Gelfand transform for a Boehmian space of analytic functions

V. Karunakaran¹ ; R. Angeline Chella Rajathi¹

¹ School of Mathematics Madurai Kamaraj Unversity Palkalai Nagar, Madurai 625 021, India

Annales Polonici Mathematici, Tome 101 (2011) no. 1, pp. 39-45

Résumé

Let $H^\infty(\mathbb{D})$ denote the usual commutative Banach algebra of bounded analytic functions on the open unit disc $\mathbb{D}$ of the finite complex plane, under Hadamard product of power series. We construct a Boehmian space which includes the Banach algebra $A$ where $A$ is the commutative Banach algebra with unit containing $H^\infty(\mathbb{D})$. The Gelfand transform theory is extended to this setup along with the usual classical properties. The image is also a Boehmian space which includes the Banach algebra $C({\mit\Delta})$ of continuous functions on the maximal ideal space ${\mit\Delta}$ (where ${\mit\Delta}$ is given the usual Gelfand topology). It is shown that every $F \in C({\mit\Delta})$ is the Gelfand transform of a suitable Boehmian. It should be noted that in the classical theory the Gelfand transform from $A$ into $C({\mit\Delta})$ is not surjective even though it can be shown that the image is dense. Thus the context of Boehmians enables us to identify every element of $C({\mit\Delta})$ as the Gelfand transform of a suitable convolution quotient of analytic functions. (Here the convolution is the Hadamard convolution).

Détail

DOI : 10.4064/ap101-1-4

Keywords: infty mathbb denote usual commutative banach algebra bounded analytic functions unit disc mathbb finite complex plane under hadamard product power series construct boehmian space which includes banach algebra where commutative banach algebra unit containing infty mathbb gelfand transform theory extended setup along usual classical properties image boehmian space which includes banach algebra mit delta continuous functions maximal ideal space mit delta where mit delta given usual gelfand topology shown every mit delta gelfand transform suitable boehmian should noted classical theory gelfand transform mit delta surjective even though shown image dense context boehmians enables identify every element mit delta gelfand transform suitable convolution quotient analytic functions here convolution hadamard convolution