When is a Field a Jacobi-Field? A Characterization of States on Tensor Algebras

Abstract

In order to analyse the positivity condition for states on the tensoralgebra E over certain (function-)spaces E more efficiently a representation of the components of the states in terms of a set of independent parameters is suggested. For this purpose the concept of a Jacobifield is introduced. In the case of a finite dimensional space E every state on E is parametrized this way. If however the basic space E is infinite dimensional additional domain problems arise related to algebras of unbounded operators which are involved naturally. It is analysed to which extent this parametrization in terms of Jacobi-fields also works in the general case, and it is shown that for many basic spaces E which occur in applications most of the states admit indeed such a parametrizati on. This then also means a corresponding decomposition for the associated algebra of unbounded operators into independent components. Several applications are indicated.