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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
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A characterization of F-algebras with all one-sided ideals closed

93%
Żelazko W.
Studia Mathematica
|
2005
|
tom 168
|
nr 2
135-145
EN
We prove that a real or complex F-algebra has all left and right ideals closed if and only if it is noetherian.
3
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Generation of B(X) two commutative subalgebras - results and open problems

93%
Żelazko W.
Banach Center Publications
|
1994
|
tom 30
|
nr 1
363-367
4
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A subfield of a complex Banach algebra is not necessarily topologically isomorphic to a subfield of ℂ

93%
Żelazko W.
Colloquium Mathematicum
|
1992
|
tom 63
|
nr 1
135-137
5
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An m-convex B₀-algebra with all left but not all right ideals closed

93%
Żelazko W.
Colloquium Mathematicum
|
2006
|
tom 104
|
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.
6
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A semitopological algebra without proper closed subalgebras

93%
Żelazko W.
Colloquium Mathematicum
|
1997
|
tom 74
|
nr 2
239-242
EN
To Czesław Ryll-Nardzewski on his 70th birthday
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When a unital F-algebra has all maximal left (right) ideals closed?

93%
Żelazko W.
Studia Mathematica
|
2006
|
tom 175
|
nr 3
279-284
EN
We prove that a real or complex unital F-algebra has all maximal left ideals closed if and only if the set of all its invertible elements is open. Consequently, such an algebra also automatically has all maximal right ideals closed.
8
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On topologization of countably generated algebras

93%
Żelazko W.
Studia Mathematica
|
1994-1995
|
tom 112
|
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.
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A characterization of Q-algebras of type F

93%
Żelazko W.
Studia Mathematica
|
2004
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tom 165
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nr 1
73-79
EN
We prove that a real or complex unital F-algebra is a Q-algebra if and only if all its maximal one-sided ideals are closed.
10
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On strongly closed subalgebras of B(X)

93%
Żelazko W.
Colloquium Mathematicum
|
1994
|
tom 67
|
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.
11
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When is L(X) topologizable as a topological algebra?

93%
Żelazko W.
Studia Mathematica
|
2002
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tom 150
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nr 3
295-303
EN
Let X be a locally convex space and L(X) be the algebra of all continuous endomorphisms of X. It is known (Esterle [2], [3]) that if L(X) is topologizable as a topological algebra, then the space X is subnormed. We show that in the case when X is sequentially complete this condition is also sufficient. In this case we also obtain some other conditions equivalent to the topologizability of L(X). We also exhibit a class of subnormed spaces X, called sub-Banach spaces, which are not necessarily sequentially complete, but for which the algebra L(X) is normable. Finally we exhibit an example of a subnormed space X for which the algebra L(X) is not topologizable.
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A characterization of commutative Fréchet algebras with all ideals closed

93%
Żelazko W.
Studia Mathematica
|
2000
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tom 138
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nr 3
293-300
EN
Let A be a commutative unital Fréchet algebra, i.e. a completely metrizable topological algebra. Our main result states that all ideals in A are closed if and only if A is a noetherian algebra
13
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Concerning topologization of real or complex algebras

93%
Żelazko W.
Colloquium Mathematicum
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1996
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tom 71
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nr 1
111-113
14
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A characterization of characters in complex m-pseudoconvex algebras

93%
Żelazko W.
Banach Center Publications
|
2005
|
tom 67
|
nr 1
405-411
EN
In this paper we extend the characterization of characters given in [1], [2] and [8] onto m-pseudoconvex algebras. As a consequence (and a generalization) we give a characterization of continuous homomorphisms from m-pseudoconvex algebras into commutative semisimple m-pseudoconvex algebras.
15
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A non-Banach in-convex algebra all of whose closed commutative subalgebras are Banach algebras.

93%
Żelazko W.
Studia Mathematica
|
1996
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tom 119
|
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].
16
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A non-locally convex topological algebra with all commutative subalgebras locally convex

93%
Żelazko W.
Studia Mathematica
|
1996
|
tom 120
|
nr 1
89-94
EN
We construct a complete multiplicatively pseudoconvex algebra with the property announced in the title. This solves Problem 25 of [6].
17
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Concerning entire functions in $B_{0}$-algebras

93%
Żelazko W.
Studia Mathematica
|
1994
|
tom 110
|
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.
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A density theorem for algebra representations on the space (s)

93%
Żelazko W.
Studia Mathematica
|
1998
|
tom 130
|
nr 3
293-296
EN
We show that an arbitrary irreducible representation T of a real or complex algebra on the F-space (s), or, more generally, on an arbitrary infinite (topological) product of the field of scalars, is totally irreducible, provided its commutant is trivial. This provides an affirmative solution to a problem of Fell and Doran for representations on these spaces.
19
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The strongest vector space topology is locally convex on separable linear subspaces

93%
Żelazko W.
Annales Polonici Mathematici
|
1997
|
tom 66
|
nr 1
275-282
EN
Let X be a real or complex vector space equipped with the strongest vector space topology $τ_{max}$. Besides the result announced in the title we prove that X is uncountable-dimensional if and only if it is not locally pseudoconvex.
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On vector spaces and algebras with maximal locally pseudoconvex topologies

59%
Kokk A., Żelazko W.
Studia Mathematica
|
1994-1995
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tom 112
|
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.
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