Rickart C.E. — General Theory of Banach Algebras :: Электронная библиотека попечительского совета мехмата МГУ

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Rickart C.E. — General Theory of Banach Algebras

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Название: General Theory of Banach Algebras

Автор: Rickart C.E.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1960

Количество страниц: 394

Добавлена в каталог: 01.11.2006

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Предметный указатель

$AW^{\star}$ -algebras      84 289—290
$A^{\star}$ -algebras      181
$A^{\star}$ -algebras, continuity of homomorphisms into      (4.1.20) 188
$A^{\star}$ -algebras, non-symmetric A $^{\star}$ -algebra      324
$A^{\star}$ -algebras, representation on Hilbert space      (4.8.12) 244
$A^{\star}$ -algebras, uniqueness of norm      188
$BQ^{\star}$ -algebras      194
$B^{\star}$ -algebras      3 180 ( 4.9) 239—260
$B^{\star}$ -algebras, adjunction of an identity      (4.1.13) 186
$B^{\star}$ -algebras, annihilator $B^{\star}$ -algebras      (4.10.14) 267 (4.10.20)
$B^{\star}$ -algebras, approximate identity      (4.8.14) 245
$B^{\star}$ -algebras, closed 2-sided ideals are $^{\star}$ -ideals      (4.9.2) 249
$B^{\star}$ -algebras, dual $B^{\star}$ -algebras      (4.10.25) 271
$B^{\star}$ -algebras, isometric isomorphisms are $^{\star}$ -isomorphisms      (4.8.19)248
$B^{\star}$ -algebras, operational calculus      (4.8.7) 241
$B^{\star}$ -algebras, representation as $C^{\star}$ -algebras      (4.8.11)244
$B^{\star}$ -algebras, structure of ideals      (4.9.6) 251 (4.9.8) (4.9.9) (4.9.13)
$B^{\star}$ -algebras, structure space properties      256—259
$B^{\star}$ -algebras, symmetry      (4.8.9) 243
$B^{\star}$ -algebras, topological implies strict irreducibility      (4.9.10)253
$B^{\star}$ -algebras, uniqueness of the involution      (4.8.18) 247
$C^{\star}$ -algebra      181
$H^{\star}$ -algebras      272—276 287
$H^{\star}$ -algebras, right $H^{\star}$ -algebras      276
$H^{\star}$ -algebras, Wedderburn structure theorems      (4.10.29) 273 (4.10.31) (4.10.32)
-algebras      (A.2.2) 296—298
$W^{\star}$ -algebras ( $AW^{\star}$ -algebras)      84 96 289—290
$\mathcal{C}$ -boundary      132
$\mathcal{C}$ -set (minimal)      132
$\mathfrak{U}$ -boundary      142
$\mathfrak{U}$ -topology      110
$^{\star}$ -algebras      178 (see also “Star algebras” “Algebras
$^{\star}$ -algebras with minimal ideals      ( $\S$ 4.10) 260—276
$^{\star}$ -homomorphism      180
$^{\star}$ -ideal      179
$^{\star}$ -isomorphism      180
$^{\star}$ -normed algebra      180
$^{\star}$ -radical      210 225—230 (4.7.15)
$^{\star}$ -radical, Kelley — Vaught characterization      (4.6.11)228
$^{\star}$ -representations      (4.3.3) 196
$^{\star}$ -representations, associated with hermitian functional      (4.3.7) 201
$^{\star}$ -representations, associated with positive functional      ( $\S$ $\S$ 4.5 4.6)
$^{\star}$ -representations, cyclic      205
$^{\star}$ -representations, direct sum of      (4.3.4) 198
$^{\star}$ -representations, essential      205 (4.10.24)
$^{\star}$ -representations, extension of      (4.7.20) 237
$^{\star}$ -representations, irreducible      205 (4.4.9) ( (4.10.24)
$^{\star}$ -representations, of a $^{\star}$ -normed algebra on a self-dual space      (4.3.9) 202
$^{\star}$ -representations, on Hilbert space      ( $\S$ 4.4) 205—212 217
$^{\star}$ -representations, unitary equivalence      205
$^{\star}$ -semi-simplicity      (4.4.9) 210 322
$^{\star}$ -semi-simplicity, implies $A^{\star}$ -algebra      (4.6.10) 226
$^{\star}$ -subalgebras      178
$^{\star}$ -subalgebras, which contain (quasi-) inverses      (4.1.9) 185
Absolute value of hermitian elements      243
Absolutely convergent Fourier series      (A.2.3) 298—300
Adjoint in an $H^{\star}$ -algebra      272
Adjoint of an operator on a self-dual space      196
Adjoint of an operator w.r.t. a bilinear form      63
Adjunction of an identity      3 4
Adjunction of an identity, to a $B^{\star}$ -algebra      (4.1.13) 186
