| eBook 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 |
| eBook 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 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 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.
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 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
-
-
RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Requests
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
