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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
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: $43.80 Click above image for expanded view Representation Theory and Numerical AF-Invariants: The Representations and Centralizers of Certain States on$\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.

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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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.

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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