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Continuous characters and joint topological spectrum

Żelazko W.

Control and Cybernetics

2007

Vol. 36, no 3

859-864

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.

Non unicité des topologies localement-convexes complètes sur certaines algèbres

Abdelali Z., Chidami M.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2002

[Z] 42(2)

125-136

We are interested, here, by the problems 1, 8 and 9 due to Wojciechowski and Żelazko (1997). We show that every algebra A generated by ℵ elements, such that ℵ > ℵ0, can be a locally-convex (resp. complete locally p-convex, p ϵ (0, 1]) semitopological algebra in ℵ +1 (resp. max{2ℵ0, ℵ +1}) different ways, which reduce the problem 9. Moreover if ℵℵ0 = ℵ, A can be a complete locally-convex semitopological algebra in ℵ +1 different ways, which solves the problem 1 for every ℵℵ0-generated algebra. On the other hand, we prove that for every algebra of polynomials generated by ℵ indeterminates, ℵ ≥ ℵ0 we can assign max{2ℵ0, ℵ +1} distinct topologies which make it a locally-convex topological algebra. Then, we get a reduction of problem 8.