arXiv subjects
Publications and source records attributed to Rudolf Rupp.
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 $ki(I)=\{0\}$.
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 $\mathbb C^n$.
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.
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.
A corona type theorem is given for the ring R of periodic distributions in R^d in terms of the sequence of Fourier coefficients of these distributions, which have at most polynomial growth. It is also shown that the Bass stable rank and the topological stable rank of R are both equal to 1.
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.
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.
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.
In this survey we determine an explicit set of generators of the maximal ideals in the ring $\mathbb R[x_1,\dots,x_n]$ of polynomials in $n$ variables with real coefficients and give an easy analytic proof of the Bass-Vasershtein theorem on the Bass stable rank of $\mathbb R[x_1,\dots,x_n]$. The ingredients of the proof stem from different publications by Coquand, Lombardi, Estes and Ohm. We conclude with a calculation of the topological stable rank of $\mathbb R[x_1,\dots,x_n]$, which seems to be unknown so far.
Let $A$ be an algebra of bounded smooth functions on the interior of a compact set in the plane. We study the following problem: if $f,f_1,\dots,f_n\in A$ satisfy $|f|\leq \sum_{j=1}^n |f_j|$, does there exist $g_j\in A$ and a constant $N\in\N$ such that $f^N=\sum_{j=1}^n g_j f_j$? A prominent role in our proofs is played by a new space, $C_{\dbar, 1}(K)$, which we call the algebra of $\dbar$-smooth functions. In the case $n=1$, a complete solution is given for the algebras $A^m(K)$ of functions holomorphic in $K^\circ$ and whose first $m$-derivatives extend continuously to $\ov{K^\circ}$. This necessitates the introduction of a special class of compacta, the so-called locally L-connected sets. We also present another constructive proof of the Nullstellensatz for $A(K)$, that is only based on elementary $\dbar$-calculus and Wolff's method.
Let A_+ be the ring of Laplace transforms of complex Borel measures on R with support in [0,+\infty) which do not have a singular nonatomic part. We compare the nu-metric d_{A_+} for stabilizable plants over A_+ given in the article by Ball and Sasane [2010], with yet another metric d_{H^\infty}|_{A_+}, namely the one induced by the metric d_{H^\infty} for the set of stabilizable plants over H^\infty given in teh article by Sasane in 2011. Both d_{A_+} and d_{H^\infty} coincide with the classical Vinnicombe metric defined for rational transfer functions, but we show here by means of an example that these two possible extensions of the classical nu-metric for plants over A_+ do not coincide on the set of stabilizable plants over A_+. We also prove that they nevertheless give rise to the same topology on stabilizable plants over A_+, which in turn coincides with the gap metric topology.
Let $Ω$ be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by $H^\infty(Ω)$ the Banach algebra of all bounded holomorphic functions on $Ω$, with pointwise operations and the supremum norm. We show that the topological stable rank of $H^\infty(Ω)$ is equal to 2. The proof is based on Suarez's theorem that the topological stable rank of $H^\infty(\D)$ is equal to 2, where $\D$ is the unit disk. We also show that for domains symmetric to the real axis, the Bass and topological stable ranks of the real symmetric algebra $H^\infty_\R(Ω)$ are 2.