On Characters of Inductive Limits of Symmetric Groups

In the paper we completely describe characters (central positive-definite functions) of simple locally finite groups that can be represented as inductive limits of (products of) symmetric groups under block diagonal embeddings. Each such group $G$ defines an infinite graph (Bratteli diagram) that encodes the embedding scheme. The group $G$ acts on the space $X$ of infinite paths of the associated Bratteli diagram by changing initial edges of paths. Assuming the finiteness of the set of ergodic measures for the system $(X,G)$, we establish that each indecomposable character $\chi :G \rightarrow \mathbb C$ is uniquely defined by the formula $\chi(g) = \mu_1(Fix(g))^{\alpha_1}...\mu_k(Fix(g))^{\alpha_k}$, where $\mu_1,...,\mu_k$ are $G$-ergodic measures, $Fix(g) = \{x\in X: gx = x\}$, and $\alpha_1,...,\alpha_k\in \{0,1,...,\infty\}$. We illustrate our results on the group of rational permutations of the unit interval.