  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. This method was proposed by A. Wassermann in terms of operator
algebras, while 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 $S(\infty)$.
