$$C^*$$-cross products and a generalized quantum mechanical $$N$$-body problem.(English)Zbl 0968.47035

Summary: For each finite-dimensional real vector space $$X$$ we construct a $$C^*$$-algebra $$C^X_0$$ graded by the lattice of all subspaces of $$X$$. Then we compute its quotient with respect to the algebra of compact operators. This allows us to describe the essential spectrum and to prove the Mourre estimate for the self-adjoint operators associated with $$C^X_0$$.

MSC:

 47L65 Crossed product algebras (analytic crossed products) 46L55 Noncommutative dynamical systems 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 81R15 Operator algebra methods applied to problems in quantum theory 81V70 Many-body theory; quantum Hall effect 46L60 Applications of selfadjoint operator algebras to physics 46N50 Applications of functional analysis in quantum physics 47L90 Applications of operator algebras to the sciences
Full Text: