  The theme of the paper is the question of existence and basic structure of
transfer operators for endomorphisms of a unital C*-algebra. We establish a
complete description of non-degenerate transfer operators, characterize
complete transfer operators and clarify their role in crossed product
constructions. Also, we give necessary and sufficient conditions for existence
of transfer operators for commutative systems, and discuss their form for
endomorphisms of B(H) which is relevant to Kadison-Singer problem.
