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Pełne teksty:
Warianty tytułu
Abstrakty
CONTENTS
Introduction........................................................................................................................... 5
1. Finite-dimensional C*-algebras.................................................................................. 8
The objects...................................................................................................................... 8
The morphisms.............................................................................................................. 9
The matrix calculus........................................................................................................ 13
The graphic representation.......................................................................................... 16
Matrix units....................................................................................................................... 17
2. Almost finite-dimensional C*-algebras..................................................................... 18
The definitions................................................................................................................. 18
Examples......................................................................................................................... 22
Bratteli's scheme............................................................................................................ 23
The separable case....................................................................................................... 25
More examples................................................................................................................ 26
3. Ideals in AFC*-algebras................................................................................................ 28
4. Bratteli diagrams and partially ordered sets............................................................. 30
Augmented posets......................................................................................................... 30
Ideals of augmented posets........................................................................................ 32
The lattice of ideals of an augmented poset............................................................. 33
AFO-aJgebras and augmented posets have equivalent ideal theories............... 34
5. The spectral theory of augmented posets................................................................. 35
Filters and prime ideals................................................................................................ 35
Hull-kernel topologies................................................................................................... 37
AFC*-algebras and augmented posets have equivalent spectral theories........... 40
Prim A characterized for separable AFC*-algebraB................................................. 45
On the center of AFC*-algebras................................................................................... 46
The isomorphy of separable AFC*-algebras reflected in their augmented posets... 48
6. Problems.......................................................................................................................... 53
Appendix. Some remarks on the distributivity of semilattices.................................... 54
References........................................................................................................................... 58
Introduction........................................................................................................................... 5
1. Finite-dimensional C*-algebras.................................................................................. 8
The objects...................................................................................................................... 8
The morphisms.............................................................................................................. 9
The matrix calculus........................................................................................................ 13
The graphic representation.......................................................................................... 16
Matrix units....................................................................................................................... 17
2. Almost finite-dimensional C*-algebras..................................................................... 18
The definitions................................................................................................................. 18
Examples......................................................................................................................... 22
Bratteli's scheme............................................................................................................ 23
The separable case....................................................................................................... 25
More examples................................................................................................................ 26
3. Ideals in AFC*-algebras................................................................................................ 28
4. Bratteli diagrams and partially ordered sets............................................................. 30
Augmented posets......................................................................................................... 30
Ideals of augmented posets........................................................................................ 32
The lattice of ideals of an augmented poset............................................................. 33
AFO-aJgebras and augmented posets have equivalent ideal theories............... 34
5. The spectral theory of augmented posets................................................................. 35
Filters and prime ideals................................................................................................ 35
Hull-kernel topologies................................................................................................... 37
AFC*-algebras and augmented posets have equivalent spectral theories........... 40
Prim A characterized for separable AFC*-algebraB................................................. 45
On the center of AFC*-algebras................................................................................... 46
The isomorphy of separable AFC*-algebras reflected in their augmented posets... 48
6. Problems.......................................................................................................................... 53
Appendix. Some remarks on the distributivity of semilattices.................................... 54
References........................................................................................................................... 58
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
174
Liczba stron
59
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CLXXIV
Daty
wydano
1980
Twórcy
autor
- Tulane University, New Orleans, La, 70118, USA
autor
- Instituto de Matematica Pura e Aplicada, Rua Liuz da Camoes, 68, CEP 20 000 Rio de Janeiro, Brazil
Bibliografia
- [1] Archbold, R. J., Prime C*-algebras and antilattices, Proc. London Math. Soc. 24 (1972), pp. 609-680.
- [2] Balbes, R., and P. Dwinger, Distributive lattices, Univ. of Missouri Press, 1974.
- [3] Behncke, H., H. Leptin and P. Krauss, C*-Algebren mil geordneten Idealfolgen, J. Funct. Anal. 10 (1972), pp. 204-211.
- [4] Behncke, H., and W. Bös, A class of C*-algebras, Proc. Amer. Math. Soc. 40 (1973), pp. 128-134.
- [5] Behncke, H., and H. Leptin, C*-algebras with finite duals, J. Funct. Anal. 14 (1973), pp. 253-268.
- [6] Behncke, H., and H. Leptin, Classification of Q*-algebras with a finite dual, ibidem 16 (1974), pp. 241-257.
- [7] Bratteli, O., Inductive limits of finite dimensional C*-algebras, Trans. Amer. Math. Soc. 171 (1972), pp. 195-234
- [8] Bratteli, O. Structure spaces of approximately finite dimensional C*-algebras, J. Funct. Anal. 16 (1974), pp. 192-204.
- [9] Bratteli, O. Structure spaces of almost finite dimensional C*-algebras II, Preprint and erratum, 1975.
- [10] Bratteli, O. The center of approximately finite-dimensional O*-algebras, Preprint 1975.
- [11] Dixmier, J., Les Q*-algèbres et leurs représentations, 2-ème éd., Ganthier-Villars, Paris 1969.
- [12] Dixmier, J., On some C*-algebras considered by Glimm, J. Funct. Anal. 1 (1967), pp. 182-203.
- [13] Dooley, A. H., The spectral theory of posets and its applications to C*-algebras, Preprint 1975.
- [14] Fell, J. M. G., The structure of algebras of operator fields, Acta Math. 106 (1961), pp. 233-280.
- [15] Gaskill, H. S., Classes of semilattices associated with an equational class of lattices, Canad. J. Math. 25 (1973), pp. 361-365.
- [16] Glimm, J., On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), pp. 318-340.
- [17] Hofmann, K. H., and K. Keimel, A general character theory for partially ordered sets and lattices, Memoir Amer. Math. Soc. 122 (1972).
- [18] Hofmann, K. H., M. Mislove and A. Stralka, The Pontryagin duality of compact Hero dimensional semilattices and its applications, Lecture Notes in Mathematics 396, Springer-Verlag, Heidelberg 1974.
- [19] Hofmann, K. H., and P. S. Mostert, Elements of compact semigroups, Charles E. Merrill, Columbus (Ohio), 1966.
- [20] Mitchell, B., Theory of categories, Academie Press, Now York 1965.
- [21] Murray, F. J., and J. von Neumann, On rings of operators IV, Ann. of Math. 44 (1943), pp. 709-808.
- [22] Powers, H. T., Representation of uniformly hyperfinite algebras and the associated von Naumann rings, Ann. of Math. 86 (1967), pp. 138-171.
- [23] Rhodes, J. B., Modular and distributive semilaltices, Trans. Amer. Math. Soc. 201 (1975), pp. 31-41.
- [24] Thayer, F. J., The Weyl-von Neumann theorem for approximately finite C*- algebras, Indiana Math. J. 24 (1975).
- [25] Varlet, J. C., On separation properties in semilattiees, Semigroup Forum 10 (1975), pp. 220-228.
- [26] Elliot, Gr. A., On the classification of inductive limits of sequences of semisimple finite dimensional algebras, J. of Alg. 38 (1976), pp. 29-44.
- [27] Elliot, Gr. A., On totally ordered groups, Copenhagen Math. Preprints 1978, no. 10.
- [28] Lazar, A. J., and D. C. Taylor, Approximately finite C*-algebras and Bratteli diagrams, Montana State Univ. Math. Preprint 1979. Amer. Math. Soc. Abstracts 1 (1980), p. 92.)
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