## Linear Operators: Spectral theory |

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Page 860

A B - algebra X is commutative in case xy = yx for all x and y in X . An involution in

a

the class of analytic functions , are commutative

A B - algebra X is commutative in case xy = yx for all x and y in X . An involution in

a

**B**-**algebra**X is a mapping x + * * of X ... with the exception of L ( - 00 , 00 ) andthe class of analytic functions , are commutative

**B**-**algebras**with involutions .Page 868

Commutative

two - sided and the quotient algebra X / I is again a commutative algebra . It will

be a

Commutative

**B**-**Algebras**In case X is a commutative**B**-**algebra**every ideal I istwo - sided and the quotient algebra X / I is again a commutative algebra . It will

be a

**B**-**algebra**if I is closed ( 1 . 13 ) . It is readily seen that every ideal I in X ...Page 869

Every homomorphism of a commutative

system is continuous . 4 LEMMA . Let M be the set of maximal ideals in the

commutative

.

Every homomorphism of a commutative

**B**-**algebra**into the complex numbersystem is continuous . 4 LEMMA . Let M be the set of maximal ideals in the

commutative

**B**-**algebra**X . Then x ( M ) = Q ( x ) and sup \ X ( M ) ] = lim ( x " | 1 / ".

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

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