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1

An infinite dimensional Banach algebra with all but one maximal abelian subalgebras of dimension two

Żelazko W.

Commentationes Mathematicae

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2008

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Vol. 48, [Z] 1

99-101

EN

I construct a unital closed subalgebra of L(H) with the property announced in the title. Moreover, for any two maxiamal abelian subalgebras of the algebra in question, their intersection consists only of scalar multiples of the unity.

2

Continuous characters and joint topological spectrum

Żelazko W.

Control and Cybernetics

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2007

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Vol. 36, no 3

859-864

EN

It is well known that there is a one-to-one correspondence between the characters of a finitely generated commutative Banach algebra and the joint spectrum of its generators. In this paper we show that this fact is also true for an arbitrary semitopological algebra and its continuous characters, provided we replace the concept of a joint spectrum by concept of a topological joint spectrum. In particular, we show that a finitely generated semitopological algebra has a continuous character if and only if the topological joint spectrum of its generators is non-void.

3

Topologically invertible elements and topological spectrum

Abel M., Żelazko W.

Bulletin of the Polish Academy of Sciences. Mathematics

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2006

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Vol. 54, no 3-4

257-271

EN

Properties of topologically invertible elements and the topological spectrum of elements in unital semitopological algebras are studied. It is shown that the inversion χ → χ-1 is continuous in every invertive Frechet algebra, and singly generated unital semi-topologicai algebras have continuous characters if and only if the topological spectrum of the generator is non-empty. Several open problems are presented,

4

On m-convexity of commutative real Waelbroeck algebras

Żelazko W.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2000

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[Z] 40

235-240

EN

We prove that the complexification of a commutative real Waelbroeck algebra is again such an algebra, and apply this result for showing that a commutative real locally convex Prechet Q-algebra must be m-convex, and, more generally, a commutative real locally convex Waelbroeck algebra must be m-convex. In this way we extend onto the real case two results known in the complex case and thus solve a problem posed in [9].