We initiate a study of linear maps on Mn(C) that have the property they factor through tracial von Neumann algebra (A,τ) via operators Z∈Mn(A) whose entries consist positive elements from von-Neumann algebra. These often arise in context non-local games especially synchronous case. establish connection with convex sets Rn containing self-dual cones and existence these maps. The Choi matrix map ...

The von Neumann entropy, named after John von Neumann, is the extension of classical entropy concepts to the field of quantummechanics and, from a numerical perspective, can be computed simply by computing all the eigenvalues of a density matrix, an operation that could be prohibitively expensive for large-scale density matrices. We present and analyze two randomized algorithms to approximate t...

In this work we investigate the notion of action or coaction of a finite quantum groupoid in von Neumann algebras context. In particular we prove a double crossed product theorem and prove the existence of an universal von Neumann algebra on which any finite groupoids acts outerly. In previous works, N. Andruskiewitsch and S.Natale define for any match pair of groupoids two C∗-quantum groupoids...

ON THE DIAGONALS OF PROJECTIONS IN MATRIX ALGEBRAS OVER VON NEUMANN ALGEBRAS Soumyashant Nayak Richard V. Kadison The main focus of this dissertation is on exploring methods to characterize the diagonals of projections in matrix algebras over von Neumann algebras. This may be viewed as a non-commutative version of the generalized Pythagorean theorem and its converse (Carpenter’s Theorem) studie...

' J. von Neumann, these PROCEEDINGS, 18, 70-82 (1932). 4E. Hopf, these PROCEEDINGS, 18, 93-100 (1932). 0G. D. Birkhoff, these PROCEEDINGS, 17, 656-660 (1931). 6 G. D. Birkhoff and P. Smith, Journal de Math., 7 [9],(1928), p. 365. 7 H. Poincar6, Calcul des Probabilitils, 2nd Edition, Paris, 1912, 320-333. 8 This remark is due to B. 0. Koopman who privately communicated it to the author two month...

In 1967, when Kadison and Ringrose began the development of continuous cohomology theory for operator algebras, they conjectured that the cohomology groups Hn(M, M), n >/= 1, for a von Neumann algebra M, should all be zero. This conjecture, which has important structural implications for von Neumann algebras, has been solved affirmatively in the type I, IIinfinity, and III cases, leaving open o...

We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. The generalization yields new classes of quantum error correcting codes that have no finite-dimensional counterparts. The error correction theory we develop begins with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of...

We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional minand max-entropy we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-di...

‎The Mod $2$ Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations‎, ‎denoted by $Sq^n$‎, ‎between the cohomology groups with $mathbb{Z}_2$ coefficients of any topological space‎. ‎Regarding to its vector space structure over $mathbb{Z}_2$‎, ‎it has many base systems and some of the base systems can also be restricted to its sub algebras‎. ‎On the contrary‎, ‎in ...

An abelian self-commutator in a C*-algebra A is an element of the form A = XX−XX, with X ∈ A, such that XX and XX commute. It is shown that, given a finite AW*-factor A, there exists another finite AW*-factor M of same type as A, that contains A as an AW*-subfactor, such that any self-adjoint element X ∈ M of quasitrace zero is an abelian self-commutator in M. Introduction According to the Murr...