No abstract

We consider the C*-algebras which contain the Weyl operators when the symplectic form which defines the C.C.R. is possibly degenerate. We prove that the C.C.R. are all obtained as a quotient of a universal C*-algebra by some of its ideals, and we characterize all these ideals.

A description of the quasi-free states on a Clifford algebra and their representations is given, and we prove that the pure quasi-free states are Fock States.

We give a complete description of the states of the C.C.R. algebra for a finite number of degrees of freedom which are invariant with respect to subgroups of the translation group of phase space. We make precise some well-known results of quantum mechanics such as Bloch theorem.

The product form of quasi-free states is outlined, and the types of the generated factors are exhibited whenever the states are translation invariant. Among these states some are shown to be involved in the study of JFermi and Bose gases.