In this article, we present a family of algorithms for linear programming based on an algorithm proposed by von Neumann. The von Neumann algorithm is very attractive due to its simplicity, but is not practical for solving most linear programs to optimality due to its slow convergence. Our algorithms were developed with the objective of improving the practical convergence of the von Neumann algo...

The purpose of this paper is to prove that a completely positive projection on a Hilbert space associated with a standard form of a von Neumann algebra induces the existence of a conditional expectation of the von Neumann algebra with respect to a normal state, and we consider the application to a standard form of an injective von Neumann algebra.

Recently Chubanov proposed a method which solves homogeneous linear equality systems with positive variables in polynomial time. Chubanov’s method can be considered as a column-wise rescaling procedure. We adapt Chubanov’s method to the von Neumann problem, and so we design a polynomial time column-wise rescaling von Neumann algorithm. This algorithm is the first variant of the von Neumann algo...

In two papers entitled “On a new foundation [Neue Begründung] of quantum mechanics,” Pascual Jordan (1927b,g) presented his version of what came to be known as the Dirac-Jordan statistical transformation theory. As an alternative that avoids the mathematical difficulties facing the approach of Jordan and Paul A. M. Dirac (1927), John von Neumann (1927a) developed the modern Hilbert space formal...

For von Neumann algebras M,N not isomorphic to C C and without type I2 summands, we show that for an order-isomorphism f : AbSub M! AbSub N between the posets of abelian von Neumann subalgebras of M and N , there is a unique Jordan ⇤-isomorphism g : M! N with the image g[S] equal to f(S) for each abelian von Neumann subalgebra S of M. The converse also holds. This shows the Jordan structure of ...