Dales, Garth; Matematični kolokvij april 2013

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Large fields and algebras of continuous functions

Garth Dales

Lancaster University, Lancaster, VB

18. april 2013

We introduce the following question of Kaplansky: Let K be a compact space, and let C(K) denote the Banach algebra of all continuous functions on K. Are are all homomorphisms from C(K) into a Banach algebra?

We answer this by introducing some `large' totally ordered fields, and thinking whether the finite elements of these fields are normable.

The only prerequisites for the talk are the knowledge that the real line is a totally ordered field that is Dedekind complete, and an awareness of the Continuum Hypothesis.

Full details of the proofs are contained in the following books:

  • H. G. Dales, Banach algebras and automatic continuity, OUP, 2001.
  • H. G. Dales and W. H. Woodin, Super-real fields: totally ordered fields with additional structure, OUP, 1996.

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