308,997 research outputs found
Valuation domains whose products of free modules are separable
It is proved that if is a valuation domain with maximal ideal and if
is countably generated for each prime ideal , then is separable
if and only is maximal, where
Maximal left ideals of the Banach algebra of bounded operators on a Banach space
We address the following two questions regarding the maximal left ideals of
the Banach algebra of bounded operators acting on an
infinite-dimensional Banach pace :
(Q1) Does always contain a maximal left ideal which is not
finitely generated? (Q2) Is every finitely-generated, maximal left ideal of
necessarily of the form \{T\in\mathscr{B}(E): Tx = 0\} (*) for
some non-zero ?
Since the two-sided ideal of finite-rank operators is not
contained in any of the maximal left ideals given by (*), a positive answer to
the second question would imply a positive answer to the first. Our main
results are: (i) Question (Q1) has a positive answer for most (possibly all)
infinite-dimensional Banach spaces; (ii) Question (Q2) has a positive answer if
and only if no finitely-generated, maximal left ideal of
contains ; (iii) the answer to Question (Q2) is positive for
many, but not all, Banach spaces.Comment: to appear in Studia Mathematic
Extending polynomials in maximal and minimal ideals
Given an homogeneous polynomial on a Banach space belonging to some
maximal or minimal polynomial ideal, we consider its iterated extension to an
ultrapower of and prove that this extension remains in the ideal and has
the same ideal norm. As a consequence, we show that the Aron-Berner extension
is a well defined isometry for any maximal or minimal ideal of homogeneous
polynomials. This allow us to obtain symmetric versions of some basic results
of the metric theory of tensor products.Comment: 13 page
Banach-valued Holomorphic Functions on the Maximal Ideal Space of H^\infty
We study Banach-valued holomorphic functions defined on open subsets of the
maximal ideal space of the Banach algebra H^\infty of bounded holomorphic
functions on the unit disk D\subset C with pointwise multiplication and
supremum norm. In particular, we establish vanishing cohomology for sheaves of
germs of such functions and, solving a Banach-valued corona problem for
H^\infty, prove that the maximal ideal space of the algebra H_{\rm comp}^\infty
(A) of holomorphic functions on \Di with relatively compact images in a
commutative unital complex Banach algebra A is homeomorphic to the direct
product of maximal ideal spaces of H^\infty and A.Comment: 30 page
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