We study completely positive module maps on $C^{*}$-algebras which are $C^*$-module over another $C^*$-algebra with compatible actions. extend several well known dilation and extension results to this setup, including the Stinespring theorem Wittstock, Arveson, Voiculescu theorems.

BACKGROUND Treatment of achalasia is focused on decreasing the resting lower esophageal sphincter by either pneumatic dilation or surgical myotomy. When patients symptomatically relapse after one or more pneumatic dilations, then one explanation is to consider the possibility of pseudoachalasia as the diagnosis. CASE REPORT We present a rare case of an elderly patient with a presentation of c...

We study the Classical Probability analogue of the dilations of a quantum dynamical semigroup in Quantum Probability. Given a (not necessarily homogeneous) Markov chain in discrete time in a finite state space E, we introduce a second system, an environment, and a deterministic invertible time-homogeneous global evolution of the system E with this environment such that the original Markov evolu...

In this paper, boundary value problems of fractional order are converted into an optimal control problems. Then an approximate solution is constructed from translations and dilations of a B-spline function such that the exact boundary conditions are satisfied. The fractional differential operators are taken in the Riemann-Liouville and Caputo sense. Several example are given and the optimal err...

We apply Hilbert module methods to show that normal completely positive maps admit weak tensor dilations. Appealing to a duality between weak tensor dilations and extensions of CP-maps, we get an existence proof for certain extensions. We point out that this duality is part of a far reaching duality between a von Neumann bimodule and its commutant in which also other dualities, known and new, f...