10 maximal ideals one way of exploiting the gelfand

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10 Maximal ideals One way of exploiting the Gelfand-Mazur theorem is to introduce the notion of maximal ideals. (From now on all our Banach algebras will be commuta- tive.) Lemma 79. Every proper ideal in a commutative algebra with unit is con- tained in a maximal ideal. (Recall that an ideal I in a commutative algebra B is a vector subspace of B such that if a B and b I then ab I . An ideal J is maximal if J 6 = B but whenever an ideal K satisfies J K B either K = J or K = B .) Lemma 80. Every maximal ideal M in a commutative Banach algebra with unit is closed. Lemma 81. If M is a maximal ideal in a commutative Banach algebra with unit then the quotient B/M is isomorphic to C as a Banach algebra. The notion of a maximal ideal is closely linked to that of a multiplicative linear functional. 23
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Definition 82. A multiplicative linear functional on a Banach algebra is a non-trivial (i.e not the zero map) linear map χ : B C such that χ ( xy ) = χ ( x ) χ ( y ) for all x, y B . Lemma 83. If B is commutative Banach algebra with identity and χ is a multiplicative linear functional then the following results hold. (i) ker χ is a maximal ideal. (ii) The map x + ker χ 7→ χ ( x ) is an algebraic isomorphism of B/ ker χ with C . (iii) χ is continuous and k χ k = 1 . Theorem 84. If B is commutative Banach algebra with identity then the mapping χ 7→ ker χ is a bijection between the set of multiplicative linear functionals on B and its maximal ideals. We now have the following useful corollary. Lemma 85. If B is commutative Banach algebra with identity then an el- ement x B is invertible if and only χ ( x ) 6 = 0 for all multiplicative linear functionals χ . The Banach algebra proof Theorem 87 was the first result to convince classical analysts of the utility of these ideas. The lemma that precedes it places the result in context. Lemma 86. If f C ( T ) has an absolutely convergent Fourier series (that is to say, -∞ | ˆ f ( n ) | < ) then f ( t ) = X -∞ ˆ f ( n ) exp( int ) . Theorem 87 (Wiener’s theorem). Suppose f C ( T ) has an absolutely convergent Fourier series. Then, if f ( t ) 6 = 0 for all t T , 1 /f also has ian absolutely convergent Fourier series. Exercise 88. Let B be any Banach space. Make it into a Banach algebra by defining xy = 0 for all x, y B . Now add an identity in the usual manner. Identify all the multiplicative linear functionals. 11 The Gelfand representation Throughout this section B will be a commutative Banach algebra with a unit e and M will be the space of maximal ideals. If x B and M ∈ M we 24
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know by Theorem 84 that there is a unique multiplicative linear functional χ M with kernel M so we may write M ( x ) = χ M ( x ) the space. We give M the weak star topology, that is to say, the smallest topology containing sets of the form { M ∈ M : | M ( x ) - M 0 ( x ) | < ² } with M 0 ∈ M and x B .
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