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 $\chi :G\to \mathbb{C}$ 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$G$-ergodicmeasures,$Fix(g) = {x\in X: gx = x}$,and$\alpha_1,…,\alpha_k\in {0,1,…,\infty}$. We illustrate our results on the group of rational permutations of the unit interval.