Related papers: Generalized Jacobson's lemma in a Banach algebra
It is a longstanding problem whether every contractible Banach algebra is necessarily finite-dimensional. In this note, we confirm this for Banach algebras acting on Banach spaces with the uniform approximation property. This generalizes a…
We introduce a class of Banach algebras of generalized matrices and study the existence of approximate units, ideal structure, and derivations of them.
In this note we shall give conditions which guarantee that $\frac{1-q^b}{1-q^a}{n\brack m}\in\mathbb{Z}[q]$ holds. We shall provide a full characterisation for $\frac{1-q^b}{1-q^a}{ka\brack m}\in\mathbb{Z}[q]$. This unifies a variety of…
We define and study the Cauchy problem for a 1-D nonlinear Dirac equation with nonlinearities concentrated at one point. Global well-posedness is provided and conservation laws for mass and energy are shown. Several examples, including…
The Jacobian conjecture involves the map $y= x - V(x)$ where $y, x$ are n-dimensional vectors, $V(x)$ is a symmetric polynomial of degree $d$ for which the Jacobian hypothesis holds: $ e^{Tr \ln(1- V'(x))} =1,\ \forall x$. The conjecture…
The Classical Jacobian Conjecture claims that any unramified endomorphism of a complex affine space is an automorphism. In order to embed this conjecture in a geometric environment, where one could enjoy the beauty and the richness of tools…
Let $d \geq 2$ be a natural number. We show that $$|A-A| \geq \left(2d-2 + \frac{1}{d-1}\right)|A|-(2d^2-4d+3)$$ for any sufficiently large finite subset $A$ of $\mathbb{R}^d$ that is not contained in a translate of a hyperplane. By a…
Under the assumption that the continuum c is a regular cardinal, we prove the existence and uniqueness of a Boolean algebra B of size c defined by sharing the main structural properties that P(N)/fin has under CH and in the aleph2-Cohen…
When a unital \ca $A$ has topological stable rank one (write $\tsr(A) = 1$), we know that $\tsr(pAp) \leq 1$ for a non-zero projection $p \in A$. When, however, $\tsr(A) \geq 2$, it is generally faluse. We prove that if a unital C*-algebra…
It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a…
A groupoid that satisfies the left invertive law: $ab\cdot c=cb\cdot a$ is called an AG-groupoid. We extend the concept of left abelian distributive groupoid (LAD) and right abelian distributive groupoid (RAD) to introduce new subclasses of…
Let A be a C*-algebra. It is shown that A is an AW*-algebra if, and only if, each maximal abelian self--adjoint subalgebra of A is monotone complete. An analogous result is proved for Rickart C*-algebras; a C*-algebra is a Rickart…
W.H. Woodin showed that if $\kappa_1 < \cdots < \kappa_n$ are strong cardinals then two-step ${\bf\Sigma}^1_{n+3}$ generic absoluteness holds after collapsing $2^{2^{\kappa_n}}$ to be countable. We show that this number can be reduced to…
A Banach space characterization of simple real or complex $C^*$-algebras is given which even characterizes the underlying field. As an application, it is shown that if $\mathfrak A_1$ and $\mathfrak A_2$ are Birkhoff-James isomorphic simple…
It is consistent that the continuum be arbitrary large and no absolute $\kappa$-Borel set $X$ of density $\kappa$, $\aleph_1<\kappa<\mathfrak{c}$, condenses onto a compact metric space. It is consistent that the continuum be arbitrary large…
We give a characterization of the ''uniform closure'' of the dual of a $C^{*}$-algebra. Some applications in harmonic analysis are given.
The $\beta$-generalized quasi-geostrophic equation is studied in the range of $\alpha \in (0, 1), \beta \in (1/2, 1), 1/2 < \alpha + \beta < 3/2$. When $\alpha \in (1/2, 1), \beta \in (1/2, 1)$ such that $1 \leq \alpha + \beta < 3/2$, using…
We prove that if a unital Banach algebra $A$ is the dual of a Banach space $\pd{A}$, then the set of weak* continuous states is weak* dense in the set of all states on $A$. Further, weak* continuous states linearly span $\pd{A}$.
We examine the condition that a complex Banach algebra $A$ have dense invertible group. We show that, for commutative algebras, this property is preserved by integral extensions. We also investigate the connections with an old problem in…
A class of numerical quadrature rules is derived, with equally-spaced nodes, and unit weights except at a few points at each end of the series, for which "corrections" (not using any further information about the integrand) are added to the…