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Larsen R. — Banach algebras: An Introduction

Larsen R. — Banach algebras: An Introduction

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Название: Banach algebras: An Introduction

Автор: Larsen R.

Аннотация:

From author: "The exposition in the following pages is an elaboration and expansion of lectures I gave to second year mathematics graduate students at Wesleyan University during the academic years 1970-71 and 1971-72. The aim of the exposition is to provide a compact introduction to the theory of Banach algebras that not only acquaints the reader with fundamental portions of the abstract theory but also illustrates the usefulness of Banach algebras in the study of harmonic analysis and function algebras and gives the reader the basic tools necessary for further work in these areas. The first half of the book is devoted primarily to the general theory of Banach algebras, while in the second half the emphasis is on various more specialized topics related to harmonic analysis and function algebras - among which are: Wiener's Tauberian Theorem, the problem of spectral synthesis, the Bishop, Choquet, and 5ilov boundaries, representing measures, Werner's Maximality Theorem, the Commutative Gel'fand-Naimark Theorem, Plancherel's Theorem, the Pontryagin Duality Theorem, almost periodic functions, and the Bohr compactification.
An intelligent reading of the book presupposes the usual mathematical equipment possessed by second year mathematics graduate students with regard to topology, algebra, and real and complex analysis, as well as a reasonably good knowledge of basic functional analysis..."

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Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1973

