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Approximately finite-dimensional C*-algebras

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Hofmann K. H., Thayer F. J.

Instytut Matematyczny Polskiej Akademii Nauk

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

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1 Hofmann K. H.

1 Thayer F. J.

1 1980

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Approximately finite-dimensional C*-algebras