Admissible positive functional      213
Adverse (= quasi-inverse)      19
Algebra of operators      ( $\S$ A.l) 277—292 (A.3.6)
Algebra of power series      (A.2.12) 317—318
Algebra of sequences      26
Algebra of set functions      (A.2.10) 312—316
Algebra with involution (= $^{\star}$ -algebra)      178
Algebra with minimal ideals      67—70 ( 96—107 (
Algebra without an involution      306
Algebra, generated by a set of elements      112
Algebraic polyhedron      151
Almost invariant functions      330
Almost periodic functions      (A.3.5) 331
Annihilator algebras      ( $\S$ 2.8) 96—107
Annihilator algebras, $^{\star}$ -algebras      266 287 289
Annihilator algebras, B $^{\star}$ -algebras      (4.10.20) 269
Annihilator algebras, necessary and sufficient condition      (2.8.23) 104
Annihilator ideals      96
Anti-symmetric algebras      304
Approximate identity      3
Approximate identity in $L^1(\mathfrak{C})$       321
Approximate identity in a B $^{\star}$ -algebra      (4.8.14) 245
Auxiliary norm      181 186
Baer rings      290
Banach algebra      2
Banach algebra of power series      317
Banach — Stone Theorem      248
Belong to A (near a point, at $\infty$ , locally)      88—90
Bilinear form      62
Bilinear form, non-degeneracy      62
Bilinear form, scalar product      195
Bohr compactification      331
Bound of a bilinear form      (2.4.8) 62
Bound of a scalar product      (4.3.1) 194
Boundary-value algebra      304
Bounded complete lattice of continuous functions      294
Canonical mapping and representation of the carrier space      113
Carrier space      (3.1.3) 110
Carrier space, approximation by finitely generated algebras      (3.4.6) 153
Carrier space, canonical representation of      113
Carrier space, homeomorphism induced by an involution      189
Carrier space, of an algebra with one generator      (3.4.5) 152
Carrier space, of an algebra with several generators      153 308
Carrier space, of subdirect sums      128—131
Carrier space, of the complexiflcation      (3.1.4) 111
Carrier space, representations of      ( $\S$ 3.4) 149—156
Carrier space, topology determined by system of generators      112
Cauchy integral formula      158
Cauchy integral formula, Herve generalization      160
Cauchy integral formula, Weil generalization      159
Cauchy — Schwartz inequality      195 213
CCR-algebras      284
Center of a normed algebra      35 61 71 85 86
Central and weakly central algebras      (2.7.6) 86
centralizer      211
Characterizations of $B(\Omega)$       295
Characterizations of $C^{\star}$ -algebras      (4.8.11) 244
Characterizations of $C_0(\Omega)$       (4.2.2) 190
Characterizations of $\mathcal{B}(\mathfrak{B})$       280
Characters of a group      325 330
Characters of a group, generalized      330
Circle operation      16
Commutative $^{\star}$ -algebras      ( $\S$ 4.2) 189—194
Commutative subset of an algebra      34 (1.6.14)
Commutativity modulo the radical      120
Compact groups      320 330
Compact operators      (4.10.17) 268 (4.10.20) (A.1.2)
Completely continuous (algebra)      284
Completely continuous (algebra), operators (= compact operators)      283
Completely regular algebras      ( $\S$ 2.7) 83—96 ( (A.2.3—A.2.5)
Complex algebras      2
Complexification of a real normed algebra      ( $\S$ 1.3) 5—9
Composition series      284
Cone      227
Continuity of the spectrum      (1.6.17) 36 282
Continuous at zero      (2.7.31)95
Continuous homomorphisms      (2.5.16) 74 (2.5.17) (4.1.20)
Continuous involution      180 (4.1.15)
Continuous subdirect sum      177
Continuously generated subalgebra      165—167 (3.7.4)
Convex set of positive functionals      (4.6.2) 222
Convolution algebras      (A.2.10—A.2.11) 312—316 (A.3.1-A.3.9)
Cross norm      287
Cyclic subalgebra (strictly, topologically)      47
Cyclic subspace w.r.t. a representation      206
Cyclic vector (strictly, topologically)      47
Decomposable algebra      167—169
Difference space (algebra)      44 144
Direct sum, decomposition      167

