Full nest algebras
For each positive integer \(n\), let \(M_n\) and \(T_n\) denote respectively the algebras of all \(n\times n\) and all upper triangular \(n\times n\) matrices. A mapping \(\nu: T_{n_1}\to T_{n_2}\) is called a nest embedding if \(\nu\) is a unital isometric homomorphism with the following properties: 1. \(\nu\) has an extension to a *-homomorphism of \(M_{n_1}\); 2. \(\nu\) maps a matrix unit in \(T_{n_1}\) to a sum of matrix units in \(T_{n_2}\); 3. \(\nu\) maps \(\text{Lat } T_{n_1}\) into \(\text{Lat } T_{n_2}\). A subalgebra \(A\) of a UHF \(C^*\)-algebra \(B\) which is a direct limit of a system \[ T_{n_1} @>\nu_1>> T_{n_2} @> \nu_2>> T_{n_3} @>\nu_3>> \cdots\to A, \] where the \(\nu\)'s are nest embeddings, is called a full nest algebra. For such a subalgebra, let \({\mathcal N}= \text{Lat } A\) and let \(D\) denote the diagonal of \(A\). Then \(D\) is the canonical masa \(\varinjlim D_n\) and \(C^* ({\mathcal N})= D\). Furthermore, \[ \text{tr} ({\mathcal N})\equiv \{\text{tr } p: p\in {\mathcal N}\}= K_0 (B)\cap [0,1 ], \] where tr is the canonical trace for \(B\), and \({\mathcal N}\) satisfies the so-called decomposition property relative to \(B\), namely, for every finite set \({\mathcal F}\) of partial isometries in the normalizer of \(D\), there is a decomposition of the identity \(1= \sum^N_{i=1} e_i\) such that (a) each \(e_i\) is an interval from \({\mathcal N}\); (b) \(\text{tr } e_i= \text{tr } e_j\) for all \(i\), \(j\); (c) if \(v\in {\mathcal F}\) and \(e_i\leq vv^*\), then \(v^* e_i v\) is an interval; (d) for any positive integer \(n\) with \(1/n\in K_0 (B)\), \(N\) may be chosen to be a multiple of \(n\). The main aim of the present paper is to show conversely that, if a nest \({\mathcal N}\) in \(B\) such that 1. \(C^* ({\mathcal N}) =D\) in a canonical masa in \(B\), 2. \(\text{tr} ({\mathcal N})= K_0 (B)\cap [0,1]\), 3. \({\mathcal N}\) satisfies the decomposition property relative to \(B\), then \(\text{Alg} ({\mathcal N})\) is a full nest subalgebra of \(B\). The remainder of the paper is primarily concerned with studying full nest algebras of the form \[ T_b @> \nu_\pi>> T_{b^2} @> \nu_\pi>> T_{b^3} @> \nu_\pi>> \cdots \to A_\pi. \] Here \(\pi\) is a fixed permutation in the permutation group \(S_b\) and each \(\nu_\pi\) is given by \[ \nu_\pi [a_{ij} ]= [a_{ij} U_\pi^{j-i} ], \] where \(U_\pi\) is the \(b\times b\) permutation unitary matrix associated with \(\pi\).
scientific article; zbMATH DE number 599245