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Abstract

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).