In this paper, we will study the theory of cyclic homology for regular multiplier Hopf algebras. We associate a cyclic module to a triple $(mathcal{R},mathcal{H},mathcal{X})$ consisting of a regular multiplier Hopf algebra $mathcal{H}$, a left $mathcal{H}$-comodule algebra $mathcal{R}$, and a unital left $mathcal{H}$-module $mathcal{X}$ which is also a unital algebra. First, we construct a para...

Consider the B-valued probability space (A, E,B), where A is a tracial von Neumann algebra. We extend the theory of operator valued free probability to the algebra of affiliated operators Ã. For a random variable X ∈ Ã we study the Cauchy transform GX and show that the von Neumann algebra (B ∪ {X}) ′′ can be recovered from this function. In the case where B is finite dimensional, we show that, ...

Let R be a von Neumann algebra acting on a Hilbert space H and let Rsa be the set of hermitean (i.e. selfadjoint) elements of R. It is well known that Rsa is a lattice with respect to the usual partial order ≤ if and only if R is abelian. We define and study a new partial order on Rsa, the spectral order ≤s, which extends ≤ on projections, is coarser than the usual one, but agrees with it on ab...

Let τ be a linear map from a unital C∗-algebra A to a von Neumann algebra B and let C be a unital C∗-algebra. A map T from a Hilbert A-module E to a von Neumann C-B module F is called a τ -map if 〈T (x), T (y)〉 = τ(〈x, y〉) for all x, y ∈ E. A Stinespring type theorem for τ -maps and its covariant version are obtained when τ is completely positive. We show that there is a bijective correspondenc...

Let M be a tracial von Neumann algebra and A be a weakly dense unital C-subalgebra of M . We say that a set X is a W -generating set for M if the von Neumann algebra generated by X is M and that X is a C-generating set for A if the unital C-algebra generated by X is A. For any finite W generating set X for M we show that δ0(X) ≤ sup{δ0(Y ) : Y is a finite C-generating set for A}. It follows tha...

The study of modules over a finite von Neumann algebra A can be advanced by the use of torsion theories. In this work, some torsion theories for A are presented, compared and studied. In particular, we prove that the torsion theory (T,P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for A. Using torsion theories, we describe the inj...

In this article, we give necessary and sufficient conditions under which the Leavitt path algebra $$L_K(\mathcal {G})$$ of an ultragraph $$\mathcal {G}$$ over a field K is purely infinite simple that it von Neumann regular. Consequently, obtain every graded either locally matricial algebra, or full matrix ring $$K[x, x^{-1}]$$ , algebra.

For a compact Hausdorff space X , the space SC(X ×X) of separately continuous complex valued functions on X can be viewed as a C∗-subalgebra of C(X)∗∗⊗C(X)∗∗, namely those elements which slice into C(X). The analogous definition for a non-commutative C∗-algebra does not necessarily give an algebra, but we show that there is always a greatest C∗-subalgebra. This thus gives a noncommutative notio...