## Gelfand transform for a Boehmian space of analytic functions.(English)Zbl 1238.46044

Summary: 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 a 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 $$C(\varDelta)$$ of continuous functions on the maximal ideal space $$\varDelta$$ (where $$\varDelta$$ is given the usual Gelfand topology). It is shown that every $$F \in C(\varDelta)$$ is the Gelfand transform of a suitable Boehmian. It should be noted that in the classical theory the Gelfand transform from $$A$$ into $$C(\varDelta)$$ 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 $$C(\varDelta)$$ as the Gelfand transform of a suitable convolution quotient of analytic functions. (Here the convolution is the Hadamard convolution.)

### MSC:

 46J10 Banach algebras of continuous functions, function algebras 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 46F99 Distributions, generalized functions, distribution spaces
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