\mathfrak I_c$.

math.CV↗
arXiv2017

One-component inner functions

We explicitely unveil several classes of inner functions $u$ in $H^\infty$ with the property that there is $η\in ]0,1[$ such that the level set $Ω_u(η):=\{z\in\mathbb D: |u(z)|<η\}$ is connected. These so-called one-component inner functions play an important role in operator theory.

math.CV↗
arXiv2016

Noncoherent uniform algebras in $\mathbb C^n$

Let $\mathbf D=\bar{\mathbb D}$ be the closed unit disk in $\mathbb C$ and $\mathbf B_n=\bar{\mathbb B_n}$ the closed unit ball in $\mathbb C^n$. For a compact subset $K$ in $\mathbb C^n$ with nonempty interior, let $A(K)$ be the uniform algebra of all complex-valued continuous functions on $K$ that are holomorphic in the interior of $K$. We give short and non-technical proofs of the known facts that $A(\bar{\mathbb D}^n)$ and $A(\mathbf B_n)$ are noncoherent rings. Using, additionally, Earl's interpolation theorem in the unit disk and the existence of peak-functions, we also establish with the same method the new result that $A(K)$ is not coherent. As special cases we obtain Hickel's theorems on the noncoherence of $A(\barΩ)$, where $Ω$ runs through a certain class of pseudoconvex domains in $\mathbb C^n$, results that were obtained with deep and complicated methods. Finally, using a refinement of the interpolation theorem we show that no uniformly closed subalgebra $A$ of $C(K)$ with $P(K)\subseteq A\subseteq C(K)$ is coherent provided the polynomial convex hull of $K$ has no isolated points.

math.FA↗
arXiv2016

On the Krull Intersection Theorem in Function Algebras

A version of the Krull Intersection Theorem states that for Noetherian domains, the Krull intersection $ki(I)$ of every proper ideal $I$ is trivial; that is $$ ki(I):=\displaystyle\bigcap_{n=1}^\infty I^n = \{0\}. $$ We investigate the validity of this result for various function algebras $R$, present ideals $I$ of $R$ for which $ ki(I)\neq \{0\}$, and give conditions on $I$ so that $ki(I)=\{0\}$.

math.CV↗
arXiv2016

Unitary stable ranks and norm-one ranks

In the context of commutative $C^*$-algebras we solve a problem related to a question of M. Rieffel by showing that the all-units rank and the norm-one rank coincide with the topological stable rank. We also introduce the notion of unitary $M$-stable rank for an arbitrary commutative unital ring and compare it with the Bass stable rank. In case of uniform algebras, a sufficient condition for norm-one reducibility is given.

math.AC↗
arXiv2016

The covering dimension of a distinguished subset of the spectrum $M(H^\infty)$ of $H^\infty$ and the algebra of real-symmetric and continuous functions on $M(H^\infty)$

We show that the covering dimension, $\dim E$, of the closure $E$ of the interval $]-1,1[$ in the spectrum of $H^\infty$ equals one. Using Suárez's result that $\dim M(H^\infty)=2$, we then compute the Bass and topological stable ranks of the algebra $C(M(H^\infty))_{\rm\scriptscriptstyle sym}$ of real-symmetric continuous functions on $M(H^\infty)$.

math.FA↗
arXiv2015

A short proof of Cartan's Nullstellensatz for entire functions in $\mathbb C^n$

Using the fact that the maximal ideals in the polydisk algebra are given by the kernels of point evaluations, we derive a simple formula that gives a solution to the Bézout equation in the space of all entire functions of several complex variables. Thus a short and easy analytic proof of Cartan's Nullstellensatz is obtained.

math.CV↗
arXiv2015

Reducibility of invertible tuples to the principal component in commutative Banach algebras

Let $A$ be a complex, commutative unital Banach algebra. We introduce two notions of exponential reducibility of Banach algebra tuples and present an analogue to the Corach-Suárez result on the connection between reducibility in $A$ and in $C(M(A))$. Our methods are of an analytical nature. Necessary and sufficient geometric/topological conditions are given for reducibility (respectively reducibility to the principal component of $U_n(A)$) whenever the spectrum of $A$ is homeomorphic to a subset of $\mathbb C^n$.

