Extension of Jones' projections
In the index theory for a pair of type \(\text{II}_ 1\)-factors, Jones' projections play an important role. A family of Jones' projections is a sequence of projections \(\{e_ i\); \(i=1,2,\dots\}\) satisfying the following condition which we call Jones' relations: (a) \(e_ i e_{i\pm 1} e_ i= \lambda e_ i\) for \(i\geq 1\) with a fixed constant \(\lambda\) \((0<\lambda<1)\), (b) \(e_ i e_ j= e_ j e_ i\) for \(| i-j|\geq 2\), (c) \(\text{tr} (e_ i\omega)= \lambda \text{tr}(\omega)\) for any word \(\omega\) on \(e_ 1,\dots, e_{i-1}\), where tr is the canonical trace on \(\{e_ i\); \(i=1,2,\dots\}''\). In the paper, we extend such a family by adding some number of projections. A necessary and sufficient condition for the existence of such a family is given by Theorems 1 and 2. For a family of extended Jones' projections \(\{e_ i,f_ j\); \(i=1,2,\dots\), \(1\leq j\leq m\}\), put \(A= \{e_ i, f_ j\); \(i=1,2,\dots\), \(1\leq j\leq m\}''\) and \(B=\{e_ i\); \(i=1,2,\dots\}''\). We calculate the index \([A:B]\) and show that the relative commutant \(B'\cap A\) is trivial. Furthermore we specify the fixed point subalgebras \(A^ \sigma\subset A\) of automorphisms \(\sigma: A\to A\), defined by permutations of \(\{f_ i\); \(1\leq i\leq m\}\), and then calculate indices \([A: A^ \sigma]\).
index theory for a pair of type \(\text{II}_ 1\)-factors
scientific article; zbMATH DE number 221212