Related papers: The Corona theorem and stable rank for the algebra…
The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space $X$, we study the Cluster Value Problem for the ball algebra $A_u(B_X)$, the Banach algebra of all uniformly continuous…
The Bohl algebra $\textrm{B}$ is the ring of linear combinations of functions $t^k e^{\lambda t}$, where $k$ is any nonnegative integer, and $\lambda$ is any complex number, with pointwise operations. We show that the Bass stable rank and…
The main result of the paper is the theorem giving a sufficient condition for the existence of a bounded analytic projection onto a holomorphic family of (generally infinite-dimensional) subspaces (a holomorphic sub-bundle of a trivial…
In this paper we extend a method of Arveson and McCullough to prove a tangential interpolation theorem for subalgebras of $H^\infty$. This tangential interpolation result implies a Toelitz corona theorem. In particular, it is shown that the…
The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a…
We study the corona problem on the unit ball and the unit polydisc in $\CC^n$. We provide affirmative solutions to both problems.
We prove the corona theorem for domains whose boundary lies in certain smooth quasicircles. These curves, which are not necessarily Dini-smooth, are defined by quasiconformal mappings whose complex dilatation verifies certain conditions.…
We denote by A_0+AP_+ the Banach algebra of all complex-valued functions f defined in the closed right half plane, such that f is the sum of a holomorphic function vanishing at infinity and a ``causal'' almost periodic function. We give a…
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…
In this paper, we introduce a notion of stable coarse algebras for metric spaces with bounded geometry, and formulate the twisted coarse Baum--Connes conjecture with respect to stable coarse algebras. We prove permanence properties of this…
We present unified proofs of several properties of the corona of $\sigma$-unital C*-algebras such as AA-CRISP, SAW*, being sub-$\sigma$-Stonean in the sense of Kirchberg, and the conclusion of Kasparov's Technical Theorem. Although our…
We study how generalized Jones $\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose boundaries are lower $d$-content regular admit…
In this paper we prove the following theorem: Suppose that $f_1,f_2\in H^\infty_\R(\D)$, with $\norm{f_1}_\infty,\norm{f_2}_{\infty}\leq 1$, with $$ \inf_{z\in\D}(\abs{f_1(z)}+\abs{f_2(z)})=\delta>0. $$ Assume for some $\epsilon>0$ and…
We show that under mild set theoretic hypotheses we have rigidity for algebras of continuous functions over Higson coronas, topological spaces arising in coarse geometry. In particular, we show that under $\mathsf{OCA}$ and $\mathsf…
Let $A$ be a separable $C^*$-algebra and let $B$ be a stable $C^*$-algebra with a strictly positive element. We consider the (semi)group $\Ext^{as}(A,B)$ (resp. $\Ext(A,B)$) of homotopy classes of asymptotic (resp. of genuine) homomorphisms…
This survey shows how, for the Nevanlinna class N of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions $ H^\infty$: interpolating…
Suppose $A$ is a $C^*$-algebra and $H$ is a $C^*$-correspondence over $A$. If $H$ is regular in the sense that the left action of $A$ is faithful and is given by compact operators, then we compute the $K$-theory of $\mathcal{O}_A(H) \rtimes…
We examine the algebra of all Dirichlet Series bounded on the right half plane. We consider the analogue of the Corona theorem in this setting, and show that it is false, i.e. the right half-plane is not dense in the maximal ideal space. We…
We extend the notion of Heyting algebra to a notion of truth values algebra and prove that a theory is consistent if and only if it has a B-valued model for some non trivial truth values algebra B. A theory that has a B-valued model for all…
In the paper we introduce a weak set theory $\mathsf{H}_{<\omega}$ . A formalization of arithmetic on finite von Neumann ordinals gives an embedding of arithmetical language into this theory. We show that $\mathsf{H}_{<\omega}$ proves a…