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Representation theory and numerical AF-invariants. The representations and centralizers of certain states on $\scr O_d$: The Representations and Centralizers of Certain States on $\mathcal {O}_d$


About this Title

Ola Bratteli, Palle E. T. Jorgensen and Vasyl′ Ostrovs′kyĭ

Publication: Memoirs of the American Mathematical Society
Publication Year 2004: Volume 168, Number 797
ISBNs: 978-0-8218-3491-6 (print); 978-1-4704-0395-9 (online)
DOI: http://dx.doi.org/10.1090/memo/0797
MathSciNet review: 2030387
MSC: Primary 46L30; Secondary 37A55, 43A65, 46L55, 46L80, 47A13

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Table of Contents


Chapters

  • A. Representation theory
  • 1. General representations of $\mathcal {O}_d$ on a separable Hilbert space
  • 2. The free group on $d$ generators
  • 3. $\beta $-KMS states for one-parameter subgroups of the action of $\mathbb {T}^d$ on $\mathcal {O}_d$
  • 4. Subalgebras of $\mathcal {O}_d$
  • B. Numerical AF-invariants
  • 5. The dimension group of $\mathfrak {A}_L$
  • 6. Invariants related to the Perron–Frobenius eigenvalue
  • 7. The invariants $N$, $D$, Prim($m_N$), Prim($R_D$), Prim($Q_{N-D}$)
  • 8. The invariants $K_0 (\mathfrak {A}_L) \otimes _{\mathbb {Z}} \mathbb {Z}_n$ and $(\operatorname {ker} \tau )\otimes _{\mathbb {Z}} \mathbb {Z}_n$ for $n = 2, 3, 4$, …
  • 9. Associated structure of the groups $K_0 (\mathfrak {A}_L)$ and $\operatorname {ker} \tau $
  • 10. The invariant $\operatorname {Ext}(\tau (K_0(\mathfrak {A}_L)), \operatorname {ker} \tau )$
  • 11. Scaling and non-isomorphism
  • 12. Subgroups of $G_0 = \bigcup ^\infty _{n=0} J^{-n}_0 \mathcal {L}$
  • 13. Classification of the AF-algebras $\mathfrak {A}_L$ with rank $(K_0 (\mathfrak {A}_L)) = 2$
  • 14. Linear algebra of $J$
  • 15. Lattice points
  • 16. Complete classification in the cases $\lambda = 2$, $N = 2, 3, 4$
  • 17. Complete classification in the case $\lambda = m_N$
  • 18. Further comments on two examples from Chapter 16