Theory of representations of a topological (non-Banach) involutory algebra

A definition is given of the $^*$-representation of a specified topological (non-Banach) involutory algebra. The concepts of a symmetric representation, a representation adjoint to a specified one, and a self-conjugate representation, are introduced. It is demonstrated that the conjugation operation on a representation exhibits properties similar to the properties of the ordinary operation of conjugation of linear operators in Hilbert space. Various forms of algebraic similarity between representations, such as isomorphism, similarity, and unitary equivalence, are defined and explored. The concepts of broad and narrow commutants of a specified symmetric representation are introduced, and it is shown that the triviality of a broad commutant is equivalent to the purity of the generating functional. It is also demonstrated that self-conjugate representations exhibit a simpler algebraic structure than do simply symmetric representations. Specifically, the concepts of broad and narrow commutants merge in the case of a self-conjugate representation.