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

ed2k: ed2k stats

Год издания: 1960

Количество страниц: 394

Добавлена в каталог: 01.11.2006

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Предметный указатель

Nilpotent      see “Topologically nilpotent”
Norm      2
Norm, $B^{\star}$ -condition (weakened)      180 (4.1.13) (4.2.2)
Norm, $\mathfrak{B}$ -admissible norms      53
Norm, Auxiliary      181 186
Norm, cross-norm      287
Norm, equivalent norms      70
Norm, incomplete norm in $C(\Omega)$       76
Norm, induced by a Banach algebra of operators      (2.4.11) 66
Norm, infimum norm in a difference space      44 (2.2.6)
Norm, majorizes another norm      53 (2.4.17)
Norm, minimal property in $C(\Omega)$       176
Norm, pseudo-norm      11 226 229
Norm, uniqueness of norm topology      ( $\S$ 2.5) 70—76 (4.1.18) (4.4.11)211
Normal element of $^{\star}$ -algebra      178
Normal positive functional      295
Normal subset of a $^{\star}$ -algebra (maximal)      179 (4.1.3)
Normed algebra      2
Normed division algebras      ( $\S$ 1.7) 38—40
Normed ring      3
Operational calculus in $B^{\star}$ -algebras      (4.8.7) 241
Operators of finite rank      64 76 102 107
Operators of scalar type      291
Operators, spectral operators      (A. 1.6) 290—292
Partition of the identity      (2.7.12) 88
Permanently singular elements      20 27
Polynomial approximation      309
Polynomial in a system of indeterminants      149
Polynomially generated subalgebra      164
Pontrjagin duality theorem      326
Positive definite functions      323
Positive definite scalar product      195
Positive functionals      212
Positive functionals and $^{\star}$ -representations      ( $\S$ $\S$ 4.5 4.6)
Positive functionals, admissible      213—215
Positive functionals, extension of      227
Positive functionals, extreme points and irreducible $^{\star}$ -representations      (4.6.4) 223
Positive functionals, included in a positive functional      220
Positive functionals, integral representation of      (4.6.15) 230
Positive functionals, normal      295
Positive functionals, on a $B^{\star}$ -algebra      (4.8.16)247
Positive functionals, relatively bounded      214—215
Positive functionals, representable      216—220
Positive functionals, weakly positive      (4.7.1) 230 (4.7.3) (4.7.8)233
Power series (Banach algebra of)      (A.2.12) 317—318
Preservation of spectrum      (1.6.9) 32 (1.6.13) (1.6.14) (3.7.6) (4.1.10) (4.8.2) (4.8.5)
Primary algebra      (2.7.26) 92
Primary completely regular algebra as sum of primary algebras      (2.7.27) 93
Primary components of $C^{(n)}(0, 1)$       301
Primary ideals in an algebra      (2.7.26) 92
Primary intersection of primary ideals      302
Primitive Banach algebras      ( $\S$ 2.4) 60—70
Primitive Banach algebras, $^{\star}$ -algebras      (4.10.5)262
Primitive Banach algebras, commutative      108
Primitive Banach algebras, fundamental isomorphism theorem      (2.5.19) 76
Primitive Banach algebras, ideals in      65 (2.4.18)69 179
Primitive Banach algebras, minimal ideals in      65 67 68
Primitive Banach algebras, representation of      (2.4.7) 62 (2.4.12)
Primitive ideals      53—55 179
Principal ideals      21
Product spaces      113
Product spaces, canonical representation of carrier space in      113
Product spaces, projections in      153
Projections      153 178
Pseudo-norm in $^{\star}$ -algebras      (4.6.9) 226 229
Quasi-inverse      16
Quasi-inverse, continuity of      (1.4.22) 19
Quasi-inverse, series expansion for      (1.4.18) 18
Quasi-regular element      16
Quasi-regular ideal      42
Quasi-singular element      16
Quotient of an ideal      50 (2.2.5)
Radical      ( $\S$ 2.3) 55—59 (3.1.8)
Radical, algebras      55 (A.2.11)
Radical, as a topologically nil ideal      (2.3.4) 56
Radical, as intersection of (maximal, primitive) ideals      (2.3.2) 55
Radical, as sum of quasi-regular ideals      (2.3.2) 55 (see $^{\star}$ -radical Strong
Radical, failure of the Wedderburn theorem      297
Rank of an operator      64
Rational functions on $\Omega$       156
Rationally generated subalgebra      164
Real algebras      2
Reduced algebras      230
Reducing ideal      230
Reduction of a representation      48
Reflexive Banach algebras      296—297
Regular algebras      83 96 173
Regular almost regular algebras      96
Regular completely regular algebras      ( $\S$ 2.7) 83—96 (
Regular elements      9
Regular elements, connected in $\mathcal{B}(\mathfrak{B})$       280
Regular elements, dense in $B(\Omega)$       295
Regular elements, in a $^{\star}$ -algebra      182 184—186
Regular elements, uncountably many components      314 (see also “Group of regular elements”)
Regular ideal (= modular ideal)      47
Regular representations      3 49 50
Regular ring      96
Relative inverse      96
Relatively bounded positive functional      214
Representable positive functionals      216—219 246
Representations      ( $\S$ 2.2) 47—54
Representations, bounded      4
Representations, canonical (of a carrier space)      113
Representations, continuous (uniformly, strongly)      4 48
Representations, cyclic (strictly, topologically)      47
Representations, equivalent (algebraically, topologically)      47 (2.4.15)
Representations, faithful      3
Representations, invariant subspaces of      47
Representations, irreducible (strictly, topologically)      48 (2.2.1) 52 108
Representations, kernel of      49
Representations, normed (norm preserving)      4
Representations, of carrier spaces      ( $\S$ 3.4) 149—156
Representations, on a difference space      50
Representations, on an ideal      49
Representations, reduction to a difference space      48
Representations, regular (left, right, extended)      3—4 49 50
Representations, restriction to a subspace      48
Representations, strongly irreducible      264 (see also “ $^{\star}$ -representations”)
Resolution of the identity      291
Resolvent (of an element, equations)      38
Restriction of a representation      48
Reverse (= quasi-inverse)      19

