Characters of the infinite alternating group ${\mathfrak{A}}_{\mathbb{N}}$ and ${\mathbb{N}}$-graded quotient graphs over involution

In 1964, German mathematician E. Thoma published the complete list of extreme characters of the infinite symmetric and alternating groups; the translation of this work and the commentary on it have been published in the current volume. Thoma has deduced the classification of extreme characters of the infinite alternating group $\mathfrak{A}_{\mathbb{N}}$ from the corresponding result for the symmetric group and general properties of countable groups that he has shown in another work. We suggest another, more direct proof of this result using different technique, – we consider the graph (Bratelli diagram), which may be viewed as a quotient of the Young graph by its natural involution.
Effectively, we prove a general result, namely, given the set of ergodic measures on a graph with an involution, we explain how to describe the set of ergodic central measures on the quotient graph. The problems of how the traces (the characters) change after various changes of a graph, have not been sufficiently explored.