Our first identification was adumbrated in our proof

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Our first identification was adumbrated in our proof of Wiener’s theorem. Example 102. Consider the space A ( T ) of continuous functions f : T C with absolutely convergent Fourier series (that is to say, -∞ | ˜ f ( n ) | < where ˜ f ( n ) is the n th Fourier coefficient). If we set k f k A = X -∞ | ˜ f ( n ) | , 26
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then ( A ( T ) , k k A ) is a commutative Banach algebra with unit 1 under point- wise multiplication. ( A ( T ) , k k A ) has maximal ideal space (identified with) T under its usual topology. We have ˆ f ( t ) = f ( t ) . Example 103. The ‘transform’ nature of the Gelfand transform is clearer if we seek the maximal ideal space and transform associated with the Banach algebra l 1 ( Z ) with standard norm and addition and multiplication given by convolution (that is a * b = c where c m = r = -∞ a m - r b r ). Here is a variation on the theme. Lemma 104. Let D = { z C : | z | < 1 } and ¯ D = { z C : | z | ≤ 1 } . Consider A ( D ) the set of continuous functions f : ¯ D C such that f is analytic in D . If f 1 , f 2 , . . . , f n A ( D ) are such that n j =1 | f j ( z ) | > 0 for all z C (that is to say that the f j do not vanish simultaneously) show that we can find g 1 , g 2 , . . . , g n A ( D ) such that n j f j =1 ( z ) g j ( z ) = 1 for all z ¯ D The next example is a key one in understanding the kind of problem we face. Example 105. Consider the sub Banach algebra A + ( T ) of A ( T ) consisting of elements f of A ( T ) with ˜ f ( n ) = 0 for n < 0 . Show that A + ( T ) has maximal ideal space (identified with) D the closed unit disc. We have ˆ f ( z ) = n =0 ˜ f ( n ) z n . Exercise 106. Consider the sub Banach algebra A - ( T ) of A ( T ) consisting of elements f of A ( T ) with ˜ f ( n ) = 0 for n > 0 . Find the maximal ideal space and associated Gelfand transform. Exercise 107. Consider the space B ( T ) of continuous functions f : T C with -∞ | n ˜ f ( n ) | < . Show that if we set k f k B = X -∞ ( | n | + 1) | ˜ f ( n ) | then ( B ( T ) , k k B ) is a commutative Banach algebra with unit 1 under point- wise multiplication. Find the maximal ideal space and associated Gelfand transform. Exercise 108. Consider the sub Banach algebra B + ( T ) of B ( T ) consisting of elements f of B ( T ) with ˜ f ( n ) = 0 for n < 0 . Find the maximal ideal space and associated Gelfand transform. 27
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Our next example is fundamental. Example 109. Let ( X, τ ) be a compact Hausdorff space. The space C ( X ) of continuous functions f : X C with the uniform norm is a commutative Banach algebra with unit 1 under pointwise operations. C ( X ) has maximal ideal space (identified with) X under its usual topology. We have ˆ f ( t ) = f ( t ) . One way of expressing many of our results is in terms of function algebras. Definition 110. Let ( X, τ ) be a compact Hausdorff space. If we consider C ( X ) as a Banach algebra in the usual way then any subalgebra A with a norm which makes it a Banach algebra is called a function algebra. Lemma 111. With the notation of Definition 110, if A is a Banach algebra with norm k k containing 1 , then k f k ≥ k f k for all f A .
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