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1
Content available Metric generalizations of Banach algebras
100%
Żelazko W.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS PRELIMINARIES § 0. Introduction.......................................................................................................................................................................3 § 1. Definitions and notation.................................................................................................................................................5 Chapter I LOCALLY BOUNDED ALGEBRAS § 2. Basic facts and examples..............................................................................................................................................6 § 3. Commutative p-normed algebras, spectral form and p-normed field..................................................................8 § 4. Commutative p-normed algebras (continued)..........................................................................................................12 § 5. Analytic functions in p-normed algebras.....................................................................................................................16 § 6. Final remarks...................................................................................................................................................................21 Chapter II F-ALGEBRAS AND TOPOLOGICAL ALGEBRAS § 7. F-algebras.........................................................................................................................................................................23 § 8. Topological division algebras.......................................................................................................................................26 Chapter III $B_0$-ALGEBRAS § 9. Basic facts.........................................................................................................................................................................29 § 10. Multiplicatively convex B_0-algebras.........................................................................................................................31 § 11. Spectra and power series in commutative m-convex $B_0$-algebras..............................................................34 § 12. Examples of non-m-convex $B_0$-algebras..........................................................................................................40 § 13. Extended spectrum; theorem on entire functions and its applications to Q-algebras and radicals.............44 § 14. Elementary properties of entire functions and characterization of commutative $B_0$-algebras with and without entire functions..................................................................................................................................................51 § 15. Entire operations in $B_0$-spaces and their applications to entire functions.................................................56 § 16. Final remarks.................................................................................................................................................................65 References...............................................................................................................................................................................68
2
Content available remote A subfield of a complex Banach algebra is not necessarily topologically isomorphic to a subfield of ℂ
90%
Żelazko W.
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nr 1
135-137
3
Content available remote A non-Banach in-convex algebra all of whose closed commutative subalgebras are Banach algebras.
90%
Żelazko W.
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1996
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tom 119
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nr 2
195-198
EN
We construct two examples of complete multiplicatively convex algebras with the property that all their maximal commutative subalgebras and consequently all commutative closed subalgebras are Banach algebras. One of them is non-metrizable and the other is metrizable and non-Banach. This solves Problems 12-16 and 22-24 of [7].
4
Content available remote Generation of B(X) two commutative subalgebras - results and open problems
90%
Żelazko W.
|
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nr 1
363-367
5
Content available remote A semitopological algebra without proper closed subalgebras
90%
Żelazko W.
|
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nr 2
239-242
EN
To Czesław Ryll-Nardzewski on his 70th birthday
6
Content available remote On topologization of countably generated algebras
90%
Żelazko W.
|
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nr 1
83-88
EN
We prove that any real or complex countably generated algebra has a complete locally convex topology making it a topological algebra. Assuming the continuum hypothesis, it is the best possible result expressed in terms of the cardinality of a set of generators. This result is a corollary to a theorem stating that a free algebra provided with the maximal locally convex topology is a topological algebra if and only if the number of variables is at most countable. As a byproduct we obtain an example of a semitopological (non-topological) algebra with every commutative subalgebra topological.
7
Content available remote On strongly closed subalgebras of B(X)
90%
Żelazko W.
|
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nr 2
289-295
EN
Let X be a real or complex Banach space. The strong topology on the algebra B(X) of all bounded linear operators on X is the topology of pointwise convergence of nets of operators. It is given by a basis of neighbourhoods of the origin consisting of sets of the form (1) U(ε;x_{1},...,x_{n}) = {T ∈ B(X): ∥ Tx_{i}∥ <ε, i=1,...,n},$ where $x_{1},...,x_{n}$ are linearly independent elements of X and ε is a positive real number. Closure in the strong topology will be called strong closure for short. It is well known that the strong closure of a subalgebra of B(X) is again a subalgebra. In this paper we study strongly closed subalgebras of B(X), in particular, maximal strongly closed subalgebras. Our results are given in Section 1, while in Section 2 we give the motivation for this study and pose several open questions.
8
Content available remote Concerning entire functions in $B_{0}$-algebras
90%
Żelazko W.
|
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nr 3
283-290
EN
We construct a non-m-convex non-commutative $B_0$-algebra on which all entire functions operate. Our example is also a Q-algebra and a radical algebra. It follows that some results true in the commutative case fail in general.
9
Content available remote An m-convex B₀-algebra with all left but not all right ideals closed
90%
Żelazko W.
|
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nr 2
317-324
EN
We construct an example as announced in the title. We also indicate all right, left and two-sided ideals in this example.
10
Content available remote Generators of maximal left ideals in Banach algebras
57%
Dales H. , Żelazko W.
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2012
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tom 212
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nr 2
173-193
EN
In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over ℂ whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement is also true if one replaces 'closed ideals' by 'maximal ideals in the Shilov boundary of A'. We give a shorter proof of this latter result, together with some extensions and related examples. We study the following conjecture. Suppose that all maximal left ideals in a unital Banach algebra A are finitely generated. Then A is finite-dimensional.
11
Content available remote On vector spaces and algebras with maximal locally pseudoconvex topologies
57%
Kokk A. , Żelazko W.
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nr 2
195-201
EN
Let X be a real or complex vector space. We show that the maximal p-convex topology makes X a complete Hausdorff topological vector space. If X has an uncountable dimension, then different p give different topologies. However, if the dimension of X is at most countable, then all these topologies coincide. This leads to an example of a complete locally pseudoconvex space X that is not locally convex, but all of whose separable subspaces are locally convex. We apply these results to topological algebras, considering the problem of uniqueness of a complete topology for semitopological algebras and giving an example of a complete locally convex commutative semitopological algebra without multiplicative linear functionals, but with every separable subalgebra having a total family of such functionals.
12
Content available remote Non-uniqueness of topology for algebras of polynomials
57%
Wojciechowski M. , Żelazko W.
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nr 1
111-121
13
Content available Concerning a problem of Arens on removable ideals in Banach algebras
40%
Żelazko W.
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nr 1
127-131
14
Content available remote A characterization of maximal ideals in commutative Banach algebras
34%
Kahane J.-P. , Żelazko W.
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nr 3
339-343
15
Content available remote Algebraic generation of B(X) by two subalgebras with square zero
34%
Żelazko W.
|
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tom 90
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nr 3
205-212
16
Content available remote On the divisors of zero of the group algebra
29%
Żelazko W.
|
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nr 1
99-102
17
Content available remote On non-removable ideals in commutative locally convex algebras
29%
Żelazko W.
|
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nr 2
133-154
18
Content available remote On maximal ideals in commutative m-convex algebras
29%
Żelazko W.
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tom 58
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nr 3
291-298
19
Content available On m-convex $B_0$-algebras of type ES
29%
Żelazko W.
|
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tom 20
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nr 2
299-304
20
Content available remote Some remarks on topological algebras
23%
Żelazko W.
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nr 1
141-149
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