659 research outputs found
Corona-type theorems and division in some function algebras on planar domains
Let be an algebra of bounded smooth functions on the interior of a
compact set in the plane. We study the following problem: if
satisfy , does there exist
and a constant such that ? A
prominent role in our proofs is played by a new space, C_{\dbar, 1}(K), which
we call the algebra of \dbar-smooth functions.
In the case , a complete solution is given for the algebras of
functions holomorphic in and whose first -derivatives extend
continuously to \ov{K^\circ}. This necessitates the introduction of a special
class of compacta, the so-called locally L-connected sets.
We also present another constructive proof of the Nullstellensatz for ,
that is only based on elementary \dbar-calculus and Wolff's method.Comment: 23 pages, 6 figure
Embedding polydisk algebras into the disk algebra and an application to stable ranks
It is shown how to embed the polydisk algebras (finite and infinite ones)
into the disk algebra .
As a consequence, one obtains uniform closed subalgebras of
which have arbitrarily prescribed stable ranksComment: 5 page
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