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7330hw5

# 7330hw5 - Math 7330 Functional Analysis Homework 5...

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Math 7330: Functional Analysis Fall 2002 Homework 5: Commutative Banach Algebras II A. Sengupta We work with a complex commutative Banach algebra B . It had been shown that the set G ( B ) of all invertible elements in B is an open subset of B . A proper ideal I in B cannot contain any invertible elements (for if x I is invertible then for any y B we would have y = ( yx - 1 ) x I , which would mean I = B ), i.e. is a subset of the closed set G ( B ) c . Zorn’s lemma shows that every proper ideal of B is contained in a maximal ideal. 1. Let J be an ideal of B . (i) Check that the closure J is also an ideal. (ii) Show that if J is a proper ideal then so is its closure J . (iii) Show that if J is a maximal ideal then J is closed. Hint: Consider the ideal J . It is an ideal which contains J . Since J , being maximal, is proper, (ii) implies that J is a proper ideal. 1

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A mapping φ : B C is a complex homomorphism if f is linear and satisfies f ( xy ) = f ( x ) f ( y ) for all x, y B . Note that then f ( x ) = f ( xe ) = f ( x ) f ( e ) for every x B , and so either f ( e
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7330hw5 - Math 7330 Functional Analysis Homework 5...

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