Unbounded *-representations of *-algebras are studied. Representations called self-adjoint representations are defined in analogy to the definition of a self-adjoint operator. It is shown that for self-adjoint representations certain pathologies associated with commutant and reducing subspaces are avoided. A class of well behaved self-adjoint representations, called standard representations, are defined for commutative *-algebras. It is shown that a strongly cyclic self-adjoint representation of a commutative *-algebra is standard if and only if the representation is strongly positive, i.e., the representations preserves a certain order relation. Similar results are obtained for ^representations of the canonical commutation relations for a finite number of degrees of freedom.

Each gauge invariant generalized free state ω A of the anticommutation relation algebra over a complex Hubert space K is characterized by an operator A on K. It is shown that the cyclic representations induced by two gauge invariant generalized free states ω A and ω B are quasi-equivalent if and only if the operators A^ -B^ and (/ -A)* -(I -B)^ are of Hubert-Schmidt class. The combination of this result with results from the theory of isomorphisms of von Neumann algebras yield necessary and sufficient conditions for the unitary equivalence of the cyclic representations induced by gauge invariant generalized free states.

The case of a noninteracting infinite Bose gas at zero temperature is studied in the formalism of local current algebras, using the representation theory of nuclear Lie groups. The class of representations describing such a system is obtained by taking an ``N / V limit'' of the finite case. These representations can also be determined uniquely from the solutions of a functional differential equation, which follows in turn from a condition on the ground state vector. Finally a system of functional differential equations is formulated for a theory with interactions, using a proposed definition of indefinite functional integration.

No abstract