Banach lecture 4

This lecture (Banach4) was given on September 2nd 2009.

Excerpts

For B a banach algebra and a \in B, if the spectral radius r(a) = 0, then a is quasi-nilpotent.

Exercise/Results

  1. If B is commutative, then the map a \mapsto r(a) is continuous.
  2. The Volterra algebra is L^1([0,1]) where multiplication is given by f*g(t) = \int_0^t f(t-s) g(s) ds, it is a nonunital commutative Banach algebra.  Prove it is a radical Banach algebra: all its elements are quasi nilpotent.

I.6 Holomorphic Functional Calculus (HFC)

Let O(sp(a)) denote the family of equivalence classes of function that are holomorphic on an open set containing sp(a).  For each f\in O(sp(a)) HFC gives an element f(a) \in B such that

  1. f(a) depends only on the equivalence class of f.
  2. if f(z) is a rational function, then f(a) is just the formal evaluation of f at a.
  3. sp(f(a)) = f(sp(a)) (spectral mapping thm)
  4. Given L \subset \mathbb{C} compact such that sp(a) \subset int (L), then \exists A_L >0 such that |f(a)| \le A_l |f|_{C(L)} where |-|_C(L) is the norm on the algebra of cont. functions on L
  5. If g is holomophic in an open nbd cont. sp(f(a) = b) then g(b) = (g \circ f) (a)

I.7 Gelfond Theory

Deals with multiplicative linear functionals B \to \mathbb{C}; i.e maps that respect the product structure, in the case B is unital these are ring homorphisms.

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