Homomorphic Conditional Expectations as Noncommutative Retractions

Homomorphic Retractions in Semigroups

The Universal A-Dynamical System

For any C∗-algebra A, an A-dynamical system is a C∗-dynamical system that contains A and can be generated by the images of A under the semigroup of nonnegative time endomorphisms. There is a universal Adynamical system that occupies a position in noncommutative dynamics that resembles the position of the tangent bundle in commutative dynamics. We describe an approach to noncommutative dilation ...

The Pythagorean Theorem: I. The finite case.

The Pythagorean Theorem and variants of it are studied. The variations evolve to a formulation in terms of noncommutative, conditional expectations on von Neumann algebras that displays the theorem as the basic result of noncommutative, metric, Euclidean Geometry. The emphasis in the present article is finite dimensionality, both "discrete" and "continuous."

Independence and Product Systems

Starting from elementary considerations about independence and Markov processes in classical probability we arrive at the new concept of conditional monotone independence (or operator-valued monotone independence). With the help of product systems of Hilbert modules we show that monotone conditional independence arises naturally in dilation theory. One of the most fundamental concepts in probab...

Noise-free Symmetric Fully Homomorphic Encryption based on noncommutative rings

In this paper, we propose a noise-free symmetric fully homomorphic encryption (FHE) based on matrices over noncommutative rings. The scheme is secure against chosen plaintext attacks based on the factorization problem of matrices over noncommutative rings as well as the hardness of an overdefined system of multivariate polynomial equations over the given non-commutative algebraic structure. Mea...