Количество страниц: 358

Добавлена в каталог: 03.12.2010

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Предметный указатель

$(t,\gamma)$       114
$AC(\Gamma)$       17
$AC(\Gamma)$ , Gel’fand representation of, maximal ideal space of      102
$AC(\Gamma)$ , Silov boundary for      228
$A\neq{o}$ , assumed after page      4
$A^{-1}$       29
$A_{-1}$       29
$A_{-1}$ , connected component of the identity in      156 158
$A_{T}$       316
$B(\sigma(T))$       317 97
$C^{n}([a,b])$       17
$C^{n}([a,b])$ , Gel’fand representation of      92
$C^{n}([a,b])$ , maximal ideal space of      92
$C^{n}([a,b])$ , Silov boundary for      228
$C^{X}$       16
$C_{O}(X)$       16
$C_{O}(X)$ , Gelf’and representation of      88
$C_{O}(X)$ , is regular      167
$C_{O}(X)$ , maximal ideal space of      88
$C_{O}(X)$ , Silov boundary for      227
$I_{E}$       11
$I_{O}(E)$       182
$J_{O}(E)$       182
$L_{1}(G)$ , closed ideals when G is compact      197
$L_{1}(G)$ , closed ideals when G is compact ilov boundary for      228
$L_{1}(G)$ , closed ideals when G is compact is regular      173
$L_{1}(G)$ , closed ideals when G is compact is semisimple      117
$L_{1}(G)$ , closed ideals when G is compact is Tauberian      184
$L_{1}(G)$ , closed ideals when G is compact maximal ideal space of      114 126
$L_{1}(G)$ , closed ideals when G is compact satisfies Ditkin's condition      214
$L_{1}(G)$ , closed ideals when G is compact sets of spectral synthesis      215
$L_{1}(G)$ , closed ideals when G is compact sets of spectral synthesis when G is compact      198
$L_{1}(G)$ , closed ideals when G is compact, closed translation-invariant linear subspace of      186
$L_{1}(G)$ , closed ideals when G is compact, condition to have an identity      107
$L_{1}(G)$ , closed ideals when G is compact, contains an approximate identity      110 188
$L_{1}(G)$ , closed ideals when G is compact, failure of spectral synthesis when G is noncompact      199
$L_{1}(G)$ , closed ideals when G is compact, Gel’fand representation of      114
$L_{p}(G)$       19
$L_{\infty}(X, S, \mu)$       18
$L_{\infty}(X, S, \mu)$ , Gelf'and representation of      93
$L_{\infty}(X, S, \mu)$ , maximal ideal space of      93
$M_{e}$       12
$N(K,\epsilon)$       293 309—310
$T^{x}$       46
$t_{f}$       280
$T_{s}(f)$       18 104
$T_{x}$       46
$U(K, \epsilon)$       123
$U(\tau; \epsilon; x_{1}, x_{2}, ..., x_{n})$       71
$x^{*}$       139 273
$x^{-1}$       6
$X^{A}$       16
$x_{-1}$       14
$\gamma$       17 21 116
$\lambda$       18
$\Lambda_{O}(G)$       282
$\mathbb{R}$       3
$\mathbb{Z}$       3
$\mu * \nu$       20
$\overline{E}$       162 71 168
$\overline{\mathbb{R}}$       115—116
$\partial A$       225
$\rho A$       221
$\sigma(x)$       54
$\tau$       66
$\tau_{e}$       69—70
$\tau_{t}$       86
$\tau_{\infty}$       70
$\triangle(A)$       69
$\triangle(A)$ , assumed nonempty after page      78
$\widehat{A}$       74
$\widehat{f}$       287
$\widehat{G}$       113
$\widehat{x}$       74
$\widehat{\Gamma}$       121
$\widehat{\mu}$       128
$\widetilde{f}$       105
$\|T\|$       22
$|\mu|$       20
$||f||_{AC}$       22
$||f||_{n}$       18
$||X||_{\Sigma}$       134
$||\mu||$       20
A(d)      17
A(D), Bishop boundary for      228
A(D), Gelfand representation of, is not regular      167—168
A(D), maximal ideal space of      97
A(D), Silov boundary for      228
A(K)      17
A(X)      150
Adjoining an identity      10
Algebra      3
Algebra of absolutely convergent Fourier series      22
Algebra, Banach      4
Algebra, commutative      3
Algebra, division      35
Algebra, normed      4
Algebra, radical      82
Algebra, separating function      220
Algebra, uniform      255
Almost periodic compactification      329
Almost periodic function      18 323
AP (G)      18
Applications of boundaries      243—249
Approximate identity      109
Approximate identity in $L_{1}(G)$       109 188
Arens      1
Arens — Royden theorem      158
Arens, Theorem      48
A[e]      9
B*-algebra      273
B*-algebra, examples of      273—275 282 323
Banach algebra      4
Banach algebra at infinity      201
Banach algebra with involution      273
Banach algebra, belong locally to I, at a point      201
Banach algebra, commutative semisimple      81
Banach algebra, conditions to have an identity      153—154 178
Banach algebra, finitely generated      98
Banach algebra, maximal ideal discrete topological group      106
Banach algebra, normal commutative      176
Banach algebra, regular commutative      166
Banach algebra, self-adjoint commutative      132 102
Banach algebra, Tauberian commutative      183
Bdy(E)      51
Beurling — Gel’fand Theorem      81
Bishop boundary for a commutative Banach algebra      226
Bishop boundary for a separating function algebra      221
Bishop boundary for A(D)      228
Bishop boundary for C(X), X compact metric      227—228
Bishop boundary, example of nonexistence of      230—232
Bishop boundary, existence for a uniform algebra      268
Bishop-deLeeuw Theorem      236—237
Bohr compactification      329
Bohr compactification, uniqueness of      328
Boundary, applications of      243—249
Boundary, applications of Bishop      221
Boundary, applications of Choquet      242
Boundary, applications of existence of      222
Boundary, applications of Silov      225
Boundary, applications of, existence of      242
Boundary, applications of, for a commutative Banach algebra      226
Boundary, applications of, for a separating function algebra      221
C(V)      23
C(X)      16
C(X) is regular      167
C(X), Bishop boundary when X is compact metric      227—228
C(X), Choquet boundary for      242
C(X), closed ideals in      195
C(X), example of $\rho C(X)=\emptyset$       230—232

