Lectures on Gleason Parts
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121 Herbert S. Bear University of Hawaii, Dept. of Mathematics, Honolulu, HI/USA
Lectures on Gleason Parts
Springer-Verlag Berlin· Heidelberg· New York 1970
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich Series: Mathematisches Institut der UniversiUit Erlangen-NOrnberg. Advisers: H. Bauer and K. Jacobs
121 Herbert S. Bear University of Hawaii, Dept. of Mathematics, Honolulu, HI/USA
Lectures on Gleason Parts
Springer-Verlag Berlin· Heidelberg· New York 1970
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § '4 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin· Heidelberg 1970. Library of Congress Catalog Card Number 74-114,,,. Printed in Germany. Title No. 3277.
CONTENTS section
o.
Standing assumptions and notation
• 1
1. Examples of Gleason parts ••
• 2
2. Proof of the d-I1-G i denti ty
4
3. Geometric description of parts in TB 4. Representing measures and parts in TC
9
.15
5. Inner parts and their associated normed spaces
.18
6. Selection of mutually absolutely continuous representing measures . .
.24
7. Integral kernels
.30
8. Geometric properties of d, and a continuous selection theorem
.36
9. Completeness of the part metric • . . . • • .
.38
10. Linear functionals as differences of positive
.41
O.
STANDING ASSUMPTIONS AND NOTATIONS
These lectures are concerned with complex function algebras, and real function spaces. We will give here the basic definitions and notations we will use throughout. Let X be a compact Hausdorf space. C(X) ,and CR(X) will denote the spaces of all continuous complex valued, and real valued, functions on X. We always use the sup-norm in C(X) , CR(X) , and subsets of these spaces: IIfll = sup {If(x) I: xeS}. A function algebra on X is a closed subalgebra A of C(X) , which contains the constant functions and separates the points of X
A function space on X is a linear subspace B of CR(X) , which contains
the constant functions and separates the points of X.
We will frequently be
concerned with the situation where A is an algebra on X and B = Re A . If A is an algebra on X , then there is a closed set r with the property that every function in A assumes its maximum modulus on r, and r is a subset of every closed set with this property.
The same statement holds
for a function space B. This set is called the Silov boundary of A or B in X.
If
B = Re A ,then A and B have the same Silov boundary.
The spectrum SA of a function algebra A is the set of all multiplicative linear functionals on A.
These homomorphisms are automatically
continuous, so SAc:. A' = the space of all
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