On the construction of fields and the topological role of Jacobi fields (Q1074827)

Given a nuclear F-space E with continuous involution * denote by T(E) the tensor algebra over E and by \(\underline E\) the completion of T(E) with respect to the locally convex direct sum topology of the projective tensor product topologies on \(E^{\otimes n}\), \(n=1,2,... \). \(\underline E\) is a nuclear *-algebra with unit. A state T on \(\underline E\) is a normalized continuous linear functional on \(\underline E\) with \(T(\underline x^*\cdot \underline x)\geq 0\) for all \(\underline x\in \underline E\). In various problems in general quantum field theory [\textit{J. Yngvason}, Commun. Math. Phys. 34, 315-333 (1973; Zbl 0269.46031) and Physica A 124, 621-628 (1984)] it is important to know when a continuous seminorm \(\underline p\) on \(\underline E\) is dominated by another such seminorm \(\underline q^ T\) which derives from a state T on \(\underline E:\) \[ \underline p(x)\leq \underline q^ T(\underline x),\quad \underline q^ T(x)^ 2=T(\underline x^*\cdot \underline x)\text{ for all }\underline x\in \underline E. \] This problem is of particular interest when \(\underline p\) has special prescribed properties and when for the class of states only a subclass of states with special (continuity) properties is admitted. We consider and solve (for \(E=S({\mathbb{R}}^ d))\) this problem for the class of Jacobi-states, e.g. those states which give rise to a ''Jacobi-field'' by the GNS-construction. This class of fields has been investigated in [\textit{E. Brüning}, Publ. Res. Inst. Math. Sci., Kyoto Univ. 22, 209-246 (1986)] and the general interest in these fields is explained there. The Jacobi-states reflect by their definition and their properties in a very direct way the fact we are dealing with states on a *-algebra with unit which has a special grading (and not merely with states on a general *-algebra).