Representation theory and numerical AF-invariants : the representations and centralizers of certain states on Od
Author(s)
Bibliographic Information
Representation theory and numerical AF-invariants : the representations and centralizers of certain states on Od
（Memoirs of the American Mathematical Society, no. 797）
American Mathematical Society, 2004
Available at 16 libraries
Note
"March 2004, volume 168, number 797 (second of 4 numbers)."
Includes bibliographical references (p. 166-169)
Description and Table of Contents
Description
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 $S_{\alpha_{<!-- -->{}}}^{<!-- -->{}}S_{\alpha_{<!-- -->{}}}^{\ast}=S_{\alpha_{1}}^{<!-- -->{}}\cdot s S_{\alpha_{k}}^{<!-- -->{}}S_{\alpha_{k}}^{\ast}\cdots S_{\alpha_{1}}^{\ast}$ 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 of this title are devoted to finding effective methods to decide isomorphism and non-isomorphism in this class of AF-algebras.
Table of Contents
Part A. Representation theory Part B. Numerical AF-invariants Bibliography List of figures List of tables List of terms and symbols.
by "Nielsen BookData"