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Representation Theory and Numerical AF-Invariants: The Representations and Centralizers of Certain States on $\mathcal{O}_d$
 
Ola Bratteli Mathematics Institute, Oslo, Norway
Palle E. T. Jorgensen University of Iowa, Iowa City, IA
Vasyl’ Ostrovs’kyĭ National Academy of Sciences of Ukraine, Kiev, Ukraine
Front Cover for Representation Theory and Numerical AF-Invariants
Available Formats:
Electronic ISBN: 978-1-4704-0395-9
Product Code: MEMO/168/797.E
List Price: $73.00
MAA Member Price: $65.70
AMS Member Price: $43.80
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  • Front Cover for Representation Theory and Numerical AF-Invariants
  • Back Cover for Representation Theory and Numerical AF-Invariants
Representation Theory and Numerical AF-Invariants: The Representations and Centralizers of Certain States on $\mathcal{O}_d$
Ola Bratteli Mathematics Institute, Oslo, Norway
Palle E. T. Jorgensen University of Iowa, Iowa City, IA
Vasyl’ Ostrovs’kyĭ National Academy of Sciences of Ukraine, Kiev, Ukraine
Available Formats:
Electronic ISBN:  978-1-4704-0395-9
Product Code:  MEMO/168/797.E
List Price: $73.00
MAA Member Price: $65.70
AMS Member Price: $43.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1682004; 178 pp
    MSC: Primary 46; 47; Secondary 43;

    Let \(\mathcal{O}_{d}\) be the Cuntz algebra on generators \(S_{1},\dots,S_{d}\), \(2\leq d<\infty\). Let \(\mathcal{D}_{d}\subset\mathcal{O}_{d}\) be the abelian subalgebra generated by monomials \(S_{\alpha_{{}}}^{{}}S_{\alpha_{{}} }^{\ast}=S_{\alpha_{1}}^{{}}\cdots S_{\alpha_{k}}^{{}}S_{\alpha_{k}}^{\ast }\cdots S_{\alpha_{1}}^{\ast}\) where \(\alpha=\left(\alpha_{1}\dots\alpha _{k}\right)\) ranges over all multi-indices formed from \(\left\{ 1,\dots,d\right\}\). In any representation of \(\mathcal{O}_{d}\), \(\mathcal{D}_{d}\) may be simultaneously diagonalized. Using \(S_{i}^{{}}\left( S_{\alpha}^{{}}S_{\alpha}^{\ast}\right) =\left( S_{i\alpha}^{{}}S_{i\alpha }^{\ast}\right) S_{i}^{{}}\), we show that the operators \(S_{i}\) from a general representation of \(\mathcal{O}_{d}\) may be expressed directly in terms of the spectral representation of \(\mathcal{D}_{d}\). We use this in describing a class of type \(\mathrm{III}\) representations of \(\mathcal{O}_{d}\) and corresponding endomorphisms, and the heart of the memoir is a description of an associated family of AF-algebras arising as the fixed-point algebras of the associated modular automorphism groups. Chapters 5–18 are devoted to finding effective methods to decide isomorphism and non-isomorphism in this class of AF-algebras.

    Readership

    Graduate students and research mathematicians interested in functional analysis and operator theory.

  • 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
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Volume: 1682004; 178 pp
MSC: Primary 46; 47; Secondary 43;

Let \(\mathcal{O}_{d}\) be the Cuntz algebra on generators \(S_{1},\dots,S_{d}\), \(2\leq d<\infty\). Let \(\mathcal{D}_{d}\subset\mathcal{O}_{d}\) be the abelian subalgebra generated by monomials \(S_{\alpha_{{}}}^{{}}S_{\alpha_{{}} }^{\ast}=S_{\alpha_{1}}^{{}}\cdots S_{\alpha_{k}}^{{}}S_{\alpha_{k}}^{\ast }\cdots S_{\alpha_{1}}^{\ast}\) where \(\alpha=\left(\alpha_{1}\dots\alpha _{k}\right)\) ranges over all multi-indices formed from \(\left\{ 1,\dots,d\right\}\). In any representation of \(\mathcal{O}_{d}\), \(\mathcal{D}_{d}\) may be simultaneously diagonalized. Using \(S_{i}^{{}}\left( S_{\alpha}^{{}}S_{\alpha}^{\ast}\right) =\left( S_{i\alpha}^{{}}S_{i\alpha }^{\ast}\right) S_{i}^{{}}\), we show that the operators \(S_{i}\) from a general representation of \(\mathcal{O}_{d}\) may be expressed directly in terms of the spectral representation of \(\mathcal{D}_{d}\). We use this in describing a class of type \(\mathrm{III}\) representations of \(\mathcal{O}_{d}\) and corresponding endomorphisms, and the heart of the memoir is a description of an associated family of AF-algebras arising as the fixed-point algebras of the associated modular automorphism groups. Chapters 5–18 are devoted to finding effective methods to decide isomorphism and non-isomorphism in this class of AF-algebras.

Readership

Graduate students and research mathematicians interested in functional analysis and operator theory.

  • 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
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