On the maximalC *-algebra of zeros of completely positive mapping and on the boundary of a dynamic semigroup

Abstract

In this paper, we study the parametrization of conservative extensions of minimal dynamic semigroups used in the quantum theory of open systems for describing irreversible processes. The particular case of extensions parametrized by normal states is discussed in [1, 2]. The construction used below is similar to that of self-ajoint extensions for densely-defined, bounded below, symmetric operators in a Hilbert space. The dynamic semigroup, unlike the unitary operator group, acts not on elements of a Hilbert space 7/but on elements of a topological operator algebra in a Hilbert space B(H); moreover, this semigroup is completely positive (see [3]). For such semigroups, direct methods of describing domains of generating mappings are absent; this complicates the parametrization of extensions of the minimal generating mapping. It turns out that the range of the resolvent of the minimal dynamic semigroup belongs to the complement of the zero set (the kernel) of the limit mapping QOO(.) associated with the formal generating mapping of the dynamic semigroup. For this reason, we first consider general properties of the zero set for a completely positive mapping (see Sec. 1). In what follows, the C*-algebra generated by positive roots of a mapping Q(-) plays the central role. In Sec. 2, we prove that this algebra has the following property: it is maximal in the class of algebras that contain an element X if and only if X* X and X X* are elements of the algebra. In Sec. 3, the mentioned property is extended to the set of positive operators that are zeros of the limit mapping QOO(.) _- s-lira Qn (.). We prove that the domain of the infinitesimal mapping of the minimal dynamic semigroup is contained in the maximal C*-Mgebra generated by positive roots of the mapping Q~(-) associated with the semigroup. In Sec. 4, we give a construction of conservative extensions of the minimal dynamic semigroup, which is based on limit properties of the resolvent mapping. In Sec. 5, we prove the characteristic theorem for completely positive mappings with given ranges, which enables us to consider the set of conservative extensions as a *-weakly compact convex manifold in/3+(/2(7-~)), whose points admit an expansion in the set of extreme elements. The barycentric expansion of elements T in extreme points of E C T establishes a one-to-one correspondence between classes of equivalent normed measures on E and conservative extensions of the minimal dynamic semigroup; in this case, E plays the role of boundary and the class of equivalent measures plays the role of boundary condition for a conservative extension.