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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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