Related papers: Corona-type theorems and division in some function…

For an analytic function $f(z)=\sum_{k=0}^\infty a_kz^k$ on a neighbourhood of a closed disc $D\subset {\bf C}$, we give assumptions, in terms of the Taylor coefficients $a_k$ of $f$, under which the number of intersection points of the…

We study the algebraic $K$-theory of the ring of continuous functions on a compact Hausdorff space with values in a local division ring, e.g., a local field: We compute its negative $K$-theory and show its $K$-regularity. The complex case…

Given commuting functions f,g, with at most a countable compact set of essential singularities, recent results for entire functions are extended to prove that Julia sets match, J(f)=J(g), in a particular case in the class K. With this…

Considering the deeper reasons of the appearance of a remarkable counterexample by J.~Kaad and M.~Skeide [17] we consider situations in which two Hilbert C*-modules $M \subset N$ with $M^\bot = \{ 0 \}$ over a fixed C*-algebra $A$ of…

This text surveys cohomological properties of pairs $(U,f)$ consisting of a smooth complex quasi-projective variety $U$ together with a regular function on~it. On the one hand, one tries to mimic the case of a germ of holomorphic function…

The most important uniform algebra is the family of continuous functions on a compact subset $K$ of the complex plane $\mathbb{C}$ which are analytic on the interior int$(K)$ For compact sets $K$ which are regular (i.e. $K =$int$(K)$ and…

We prove that for any function $f$ satisfying certain mild conditions and any Cantor set $K$ with Newhouse thickness greater than $1$, there exists $x\in K$ and $t>0$ such that \[ \{x-t,x,x+f(t)\}\subset K. \] This is an extension of…

We show that several spaces of holomorphic functions on a Riemann domain over a Banach space, including the nuclear and Hilbert-Schmidt bounded type, are locally $m$-convex Fr\'echet algebras. We prove that the spectrum of these algebras…

In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the…

Function (linear) spaces on which an arbitrary function operates (i.e. the space is stable w.r.t. the pointwise unary operation defined by the function) were investigated, for continuous real or complex operations, by deLeeuw-Katznelson,…

Imposing some conditions on derivatives of the known functions, using the Fiber Contraction Theorem we prove the existence of $C^1$ solutions of a class of iterative functional equations which involves iterates of the unknown functions and…

We define the algebra of Colombeau generalized functions on a subset A of the space of d-dimensional generalized points. If the domain A is open, such generalized functions can be identified with pointwise maps from A into the ring of…

Let $D$ be a domain in the complex plane, $M$ be an extended real function on $D$. If $f$ is a non-zero holomorphic function on $D$ with an upper constraint $|f|\leq \exp M$ on this domain $D$, then it is natural to expect that there must…

Let $M$ be a non-compact connected Riemann surface of finite type, and $R\subset\subset M$ be a relatively compact domain such that $H_{1}(M,\Z)=H_{1}(R,\Z)$. Let $\tilde R\longrightarrow R$ be a covering. We study the algebra…

We establish an equivalency of the Corona problem (1962) and Gleason problem (1964) in the theory of several complex variables. As an application, we give an affirmative solution of the Corona problem for certain bounded pseudoconvex…

It is shown that if $A$ is a uniform algebra generated by real-analytic functions on a suitable compact subset $K$ of a real-analytic variety such that the maximal ideal space of $A$ is $K$, and every continuous function on $K$ is locally a…

For a class $F$ of complex-valued functions on a set $D$, we denote by $g_n(F)$ its sampling numbers, i.e., the minimal worst-case error on $F$, measured in $L_2$, that can be achieved with a recovery algorithm based on $n$ function…

We prove Banach, Newton-Raphson and Brouwer fixed point theorems in the framework of generalized smooth functions, a minimal extension of Colombeau's theory (and hence of classical distribution theory) which makes it possible to model…

We prove a commutative Gelfand--Naimark type theorem, by showing that the set $C_s(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space $X$ (provided with the supremum…

We deal with a family of functionals depending on curvatures and we prove for them compactness and semicontinuity properties in the class of closed and bounded sets which satisfy a uniform exterior and interior sphere condition. We apply…