Monotone operator functions on C*-algebras

The article is devoted to investigation of classes of functions monotone as functions on general C-*-algebras that are not necessarily the C-*-algebra of all bounded linear operators on a Hilbert space as in classical case of matrix and operator monotone functions. We show that for general C-*-algebras the classes of monotone functions coincide with the standard classes of matrix and operator monotone functions. For every class we give exact characterization of C-*-algebras with this class of monotone functions, providing at the same time a monotonicity characterization of subhomogeneous C-*-algebras. We use this result to generalize characterizations of commutativity of a C-*-algebra based on monotonicity conditions for a single function to characterizations of subhomogeneity. As a C-*-algebraic counterpart of standard matrix and operator monotone scaling, we investigate, by means of projective C-*-algebras and relation lifting, the existence of C-*-subalgebras of a given monotonicity class