Ring of operators      289
Rouche’s Theorem      (3.3.22) 145
SBI-ring      58
Scalar product      195
Scalar product, a bound for      194
Scalar product, degenerate and non-degenerate      195
Scalar product, positive definite      195
Schmidt-class      285
Schur’s lemma      61
Schur’s lemma, continuous analogue of      212
Schur’s lemma, generalization of      (4.4.12) 211
Self-adjoint algebra      (3.2.6) 122 178
Self-dual vector spaces (normed, equivalent)      (4.3.1) 194 195
Semi-simplicity      (2.3.1) 55
Semi-simplicity, $^{\star}$ -semi-simplicity      210
Semi-simplicity, $^{\star}$ -subalgebras of an $A^{\star}$ -algebra      (4.1.19) 188
Semi-simplicity, strong, weak semi-simplicity      59
Semi-simplicity, subalgebra of commutative algebra      (2.3.7) 57
Separate points (strongly)      120
Separating (element, function, ideal)      70—73 (4.4.11)211
Sets of constancy      (3.2.14) 125 126
Silov boundary      ( $\S$ 3.3) 132—148
Silov boundary, existence      (3.3.1) 133
Silov boundary, for $^{\star}$ -algebras      189
Silov boundary, for subalgebras      146—147
Silov boundary, non-existence      (A.2.9) 311
Singular elements      9
Singular elements, interior points of      278
Singular elements, of a $^{\star}$ -algebra      185
Singular elements, permanently singular elements      20 27
Singular elements, which are topological divisors of zero      (1.5.4)22 279 295
Socle (left, right)      46
Socle (left, right), existence of      (2.1.11) 46
Socle (left, right), of a $^{\star}$ -algebra      (4.10.2) 261 (4.10.4)262 (4.10.16)267
Socle (left, right), of a strictly dense algebra      65 (2.4.12)67
Socle (left, right), of an annihilator algebra      100—102
Special maximum set      138—140
Special set      139
Special set, minimal special set      139
Spectral mapping theorem      33 (3.5.1)
Spectral measure      290
Spectral operators      (A. 1.6) 290—291
Spectral operators of scalar type      291
Spectral part of a spectral operator      291
Spectral radius (formula)      (1.4.1) 10 (1.6.5) 38
Spectral synthesis      327
Spectrum      ( $\S$ 1.6) 27—38
Spectrum, characterization for real algebras      (1.6.6) 30
Spectrum, continuity of      35—37 282
Spectrum, existence      (1.6.4) 30
Spectrum, for $^{\star}$ -algebras      (4.1.1) 182
Spectrum, for commutative algebras      (3.1.6) 111
Spectrum, joint spectrum      150
Spectrum, of an algebra      81
Spectrum, properties of the boundary      (1.6.12)33 (3.3.18)142 (3.3.19)
Square roots of elements      (4.1.4) 183 (4.7.2)231 243
Square roots of elements, elements without square roots      280
Stone — Cech compactification      123
Stone — Weierstrass theorem      (3.2.12) 124
Stone — Weierstrass, theorem, Silov generalization      (3.2.20) 127
Stonian space      294
Strictly dense algebra of operators      (2.4.2) 60
Strictly dense algebra of operators with minimal ideals      (2.4.14) 68
Strong boundary (points)      (3.3.9) 138 141
Strong irreducibility      264
Strong radical and semi-simplicity      (2.3.10) 59
Strong structure space      (2.6.3) 78 179
Structure spaces      (2.6.3) 78
Structure spaces, compactness properties of      79 258
