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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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.
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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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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.
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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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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
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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.
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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].
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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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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.
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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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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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