Related papers: Noncommutative Approximation: Inverse-Closed Subal…

Using principles of the theory of smoothness spaces we give systematic constructions of scales of inverse-closed subalgebras of a given Banach algebra with the action of a d-parameter automorphism group. In particular we obtain the…

It is a well-known fact in K-theory that the rapidly decreasing matrices of countable size form an associative topological algebra whose set of quasi-invertible elements is open, and such that the quasi-inversion map is continuous. We…

We give a systematic construction of inverse-closed (Banach) subalgebras in general higher-dimensional non-commutative tori

We give a self-contained proof of a recently established $\mathcal{B}(\mathcal{H})$-valued version of Jaffards Lemma. That is, we show that the Jaffard algebra of $\mathcal{B}(\mathcal{H})$-valued matrices, whose operator norms of their…

In this dissertation I establish that a broad class of Banach *-algebras of infinite integral operators, defined by the property that the kernels of the elements of the algebras possess subexponential off-diagonal decay, is inverse closed…

We construct various examples of non-trivial closed ideals of the compact-by-approximable algebra $\mathfrak{A}_X =:\mathcal K(X)/\mathcal A(X)$ on Banach spaces $X$ failing the approximation property. The examples include the following:…

We define new symbol classes for pseudodifferntial operators and investigate their pseudodifferential calculus. The symbol classes are parametrized by commutative convolution algebras. To every solid convolution algebra over a lattice we…

Let $A$ be a Banach algebra, not necessarily unital, and let $B$ be a closed subalgebra of $A$. We establish a connection between the Banach cyclic cohomology group $ {\cal{HC}}^n(A)$ of $A$ and the Banach $B$-relative cyclic cohomology…

In previous work we have shown that classical approximation theory provides methods for the systematic construction of inverse-closed smooth subalgebras. Now we extend this work to treat inverse-closed subalgebras of ultradifferentiable…

Arbitrary operator A on a Banach space X which is the generator of C_0-group with certain growth condition at infinity is considered. The relationship between its exponential type entire vectors and its spectral subspaces is found. Inverse…

For every Banach space $X$ with a Schauder basis consider the Banach algebra $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$ of all diagonal operators that are of the form $\lambda I + K$. We prove that $\mathbb{R}…

We exhibit a Banach space $Z$ failing the approximation property, for which there is an uncountable family $\mathscr F$ of closed subideals contained in the Banach algebra $\mathcal K(Z)$ of the compact operators on $Z$, such that the…

We prove that the weighted quasi-Banach algebras of operator valued matrices satisfying Schur and Baskakov-Gohberg-Sj\"ostrand (BGS) conditions are inverse-closed in the Banach algebra $B(\ell^2(X,\mathcal{H}))$ whenever the weight is…

The method of compatible sequences is introduced in order to produce non-trivial (closed) invariant subspaces of (bounded linear) operators. Also a topological tool is used which is new in the search of invariant subspaces: the extraction…

Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to…

In a recent paper of Benson and Symonds, a new invariant was introduced for modular representations of a finite group. An interpretation was given as a spectral radius with respect to a Banach algebra completion of the representation ring.…

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 study infinite matrices $A$ indexed by a discrete group $G$ that are dominated by a convolution operator in the sense that $|(Ac)(x)| \leq (a \ast |c|)(x)$ for $x\in G$ and some $a\in \ell ^1(G)$. This class of "convolution-dominated"…

We show that smoothness implies norm-controlled inversion: the smoothness of an element $a$ in a Banach algebra with a one-parameter automorphism group is preserved under inversion, and the norm of the inverse $a^{-1}$ is controlled by the…

Let $A$ and $B$ be unital Banach algebras with $A$ a subalgebra of $B$. Denote the algebra of all $n\times n$ matrices with entries from $A$ by $M_{n}(A)$. In this paper we prove some results concerning the open question: If $A$ is inverse…