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Fyrirlesari: Eggert Briem, University of Iceland

Titill: Gelfand Theory for Real Banach Algebras

Staðsetning: Naustið (Endurmenntun)
Tími: Föstudagur 19. Óktober kl. 11.40

Ágrip:

A real Banach algebra is a Banach algebra over the reals. We will only consider commutative Banach algebras with unit. An example is the algebra of continuous functions, f , on the unit disc, analytic in the interior of the disc, satisfying f (z) = f (z). The norm on the algebra is the sup-norm.

Another example is the algebra of continuously differentiable real-valued functions on the unit interval with the norm given by

∥f∥ = ∥f∥∞ +∥f′∥∞

According to Gelfand theory, a commutative Banach algebra A with unit, over the complex numbers, can be represented as an algebra of continuous complex valued functions on a compact Hausdorff space X, with

sup_{x ∈ X}|ã(x)| = r(a) := lim∥a^n∥^1/n

for a ∈ A. Here X is the space of multiplicative linear functionals on A, equipped with the w∗-topology, and ã(x) = x(a) for x ∈ X.