Electronic ISBN:  9781470403959 
Product Code:  MEMO/168/797.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 168; 2004; 178 ppMSC: 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 multiindices 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 AFalgebras arising as the fixedpoint algebras of the associated modular automorphism groups. Chapters 5–18 are devoted to finding effective methods to decide isomorphism and nonisomorphism in this class of AFalgebras.
ReadershipGraduate 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 oneparameter subgroups of the action of $\mathbb {T}^d$ on $\mathcal {O}_d$

4. Subalgebras of $\mathcal {O}_d$

B. Numerical AFinvariants

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($Q_{ND}$)

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 nonisomorphism

12. Subgroups of $G_0 = \bigcup ^\infty _{n=0} J^{n}_0 \mathcal {L}$

13. Classification of the AFalgebras $\mathfrak {A}_L$ 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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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 multiindices 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 AFalgebras arising as the fixedpoint algebras of the associated modular automorphism groups. Chapters 5–18 are devoted to finding effective methods to decide isomorphism and nonisomorphism in this class of AFalgebras.
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 oneparameter subgroups of the action of $\mathbb {T}^d$ on $\mathcal {O}_d$

4. Subalgebras of $\mathcal {O}_d$

B. Numerical AFinvariants

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($Q_{ND}$)

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 nonisomorphism

12. Subgroups of $G_0 = \bigcup ^\infty _{n=0} J^{n}_0 \mathcal {L}$

13. Classification of the AFalgebras $\mathfrak {A}_L$ 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