C(X), Gel’fand representation of      87—88
C(X), maximal ideal space of      86 90
C(X), sets of spectral synthesis for      196
C(X), Silov boundary for      227
C*-algebra      275
CHARACTER      331
Character, continuous      113
Characterization of singular elements in C(X)      43
Characterization of singular elements in commutative Banach algebras      81
Characterization of singular elements in self-adjoint semisimple commutative Banach algebras      141
Choquet boundary, characterization of      257 263
Choquet boundary, characterization of existence of      242
Choquet boundary, characterization of, for A(D)      242
Choquet boundary, characterization of, for C(X)      242
Choquet boundary, characterization of, for separating function algebras      242
Choquet — Bishop-deLeeuw Theorem      271
Closed convex hull      233
Closed ideals in C(X)      195
Closed ideals in C(X) in $L_{1}(G)$ , G compact      197
Closure operation      162
Commutative algebra      3
Commutative Gel’fand — Naimark Theorem      277
Compactification      323
Compactification, almost periodic      329
Compactification, Bohr      329
Compactification, Stone — Cech      90
Complex homomorphism      67
Connected component      1551
Connected component of the identity in $A^{-1}$       156 158
Continuity of inversion      31
Continuity of inversion of quasi-inversion      29
Continuous character      113
Contour, regular      143
Contour, spectral      144
Convex hull      233
Convex hull, closed      233
Convolution      19—20
Ditkin's condition      204 340
Division algebra      35
Dual group      127
EXP      156
Ext(E)      233
Extreme point      233
Finitely generated Banach algebra      98
Fourier coefficient      21
Fourier coefficient, homomorphism, complex      67
Fourier coefficient, transformation      21 115 277
Fourier — Stieltjes transform      128
Fourier — Stieltjes transform, homomorphism space      69
Function, almost periodic      18 323
Function, hull      160
Function, hull-kernel closure      162
Function, slowly oscillating      190
Functions which operate      151
Functions which operate, topology      163
Gel’fand representation      76
Gel’fand representation of      99
Gel’fand representation of $C^{n}([a, b])$       92
Gel’fand representation of $C_{O}(X)$       88
Gel’fand representation of $L_{1}(G)$       114
Gel’fand representation of $L_{\infty}(X, S, \mu)$       93
Gel’fand representation of a finitely generated commutative Banach algebra      99
Gel’fand representation of A(D)      97
Gel’fand representation of AC( $\Gamma$ )      102
Gel’fand representation of C(X)      87—88
Gel’fand Representation Theorem      74
Gel’fand topology      71
Gel’fand transform      76
Gel’fand transformation      76
Gel’fand — Mazur theorem      35
H(I)      160
Haar measure      18
Ideal      4
Ideal, left      4
Ideal, maximal      4
Ideal, modular      7
Ideal, primary      216
Ideal, proper      4
Ideal, regular      7
Ideal, right      4
Ideal, two-sided      4
Identity modulo I      7
Int(K)      17
Inverse      5
Inverse, left      5
Inverse, right      5
Inversion theorem      298 301
Invertible      5
Involution      273
Isomorphism, topological      131
Joint spectrum      98
K(E)      160
Kernel      160
Kronecker approximation theorem      332
L(v)      22
Left ideal      4
Left ideal, inverse      5
Left ideal, modulus of integrity      46
Left ideal, quasi-inverse      13
Left ideal, topological zero divisor      40
Left ideal, zero divisor      40
Linear functional, multiplicative      67
Linear functional, positive      259
Local Maximum Modulus Theorem      220
M(G)      20
M(G) is semisimple      315
M(G), maximal ideal space of      128
Malliavinfs Theorem      199
Maximal ideal      4
Maximal ideal space      69
Maximal ideal space of $AC(\Gamma)$       102
Maximal ideal space of $C^{n}([a,b])$       92
Maximal ideal space of $C_{O}(X)$       88
Maximal ideal space of $L_{1}(G)$       114 126
Maximal ideal space of $L_{\infty}(X, S, \mu)$       93
Maximal ideal space of $M^{\infty}(G)$       128
Maximal ideal space of a finitely generated commutative Banach algebra      99
Maximal ideal space of A(D)      97
Maximal ideal space of C(X)      86 90
Maximum modulus set      221
Maximum modulus theorem      218—219
Mergelyanfs Theorem      101
Modular ideal      7 342
Modulus of integrity      46
Modulus of integrity, left      46
Modulus of integrity, right      46
Multiplicative linear functional      67
Nilpotent      33
Nilpotent, topological      33
Noncommutative Gel’fand — Naimark Theorem      280
Normal commutative Banach algebra,quasi-invertible      13 176
Normed algebra      4
Plancherel transform      169 305
Plancherel transformation      169 305
Plancherelfs Theorem      169 301 307—308
Polynomial Spectral Mapping Theorem      57
Pontryagin duality theorem      312
Positive linear functional      259
Primary ideal      216
Proper ideal      4
Quasi-inverse      13
Quasi-inverse, left      13
Quasi-inverse, right      13
Quasi-regular      13
Quasi-singular      13
r(f)      55
R(K)      17
Rad(A)      82
Radical      82
Radical algebra      82

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