Lecture Details
Name: Luiza de Moraes (Universidade Federal do Rio de Janeiro)Speaking: Fri 10th May 15:00 - 15:25Title: Algebras of Lorch analytic mappingsAbstract: If $E$ is a unitary commutative complex Banach algebra, a mapping $f: E \rightarrow E$ is analytic in $E$ in the sense of Lorch if given any $a \in E$ there exist unique elements $a_n \in E,$ such that $f(z)=\sum _{n=0}^\infty a_n(z-a)^n,$ for all $z\in E.$
The definition of Lorch analytic mapping extends to unitary commutative complex Banach algebras the classical definition of analytic function on $\mathbb{C}$ in a very natural way and a considerable portion of the classical theory of analytic functions carries over to the Lorch analytic mappings.
Let $\mathcal{H}_L(E)$ denote the set of all $f:E \rightarrow E$ that are Lorch analytic in $E,$ considered as a subalgebra of the algebra $C(E,E)$ of continuous mappings from $E$ into $E.$ Endowed with the topology $\tau_b$ of uniform convergence on the bounded subsets of $E, \; \mathcal{H}_L(E)$ is a unitary commutative Fr\'echet algebra.
An interesting result of the classical theory of analytic functions states that the closed ideals in $(\mathcal{H}(\mathbb{C}),\tau_0)$ are precisely the principal ideals (see [1], p.109, Theorem 13.7).
As $(\mathcal{H}_L(E), \tau_b)=(\mathcal{H}(\mathbb{C}), \tau_0)$ when $E=\mathbb{C}$, it is natural to ask how the closed ideals and the principal ideals of $(\mathcal{H}_L(E), \tau_b)$ are related
In this talk we will be concerned with this question.
The main results presented in this talk were obtained in collaboration with Guilherme V. S. Mauro, from the Universidade Federal da Integração Latino-Americana (UNILA), Brazil.
[1] D.H. Luecking and L.A. Rubel, Complex analysis. A Functional Analysis Approach, Springer-Verlag New York, 1984.
[2] G. V.S.Mauro, Espectros de Álgebras de Aplicações Analíticas no sentido de Lorch, Tese de Doutorado, UFRJ (2016)