Exercise 62 if you know measure theory you ought to

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Exercise 62. If you know measure theory you ought to work through this exercise. We work in L 1 the space of Lebesgue integrable functions f : R C . (i) Use Fubini’s theorem to show that, if f, g L 1 , then f * g ( x ) = Z -∞ f ( x - t ) g ( t ) dx is well defined almost everywhere and that f * g L 1 with k f * g k 1 ≤ k f k 1 k g k 1 (ii) Use Fubini’s theorem to show that, if f, g L 1 then [ f * g ( λ ) = ˆ f ( λ g ( λ ) for all λ R . (iii) If e a ( t ) = e - iat compute ˆ e a f for f L 1 . Show that if e L 1 is a unit ˆ e = 1 . (iv) Show that if f L 1 then sup λ R | ˆ f ( λ ) | ≤ | f k 1 . Show that if f is once continuously differentiable with f, f 0 L 1 and f ( t ) , f 0 ( t ) 0 as | t | → ∞ then ˆ f ( λ ) 0 as | λ | → ∞ . Use a density argument to show that ˆ g ( λ ) 0 as | λ | → ∞ whenever g L 1 (this is the Lebesgue-Riemann lemma). (v) Use (iii) and (iv) to show that ( L 1 , * ) has no unit. 19
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Lemma 63. If B is a Banach algebra without unit we can find ˜ B a Banach algebra with a unit e such that (i) B is a sub Banach algebra of ˜ B , (ii) B is closed in ˜ B , (iii) ˜ B = span( B, e ) in the algebraic sense. Exercise 64. (i) Suppose we apply the construction of Lemma 63 to a Ba- nach algebra B with unit u . Is u a unit of the extended algebra ˜ B ? Does ˜ B have a unit? (ii) (Needs measure theory.) Can you find a natural identification for the unit of f L 1 where L 1 is the Banach algebra of Exercise 62. Thus any Banach algebra B without a unit can be studied by ‘adjoining a unit and then removing it’. This is our excuse for only studying Banach algebras with a unit. The following result is easy but fundamental. Lemma 65. Let B be a Banach algebra with unit e . (i) If k e - a k < 1 then a is invertible (that is has a multiplicative inverse). (ii) If E is the set of invertible elements in B then E is open. Lemma 65 (i) can be improved in a useful way. Theorem 66. (i) If B is a Banach algebra and b B then, writing ρ ( b ) = inf n k b n k 1 /n we have k b n k 1 /n ρ ( b ) as n → ∞ . (ii) If B is a Banach algebra with unit e and ρ ( e - a ) < 1 then a is invertible. We call ρ ( a ) the spectral radius of a . Exercise 67. Consider the space M n of n × n matrices over C with the operator norm. (i) Show that M n is a Banach algebra with unit. For which values of n is it commutative? (ii) Give an example of an A M 2 with A 6 = 0 but ρ ( A ) = 0 . (iii) If A is diagonalisable show that ρ ( A ) = max {| λ | : λ an eigenvalue of A } . (iv) (Harder and not essential.) Show that the formula of (iii) holds in general. 20
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8 Maximal ideals We now embark on a line of reasoning which will eventually lead to a char- acterization of a large class of commutative Banach algebras. Initially we continue to deal with Banach algebras which are not neces- sarily commutative. The generality is more apparent than real as the next exercise reveals.
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  • Fall '08
  • Groah
  • Math, Compact space, Banach space, Banach, Banach algebra, commutative Banach

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