Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

Non-commutative Gröbner Bases for Commutative Algebras

An ideal I in the free associative algebra k〈X1, . . . ,Xn〉 over a field k is shown to have a finite Gröbner basis if the algebra defined by I is commutative; in characteristic 0 and generic coordinates the Gröbner basis may even be constructed by lifting a commutative Gröbner basis and adding commutators.

Acceptable random variables in non-commutative probability spaces

Acceptable random variables are defined in noncommutative (quantum) probability spaces and some of probability inequalities for these classes  are obtained. These results are a generalization of negatively orthant dependent random variables in probability theory. Furthermore, the obtained results can be used for random matrices.

Gelfand theory for non-commutative Banach algebras

Let A be a Banach algebra. We call a pair (G,A) a Gelfand theory for A if the following axioms are satisfied: (G 1) A is a C∗-algebra, and G : A → A is a homomorphism; (G 2) the assignment L 7→ G−1(L) is a bijection between the sets of maximal modular left ideals of A and A, respectively; (G 3) for each maximal modular left ideal L of A, the linear map GL : A/G−1(L) → A/L induced by G has dense...

Non-Commutative Dimension for C*-Algebras

Non Commutative Arens Algebras and Their Derivations

Given a von Neumann algebra M with a faithful normal semi-finite trace τ, we consider the non commutative Arens algebra Lω(M, τ) = ⋂ p≥1 Lp(M, τ) and the related algebras L2 (M, τ) = ⋂ p≥2 Lp(M, τ) and M + L2 (M, τ) which are proved to be complete metrizable locally convex *-algebras. The main purpose of the present paper is to prove that any derivation of the algebra M + L2 (M, τ) is inner and...