On the maximal \(C^*\)-algebra of zeros of completely positive mapping and on the boundary of a dynamic semigroup (Q1905334)

The author considers the parametrization of conservative extensions of minimal dynamic semigroups used in the quantum theory of open systems for description of irreversible processes. For such semigroups no direct method exists for describing domains of generating mappings and this complicates the parametrization of extensions of the minimal generating mapping. The range of the resolvent of the minimal dynamic semigroup belongs to the complement of the kernel of the limit mapping \(Q^\infty( )\) associated with the formal generating mapping of the semigroup. For this reason the author considers general properties of kernels for a completely positive mapping (Section 1). In Section 2 he proves that the \(C^*\)-algebra generated by positive roots of a mapping \(Q( )\) has the following property: it is maximal in the class of algebras that contain an element \(x\) if and only if \(x^*x\) and \(xx^*\) are elements of the algebra. This property is also valid for positive kernels of the limit mapping \(Q^\infty( )=s-\lim Q^n( )\). It is proved that the domain of the infinitesimal mapping of the minimal dynamic semigroup is contained in the maximal \(C^*\)-algebra generated by positive roots of the mapping \(Q^\infty( )\) associated with the semigroup (Section 3). Section 4 is devoted to a construction of conservative extensions of the minimal dynamic semigroup, which is based on limit properties of the resolvent mapping. In Section 5 the characterization theorem is proved for completely positive mappings with given ranges which enable us to consider the set of conservative extensions as \(a^*\)-weakly compact convex manifold in \(B_+(\ell_2(H))\), whose points admit an expansion in the set of extreme elements. The barycentric expansion of elements \(T\) in the set \(E\) of extreme points of \(T\) establishes a one-to-one correspondence between classes of equivalent normed measures on \(E\) and conservative extensions of the minimal dynamical semigroups; 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.