Direct sum, normed full direct sum      77 256
Direct sum, of $B^{\star}$ -algebras      271—272
Direct sum, of $^{\star}$ -representations      (4.3.4) 198 (4.4.7) (4.4.8) (4.10.25) 284 “Subdirect
Direct sum, of cyclic representations      (4.4.8) 209
Direct sum, of ideals      46 (2.8.15) (4.10.12) (4.10.14) (4.10.31)
Direct topological sum of ideals      46
Disc algebra      23 303
Division algebras      38—40
Divisors of zero      see “Topological divisors of zero”
Dual algebras      (2.8.1) 76 (2.8.27 2.8.28) (2.8.29) (4.10.20)269 (4.10.25) (4.10.26) 330
Dual mapping      116 175
Dual vector spaces      62—67
Ecto-ideal      318
Endo-ideal      318
Equivalence of holomorphic functions on $\Omega$       157
Equivalence of normed self-dual vector spaces      195
Equivalence of representations (algebraic, topological)      47
Equivalence, unitary equivalence      205—209
Essential $^{\star}$ -representation      205 (4.4.8)
Existence of an identity in a commutative Banach algebra      (3.6.6) 169
Exponential function      13
Extreme points      223 225 229 235 251 254
Extremely disconnected space      294
Finitely generated algebras      153
Frobenius’ theorem      40
Full subalgebra generated by a set of elements      291
Functionals      see “Hermitian functional” “Positive
Functions of bounded variation      (A.2.5) 302—303
Functions of class $C^{(n)}$       (A.2.4) 300—302
Functions, belong to $\mathfrak{C}$ on a set      (3.2.19) 127
Functions, belong to A (near a point, at oo, locally)      (2.7.13) 88
Functions, which operate on      51 167
Fundamental isomorphism theorem      (2.5.19) 76
Fundamental isomorphism theorem for $^{\star}$ -algebras      263 (4.10.9)
GCR-algebras      284
Gelfand representation theorem      (3.1.20) 119
Gelfand — Naimark Theorems      (4.2.2) 190 (4.8.11)244
Generalized analytic functions      148
Generalized divisors of zero      27
Generators of an Algebra (subalgebra)      112
Germ of a holomorphic function      157
Group algebras      ( $\S$ A.3) 318—332
Group algebras, commutative      (A.3.2) 325—328
Group algebras, compact groups      (A.3.4) 330—331
Group algebras, involution in $L^1(\mathfrak{C})$       322
Group algebras, non-symmetric group algebras      324—325
Group algebras, of operators      (A.3.6) 331—332
Group algebras, semi-simplicity of      323
Group of regular (quasi-regular) elements      ( $\S$ 1.4) 9—19
Group of regular (quasi-regular) elements, boundary consists of topological divisors of zero      (1.5.4) 22
Group of regular (quasi-regular) elements, number of components      (1.4.14) 15 294 314
Group of regular (quasi-regular) elements, principal component      (1.4.9) 13 (1.4.10) 19
Haar measure      319
Hausdorff structure space      82 83
Hermitian components of elements in a $^{\star}$ -algebra      179
Hermitian components of functionals on a $^{\star}$ -algebra      (4.3.6) 200
Hermitian elements      178
Hermitian elements, absolute value of      243
Hermitian elements, partial ordering of      231
Hermitian elements, positive and negative parts of      243
Hermitian functional      198
Hermitian functional, associated $^{\star}$ -representation      (4.3.7) 201
Hermitian functional, included in another      220
Hermitian functional, not a linear combination of positive functionals      305
Hermitian involution      (4.1.6) 184
Holomorphic functions      (A.2.6 A.2.7)
Holomorphic on $\Omega$       157
Holomorphically generated subalgebra      164
Holomorphically generated subalgebra, $\Theta$ -holomorphically generated subalgebra      165
Homomorphisms, $^{\star}$ -homomorphism      180
Homomorphisms, extensions of      (3.3.26) 147 156 163
