Quantum operations frequently occur in quantum measurement theory, quantum probability, quantum computation and quantum information theory. If an operator A is invariant under a quantum operation φ we call A a φ-fixed point. Physically, the φ-fixed points are the operators that are not disturbed by the action of φ. Our main purpose is to answer the following question. If A is a φ-fixed point, is A compatible with the operation elements of φ ? We shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes. Our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von Neumann algebras.

We consider positive semidefinite kernels valued in the * -algebra of continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of * -semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist * -representations of the underlying * -semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally C * -algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring-Kasparov type dilation theorem for completely positive maps on locally C *algebras and with values adjointable operators on Hilbert modules over locally C * -algebras.

In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels that realize the given action by bounded operators on a Kreȋn space. Applications to the GNS representation of * -algebras associated to hermitian functionals are given. We explain the key role played by the Kolmogorov decomposition in the construction of Weyl exponentials associated to an indefinite inner product and in the dilation theory of hermitian maps on C * -algebras.

Motivated by questions related to embeddings of homogeneous Sobolev spaces and to comparison of function spaces and operator ranges, we introduce the notion of closely embedded Hilbert spaces as an extension of that of continuous embedding of Hilbert spaces. We show that this notion is a special case of that of Hilbert spaces induced by unbounded positive selfadjoint operators that corresponds to kernel operators in the sense of L. Schwartz. Certain canonical representations and characterizations of uniqueness of closed embeddings are obtained. We exemplify these constructions by closed, but not continuous, embeddings of Hilbert spaces of holomorphic functions. An application to the closed embedding of a homogeneous Sobolev space on Rn in L2(Rn), based on the singular integral operator associated to the Riesz potential, and a comparison to the case of the singular integral operator associated to the Bessel potential are also presented. As a second application we show that a closed embedding of two operator ranges corresponds to absolute continuity, in the sense of T. Ando, of the corresponding kernel operators. © 2010 Elsevier Inc

We present certain existence criteria and parameterizations for an interpolation problem for completely positive maps that take given matrices from a finite set into prescribed matrices. Our approach uses density matrices associated to linear functionals on (Formula presented.) -subspaces of matrices, inspired by the Smith-Ward linear functional and Arveson’s Hahn-Banach Type Theorem. A necessary and sufficient condition for the existence of solutions and a parametrization of the set of all solutions of the interpolation problem in terms of a closed and convex set of an affine space are obtained. Other linear affine restrictions, like trace preserving, can be included as well, hence covering applications to quantum channels that yield certain quantum states at prescribed quantum states. We also perform a careful investigation on the intricate relation between the positivity of the density matrix and the positivity of the corresponding linear functional. © 2014, © 2014 Taylor & Francis