Related papers: Algebraic operations on the space of Hausdorff con…

We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural…

In this note, we continue to highlight some applications of Theorem 1 of [3]. Here is a sample: Let $X$ be an open set in ${\bf C}^n$, $\Omega$ an open convex set in ${\bf C}$ and $f, g : X\to {\bf C}$ two holomorphic functions such that…

This paper emphasizes the ubiquitous role of moduli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy Gerstenhaber (i.e., homotopy…

The duality between partial actions (partial $H$-module algebras) and co-actions (partial $H$-comodule algebras) of a Hopf algebra $H$ is fully explored in this work. A connection between partial (co)actions and Hopf algebroids is…

We investigate H\"ormander spectral multiplier theorems as they hold on $X = L^p(\Omega),\: 1 < p < \infty,$ for many self-adjoint elliptic differential operators $A$ including the standard Laplacian on $\R^d.$ A strengthened matricial…

Algebra extensions A < B where A is a left B-module such that the B-action extends the multiplication in A are ubiquitous. We encounter examples of such extensions in the study of group actions, group gradings or more general Hopf actions…

Given a set $A$ and an abelian group $B$ with operators in $A$, in the sense of Krull and Noether, we introduce the Ore group extension $B[x; \sigma_B, \delta_B]$ as the additive group $B[x]$, with $A[x]$ as a set of operators. Here, the…

We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of…

We study the *-double functor between the categories of associative and involutive algebras. It is proved that an associative algebra is isomorphic to a subalgebra of a $C\sp*$-algebra if and only if its *-double is *-isomorphic to a…

A necessary and sufficient condition for an operator space to support a multiplication making it completely isometric and isomorphic to a unital operator algebra is proved. The condition involves only the holomorphic structure of the Banach…

In this note, as a particular case of a more general result, we obtain the following theorem: Let $\Omega\subseteq {\bf R}^n$ be a non-empty bounded open set and let $f:\overline {\Omega}\to {\bf R}^n$ be a continuous function which is…

This paper is concerned with the algebraic dual D*(\Omega) of the space of test functions D(\Omega). The emphasis is on failures and successes of D*(\Omega) as compared to the continuous dual D'(\Omega), the space of distributions.…

Let $\Omega_n$ be the ring of polynomial-valued holomorphic differential forms on complex $n$-space, referred to in physics as the superspace ring of rank $n$. The symmetric group $\mathfrak{S}_n$ acts diagonally on $\Omega_n$ by permuting…

It is well known that in the commutative case, i.e. for $A=C(X)$ being a commutative C*-algebra, compact selfadjoint operators acting on the Hilbert C*-module $H_A$ (= continuous families of such operators $K(x)$, $x\in X$) can be…

The symmetric homology of a unital associative algebra $A$ over a commutative ground ring $k$, denoted $HS_*(A)$, is defined using derived functors and the symmetric bar construction of Fiedorowicz. In this paper we show that $HS_*(A)$…

Given a complex domain $\Omega$ and analytic functions $\varphi_1,\ldots,\varphi_n : \Omega \to \mathbb{D}$, we give geometric conditions for $H^\infty(\Omega)$ to be generated by functions of the form $g \circ \varphi_k$, $g \in…

With view to applications, we establish a correspondence between two problems: (i) the problem of finding continuous positive definite extensions of functions $F$ which are defined on open bounded domains $\Omega$ in $\mathbb{R}$, on the…

Let $\Omega \subset \mathbb{R}$ be a nonempty and open set, then for all $f, g, h\in \mathscr{C}^{2}(\Omega)$ we have \begin{multline*} \diff{2}{x}(f\cdot g\cdot h) -f\diff{2}{x}(g\cdot h)-g\diff{2}{x}(f\cdot h)-h\diff{2}{x}(f\cdot g) +…

In this work we study how to extend a partial action of a Hopf Algebra $A$ on an algebra $R$ to a partial action of a Hopf-Ore extension of $A$ on $R$. As consequence, we characterize all partial actions of rank one Hopf algebras (in…

Let $B(\Omega)$ be the Banach space of holomorphic functions on a bounded connected domain $\Omega$ in $\mathbb C^n$, which contains the ring of polynomials on $\Omega $. In this paper, we first establish a criterion for $B(\Omega )$ to be…