Semifinite harmonic functions on branching graphs

We study semifinite harmonic functions on arbitrary branching graphs. We give a detailed exposition of an algebraic method which allows one to classify semifinite indecomposable harmonic functions on some multiplicative branching graphs. It was suggested by A. Wassermann in terms of operator algebras, but we rephrase, clarify, and simplify the main arguments working only with combinatorial objects. This work was inspired by the theory of traceable factor representations of the infinite symmetric group $S(\infty)$.