Riesz Spaces

Riesz Spaces
Volume I
W. A. J. LUXEMBURG
California Institute of Technology, Pasadena, California
and
A. C. ZAANEN
Leiden University, Leiden, The Netherlands
1971
NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM • LONDON
Table of Contents
PREFACE
v
CHAPTER 1. DISTRIBUTIVE LATTICES AND NORMED FUNCTION SPACES
1
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Partial ordering
Lattices
Boolean algebras
Boolean rings
Prime ideals
Hulls and kernels; Stone's representation theorem
The hull-kernel topology
Compactness and separation properties of the hull-kernel topology
Normed Kothe spaces
Orlicz spaces
1
3
5
7
19
24
29
34
41
45
CHAPTER 2. ELEMENTARY PROPERTIES OF ORDERED VECTOR
SPACES AND RIESZ SPACES
48
11.
12.
13.
14.
15.
16.
Ordered vector spaces and Riesz spaces
Inequalities and distributive laws in a Riesz space
Suprema and inflma of subsets of an ordered vector space
. . .
Disjointness
Monotone sequences and directed sets
Order convergence and relatively uniform convergence. . . . .
CHAPTER 3. IDEALS AND BANDS; ARCHIMEDEAN RIESZ
SPACES
17.
18.
19.
20.
21.
48
62
66
67
70
78
92
Riesz subspaces, ideals and bands
93
Riesz homomorphisms and quotient spaces
98
Disjoint complements
105
The band generated by an ideal
107
Order dense ideals, quasi order dense ideals and super order dense
ideals
109
22. Archimedean Riesz spaces
;
112
IX
X
TABLE OF CONTENTS
CHAPTER 4. DEDEKIND COMPLETENESS AND PROJECTION
PROPERTIES
122
23.
24.
25.
26.
27.
28.
29.
Dedekind completeness
Projection properties
The main inclusion theorem
Atoms and flnitedimensional Riesz spaces
Maximal ideals
Order bases and disjoint systems
Bands or ideals with a countable order basis and the relation with order
separability
• • 30. The Boolean algebra of all projection bands.
31. Principal projection bands
32. Dedekind completion of an Archimedean Riesz space
124
131
137
145
153
160
CHAPTER 5. PRIME IDEALS
199
33.
34.
35.
36.
37.
164
171
181
185
Prime ideals
200
Prime ideals in C(X) for X compact and Hausdorff
207
Hulls and kernels
2\2
The hull-kernel topology
218
Compactness and separation properties of the hull-kernel topology 223
CHAPTER 6.
FREUDENTHAL'S SPECTRAL THEOREM .
.
.
.
247
38.
39.
40.
41.
42.
Spectral systems
. 249
Uniform Cauchy sequences
252
Freudenthal's spectral theorem
253
The Radon-Nikodym theorem and the Poisson formula . . . .
269
Uniform completeness, Dedekind completeness and the spectral
theorem
276
43. The spaces C(X) and Cb(X)
. . .
282
CHAPTER 7.
SPECTRAL REPRESENTATION THEORY . . . .
44. The Johnson-Kist representation theorem
45. The Yosida spectral representation theorem in an Archimedean Riesz
space with strong unit
46. The Yosida spectral representation theorem in an Archimedean Riesz
space
. 47. The space C"* (X) for X. extremally disconnected
48. Subsets of the set of all bands
. 49. The Ogasawara-Maeda representation theorem
50. The Ogasawara-Maeda representation theorem for an Archimedean
space with a weak unit or with a disjoint order basis
51. The Nakano representation theorem
52. Prime ideal extension in a Riesz space . . .
292
294
305
319
322
326
329
337
341
358
TABLE OF CONTENTS
CHAPTER 8.
53.
54.
55.
56.
57.
58.
HERMITIAN OPERATORS IN HILBERT SPACE
XI
.
.
369
The ordered vector space of Hermitian operators in a Hilbert space 370
The square root of a positive Hermitian operator
377
The second commutant of a commuting set of Hermitian operators 382
The spectral theorem for Hermitian operators and normal operators 388
The structure of the second commutant
405
Kadison's anti-lattice theorem
415
CHAPTER 9.
RIESZ HOMOMORPHISMS AND QUOTIENT SPACES
419
59. Additional results on Riesz homomorphisms
60. Archimedean quotient spaces
61. Riesz spaces with the property that every quotient space is Archimedean
62. The ideal of the infinitely small elements and Riesz seminorms with
their quotient seminorms, in particular in spaces with a strong unit .
63. Riesz homomorphisms and the relatively uniform topology . . .
64. Images of principal bands under a Riesz homomorphism . . . .
65. Riesz a-homomorphisms and the order topology
66. Normal Riesz homomorphisms
421
426
429
435
441
446
449
456
CHAPTER 10. THE EGOROFF PROPERTY AND THE DIAGONAL
PROPERTY
458
67.
68.
69.
70.
71.
72.
73.
74.
75.
The Egoroff property
460
The diagonal property
470
The Egoroff property and the diagonal property in order bounded sets 475
Stability of order convergence and regular Riesz spaces . . . .
477
Function spaces possessing the diagonal property
481
The diagonal property for relatively uniform c o n v e r g e n c e . . . . 487
The Egoroff property in Boolean rings
491
Egoroff's theorem
494
The Egoroff property and Dedekind completeness
498
BIBLIOGRAPHY
508
SUBJECT INDEX
512

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