Structure spaces, example of non-Hausdorff      82
Structure spaces, of $B^{\star}$ -algebras      255—260
Structure spaces, of annihilator algebras      101 267
Structure spaces, of completely regular algebras      83 (3.7.1)
Structure spaces, of ideals and difference algebras      (2.6.6) 79
Structure theorems      (2.6.1) 77 (2.7.27) (2.8.15) (3.1.20) (3.6.5) 176 (4.4.8) (4.9.23) (4.10.12) (4.10.14) (4.10.24) (4.10.29) (4.10.31)274 (4.10.32)275
Subalgebra, contains inverses      (1.6.14) 35 (4.1.3) (4.1.9) (4.7.18)
Subalgebra, generated by a system of generators      112
Subdirect sum, carrier space of      128
Subdirect sum, closed under multiplication by functions      (2.7.28) 93
Subdirect sum, continuous, normed      177 (see also “Direct sum” “Sum”)
Sum of ideals      46 81 “Topological
Symmetric $^{\star}$ -algebras      ( $\S$ 4.7) 230—239
Symmetric $^{\star}$ -algebras, $B^{\star}$ -algebras      (4.8.9) 243
Symmetric $^{\star}$ -algebras, adjunction of an identity to      (4.7.9) 233
Symmetric $^{\star}$ -algebras, annihilator $^{\star}$ -algebras      (4.10.11) 266
Symmetric $^{\star}$ -algebras, closed $^{\star}$ -subalgebra      (4.7.7) 233
Symmetric $^{\star}$ -algebras, criterion for      (4.7.21) 238
Symmetric $^{\star}$ -algebras, hermitian involution      (4.7.6) 233
System of generators      112
Tauberian algebra      92
Taylor’s theorem (generalized)      318
Topological algebra (ring)      2
Topological divisors of zero      ( $\S$ 1.5) 19—27 57 (2.5.6) 137 185
Topological sums of ideals      46 (2.6.8) (2.8.15) (4.10.12) (4.10.14) (14.10.31)
Topologically nil ideal      56
Topologically nilpotent elements      (1.4.3) 12 278 282
Topologically simple algebras      (2.8.20) 101 107 (4.10.20)
Topology, $\mathfrak{U}$ -topology      110 222
Topology, hull-kernel topology      78 115
Totally symmetric (= symmetric)      239
Trace-class (-function)      288
Transitivity      (2.4.2) 60
Type C      177 301
Unimodular groups      320
Unique maximum point      141
Uniqueness of norm topology in $^{\star}$ -algebras      (4.1.18) 188 (4.4.11)
Uniqueness of norm topology in a Banach space under an algebra of operators      (2.2.7) 53
Uniqueness of norm topology in Banach algebras      ( $\S$ 2.5) 70—76 (2.8.17)
Unitarily equivalent $^{\star}$ -representations      205—209 (4.10.22) (4.10.23)271 284
Unitary elements (span a $^{\star}$ -algebra)      (4.1.5) 184
Upper semi-continuous decomposition      125 (3.2.15)
Upper semi-continuous functions      129 (4.9.17)
Upper semi-continuous spectrum Sp(x) (as a function of x)      (1.6.17) 36
Vanish at infinity      (2.7.31) 95 118 (4.9.18)258
Von Neumann algebras      289
Wedderburn structure theorems, failure of the principal theorem for $H^{\star}$ -algebras      (4.10.29) 273 (4.10.31) (4.10.32)
Wedderburn structure theorems, failure of the principal theorem for Banach algebras      297
Wedderburn structure theorems, principal theorem      59 292
Weierstrass approximation theorem      124
Weight function      312
Wiener algebras      299
Wiener lemma      299
Wiener Tauberian theorem      92 326

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