On well-behaved unbounded representations of *-algebras

A general approach to well-behaved unbounded (*)-representations of a (*)-algebra X is proposed. Let B be a normed (*)-algebra equipped with a left action > of X on B such that (x > a) (+)b = a(+) (x(+) > b) for a, b is an element of B and x is an element of X. Then the pair (X, B is called a compatible pair. For any continuous non-degenerate (*)-representation rho of B there exists a closed *-representation rho' of X such that rho'(x)rho(b) = p(x > b), where x is an element of X and b is an element of B. The (*)-representations rho' are called the well-behaved (*)-representations associated with the compatible pair (X, B). A number of examples illustrating this concept are developed in detail.