Homomorphisms, into the complex field      109
Homomorphisms, natural homomorphism      44
Homomorphisms, of function algebras into Banach algebras      ( $\S$ 3.5) 156—167
Hull      (2.6.2) 78 81 (3.1.14) 116
Hull, compact hulls      (2.6.4 2.6.5) (2.7.10) (3.6.7) (3.6.14)
Hull-kernel topology      78
Hull-kernel topology, not equivalent to $\mathfrak{U}$ -topology      115 304
Hull-kernel topology, of the carrier space      115
Hull-kernel topology, of the structure spaces      78 82
Ideals (left, right      2-sided proper) 41
Ideals, contained in modular ideals      (2.7.25) 91 (4.2.6) 327
Ideals, dense in the algebra      (4.2.6) 194
Ideals, in $C_0(\Omega)$       (4.2.4) 193
Ideals, intersections of maximal modular ideals      327
Ideals, intersections of primary ideals      302
Idempotents      35 71 118 261 294 295
Idempotents, given by a decomposition of carrier space      (3.6.3) 168
Idempotents, minimal      (2.1.7) 45 261
Idempotents, modulo the radical      (2.3.9) 58
Idempotents, orthogonal (maximal family)      (2.1.7) 45 266
Idempotents, SBI-rings      58
Integral operators      285
Invariant, almost invariant functions      330
Invariant, family of functions on a group      319
Invariant, functional (integral)      319 (see also “Representations” “
Inverse (left, right)      9
Inverse (left, right), continuity of      (1.4.8) 13
Inverse (left, right), relative inverse      96
Inverse (left, right), series representation of      (1.4.4) 12 (see also “Quasi-inverse”)
Involution      178
Involution, $B^{\star}$ -condition      180
Involution, an algebra with no involution      306
Involution, continuity (local)      180 (4.1.5)
Involution, hermitian      (4.1.6) 184
Involution, in a symmetric algebra      (4.1.5) 233
Involution, in an algebra of operators      264
Involution, uniqueness in a $B^{\star}$ -algebra      (4.8.18) 247 296
Irreducibility of representations      48 (4.4.9) 221—224
Irreducibility, equivalent to 1-fold transitivity      60
Irreducibility, strictly irreducible      48
Irreducibility, strpngly irreducible      264
Irreducibility, topologically irreducible      48 205 (4.4.12)211
Isomorphism of group algebras      322
Joint spectrum      150
Kadison theorem      (4.9.10) 253
Kelley — Vaught characterization of $^{\star}$ -radical      (4.6.11)228
Krein extension theorem      227
Krein — Millman theorem      225 229 252
Left (right) primitive ideals      179
Linear functional      see “Hermitian functionals” “Positive
Liouville theorem      40
Local continuity      180
Local identities      170
Logarithm of elements in a Banach algebra      14
Matrix algebra (full)      275
Matrix algebra (full), representation of simple $H^{\star}$ -algebra      (4.10.32) 275
Matrix Units (complete system)      (4.10.21) 269 270
Maximal (-closed, modular) ideals      42—43 (2.8.5) (2.8.7) (3.1.2)
Maximal ideal space      82 120
Maximal subalgebras      305 306
Maximal subalgebras, commutative      (1.6.14) 35
Maximal subalgebras, normal      (4.1.3) 182
Maximizing set      132
Maximum Modulus Principle      132
Maximum modulus principle, local      148
Maximum point (unique)      141 (3.3.17)
Maximum set for a function (special)      138
Mazur — Gelfand theorem      1 (1.7.1) 40
Meso-ideal      318
Metric ring      3
Minimal $\mathcal{C}$ -set (special)      132 139
Minimal (-closed) ideals      41 43 46 98 99 262 267
Minimal boundary      142
Modular ideals      42 47
Multiplicative condition      2
N-algebra ( $N^{\star}$ -algebra)      92—93 (2.7.30) 300—303 326
Natural embedding      120—123
Natural homomorphism      44
Nil ideal      see “Topologically nil ideal”

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