math.FA↗
arXiv2015

The cone and cylinder algebra

In this exposition-type note we present detailed proofs of certain assertions concerning several algebraic properties of the cone and cylinder algebras. These include a determination of the maximal ideals, the solution of the Bézout equation and a computation of the stable ranks by elementary methods.

math.RA↗
arXiv2015

On a family of pseudohyperbolic disks

Hyperbolic geometry plays an important role within function theory of the disk. For example, via the Schwarz-Pick Lemma, the isometries of the unit disk $\mathbb D$ with respect to this geometry are the conformal self-maps of $\mathbb D$. In this elementary classroom note, we are interested in the collection of the pseudohyperbolic disks $D_ρ(x,r)$ (with fixed radius $r$ and variable hyperbolic centers $-1<x<1$) and determine explicitely with function theoretic tools the enveloppe of these disks.

math.CV↗
arXiv2015

Some curiosities of the algebra of bounded Dirichlet series

It is shown that the algebra of bounded Dirichlet series is not a coherent ring, and has infinite Bass stable rank. As corollaries of the latter result, it is derived that the algebra of bounded Dirichlet series has infinite topological stable rank and infinite Krull dimension.

math.CV↗
arXiv2014

Partial regularity and t-analytic sets for Banach function algebras

In this note we introduce the notion of $t$-analytic sets. Using this concept, we construct a class of closed prime ideals in Banach function algebras and discuss some problems related to Alling's conjecture in $H^\infty$. A description of all closed $t$-analytic sets for the disk-algebra is given. Moreover, we show that some of the assertions in Daoui et al. (Proc. Am. Math. Soc. 131:3211-3220, 2003) concerning the $O$-analyticity and $S$-regularity of certain Banach function algebras are not correct. We also determine the largest set on which a Douglas algebra is pointwise regular.

math.FA↗
arXiv2014

Logarithms and exponentials in Banach algebras

Let $A$ be a complex Banach algebra. If the spectrum of an invertible element $a\in A$ does not separate the plane, then $a$ admits a logarithm. We present two elementary proofs of this classical result which are independent of the holomorphic functional calculus. We also discuss the case of real Banach algebras. As applications, we obtain simple proofs that every invertible matrix over $\mathbb C$ has a logarithm and that every real matrix $M$ in $M_n(\mathbb R)$ with $\det M>0$ is a product of two real exponential matrices.

math.FA↗
arXiv2014

Holomorphic injective extensions of functions in P(K) and algebra generators

We present necessary and sufficient conditions on planar compacta $K$ and continuous functions $f$ on $K$ in order that $f$ generates the algebras $P(K), R(K), A(K)$ or $C(K)$. We also unveil quite surprisingly simple examples of non-polynomial convex compacta $K\subseteq\mathbb C$ and $f\in P(K)$ with the property that $f \in P(K)$ is a homeomorphism, but for which $f^{-1}\notin P(f(K))$. As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull $\widehat K$. On the other hand, it will be shown that the restriction $f^*|_G$ of the Gelfand-transform $f^*$ of an injective function $f\in P(K)$ is injective on every regular, bounded complementary component $G$ of $K$. A necessary and sufficient condition in terms of the behaviour of $f$ on the outer boundary of $K$ is given in order $f$ admits a holomorphic injective extension to $\widehat K$. We also include some results on the existence of continuous logarithms on punctured compacta containing the origin in their boundary.

math.CV↗
arXiv2014

The Bass and topological stable ranks of the Bohl algebra are infinite

The Bohl algebra $\textrm{B}$ is the ring of linear combinations of functions $t^k e^{λt}$, where $k$ is any nonnegative integer, and $λ$ is any complex number, with pointwise operations. We show that the Bass stable rank and the topological stable rank of $\textrm{B}$ (where we use the topology of uniform convergence) are infinite.

math.RA↗
arXiv2014

The Bass and topological stable ranks for algebras of almost periodic functions on the real line

Let $Λ$ be a sub-semigroup of the reals. We show that the Bass and topological stable ranks of the algebras ${\rm AP}_Λ=\{f\in {\rm AP}: σ(f)\subseteq Λ\}$ of almost periodic functions on the real line and with Bohr spectrum in $Λ$ are infinite whenever the algebraic dimension of the $\mathbb Q$-vector space generated by $Λ$ is infinite. This extends Suárez's result for ${\rm AP}_\mathbb R={\rm AP}$. Also considered are general subalgebras of AP.

math.FA↗