Exercise 68 let b be a banach algebra with unit e let

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Exercise 68. Let B be a Banach algebra with unit e . Let A be the closed Banach algebra generated by e and some a B . (Formally, A is the smallest closed sub Banach algebra containing e and a .) Then A is commutative. Definition 69. Let B be a Banach space with unit e . If x B the resolvent R ( x ) of x is defined by R ( x ) = { λ C : x - λe is invertible } . Lemma 70. We use the notation of Definition 69. (i) C \ R ( x ) is bounded. (ii) R ( x ) is open. (iii) If μ R ( x ) we can find a δ > 0 and a 0 , a 1 , . . . B such that j =0 a j z j converges for all | z | < δ and ( x - λe ) - 1 = X j =0 a j ( λ - μ ) j for λ C and | λ - μ | < δ . (iv) R ( x ) 6 = C . Lemma 70 gives us our first substantial result on the nature of commu- tative Banach algebras. Theorem 71 (Gelfand-Mazur). Any Banach algebra which is also a field is isomorphic as a Banach algebra to C . 9 Analytic functions In order to extract more information on the resolvent we take a detour through a little (easy) integration theory and complex variable theory. Theorem 72. Let U be a Banach space, [ a, b ] a closed bounded interval in R Then we can define an integral R b a F ( t ) dt for every F : [ a, b ] U a 21
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continuous function having the following properties (here F, G : [ a, b ] U are continuous and λ, μ C ). (i) Z b a λF ( t ) + μG ( t ) dt = λ Z b a F ( t ) dt + μ Z b a G ( t ) dt . (ii) If a < c < b then Z b a F ( t ) dt = Z c a F ( t ) dt + Z b c F ( t ) dt. (iii) Z b a F ( t ) dt Z b a k F ( t ) k dt. (iv) If T : U C is a continuous linear functional Z b a T ( F ( t )) dt = T Z b a F ( t ) dt. Using the integral just defined we can define contour integrals as we did in the complex variable course. Definition 73. If γ : [ a, b ] C is continuously differentiable with γ ( a ) = γ ( b ) and F : [ a, b ] U a continuous function we define Z γ F ( z ) dz = Z b a F ( γ ( t )) γ 0 ( t ) dt. (We shall talk about the ‘closed contour’ γ .) We can now introduce the notion of an analytic Banach algebra valued function. Definition 74. Let B be a Banach algebra and Ω a simply connected 7 open set in C . A function f : Ω B is said to be analytic on Ω if there exists an f 0 : Ω B such that, for all z Ω f ( z + h ) - f ( z ) h - f 0 ( z ) 0 as h 0 through values of h such that z + h Ω . Theorem 75. Let B be a Banach algebra, Ω an open simply connected set in C , and γ a closed contour in Ω . Then Z γ f ( z ) dz = 0 . 7 Informally ‘with no holes’. 22
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We can follow a first undergraduate complex variable course and show. Lemma 76. Let B be a Banach algebra with a unit e , Ω an open set in C containing a disc D ( z 0 , R ) , and γ a contour describing a circle centre z 0 radius 0 < r < R . If | z 0 - z | < r then f ( z ) = 1 2 πi Z γ f ( w ) z - w dw. Lemma 77. Let B be a Banach algebra with a unit e and Ω an open set in C containing a disc D ( z 0 , R ) . There exist unique a 0 , a 1 , a 2 , . . . B such that j =0 a r ( z - z 0 ) r converges and f ( z ) = X j =0 a r ( z - z 0 ) r for all | z - z 0 | < R . Theorem 78. If B is a Banach algebra with unit sup {| λ | : λ / R ( x ) } = ρ ( x ) .
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  • Fall '08
  • Groah
  • Math, Compact space, Banach space, Banach, Banach algebra, commutative Banach

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