Finite quantum groupoids can be described in many equivalent ways [8, 11, 16]: In terms of the weak Hopf C -algebras of Böhm, Nill, and Szlachányi [2] or the finite-dimensional Hopf-von Neumann bimodules of Vallin [14], and in terms of finite-dimensional multiplicative partial isometries [4] or the finite-dimensional pseudo-multiplicative unitaries of Vallin [15]. In this note, we show that in ...

Murray-von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra A of the Murray-von Neumann algebra A(f)(R) associated with a finite von Neumann algebra R is the Murray-von Neumann algebra A(f)(A(0)), wh...

The aim of this note is to study the submajorization inequalities for $tau$-measurable operators in a semi-finite von Neumann algebra on a Hilbert space with a normal faithful semi-finite trace $tau$. The submajorization inequalities generalize some results due to Zhang, Furuichi and Lin, etc..

In this paper we characterize the left Jordan derivations on Banach algebras. Also, it is shown that every bounded linear map $d:mathcal Ato mathcal M$ from a von Neumann algebra $mathcal A$ into a Banach $mathcal A-$module $mathcal M$ with property that $d(p^2)=2pd(p)$ for every projection $p$ in $mathcal A$ is a left Jordan derivation.

We introduce C∗-pseudo-multiplicative unitaries and concrete Hopf C∗-bimodules for the study of quantum groupoids in the setting of C∗-algebras. These unitaries and Hopf C∗-bimodules generalize multiplicative unitaries and Hopf C∗-algebras and are analogues of the pseudo-multiplicative unitaries and Hopf–von Neumann-bimodules studied by Enock, Lesieur and Vallin. To each C∗-pseudo-multiplicativ...

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.

We prove that the relative commutant of a diffuse von Neumann subalgebra in a hyperbolic group von Neumann algebra is always injective. It follows that any non-injective subfactor in a hyperbolic group von Neumann algebra is non-Γ and prime. The proof is based on C∗-algebra theory.

It is well known that every derivation of a von Neumann algebra into itself is an inner derivation and that every derivation of a von Neumann algebra into its predual is inner. It is less well known that every triple derivation (defined below) of a von Neumann algebra into itself is an inner triple derivation. We examine to what extent all triple derivations of a von Neumann algebra into its pr...

We prove that for all 1 ≤ p ≤ ∞, p 6= 2, the L spaces associated to two von Neumann algebras M, N are isometrically isomorphic if and only if M and N are Jordan *-isomorphic. This follows from a noncommutative L Banach-Stone theorem: a specific decomposition for surjective isometries of noncommutative L spaces.

It was observed by Gilfeather, Hopenwasser, and Larson in [1] that Tomita's commutation formula for tensor products of von Neumann algebras can be rewritten in a way that makes sense for tensor products of arbitrary reflexive algebras. The tensor product problem for reflexive algebras is to decide for which pairs of reflexive algebras this tensor product formula is valid. Recall that a subalgeb...