## Linear Operators: Spectral theory |

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Page 868

If I is a closed ideal in the commutative B - algebra X then the quotient algebra X /

? is isometrically isomorphic to the

maximal . PROOF . If I is not maximal it is properly contained in an ideal and so X

/ I ...

If I is a closed ideal in the commutative B - algebra X then the quotient algebra X /

? is isometrically isomorphic to the

**field**of complex numbers if and only if I ismaximal . PROOF . If I is not maximal it is properly contained in an ideal and so X

/ I ...

Page 1048

Each point æ in E " may be written uniquely as x = rw , where re R , WES , and

the mapping [ r , w ] + x = rw is evidently a homeomorphism of RXS onto Em .

Thus the o -

BR of ...

Each point æ in E " may be written uniquely as x = rw , where re R , WES , and

the mapping [ r , w ] + x = rw is evidently a homeomorphism of RXS onto Em .

Thus the o -

**field**B . of Borel subsets of E " is the product o -**field**of the o -**field**BR of ...

Page 1153

Thus the product group has a Haar measure 2 ( 2 ) defined on its Borel

. It is natural to expect that the product measure 2 x a coincides , up to a constant

multiple , with 2 ( 2 ) . This fact will be established in Theorem 7 . 6 LEMMA .

Thus the product group has a Haar measure 2 ( 2 ) defined on its Borel

**field**{ ( 2 ). It is natural to expect that the product measure 2 x a coincides , up to a constant

multiple , with 2 ( 2 ) . This fact will be established in Theorem 7 . 6 LEMMA .

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero