In this essay we construct a noncommutative von Neumann dynamical system with certain pathological properties from the viewpoint of analogies with classical ergodic theory. These relate to obstructions to the existence of special kinds of subsystem or subextension. We will build our example as an extension of a commutative dynamical system via a unitary cocycle acting on a Hilbert bundle, and w...

In this paper we survey the field of Hochschild cohomology for von Neumann algebras. We describe the basic definitions and results without assuming background knowledge beyond some familiarity with von Neumann algebra theory. We offer no formal proofs of results, although we give detailed references to the literature. Instead, we concentrate on explaining the main techniques and how they are ap...

We observe that the von Neumann (for short, W*-)envelope of the quantum algebra of functions on the normalizer of the group SU(1, 1) ∼= SL(2,R) in SL(2,C) via deformation quantization contains the von Neumann algebraic quantum normalizer of SU(1, 1) in the frame work of Waronowicz-Korogodsky. We then use the technique of reduction to the maximal subgroup to compute the K-theory, the periodic cy...

In this paper we analyze the structure of some sets of non-commutative moments of elements in a finite von Neumann algebra M . If the fundamental group of M is R+\{0}, then the moment sets are convex, and if M is isomorphic to M ⊗M , then the sets are closed under pointwise multiplication. We introduce a class of discrete groups that we call hyperlinear. These are the discrete subgroups (with i...

Let X be a finite connected graph, each of whose vertices has degree at least three. The fundamental group Γ of X is a free group and acts on the universal covering tree ∆ and on its boundary ∂∆, endowed with a natural topology and Borel measure. The crossed product C∗-algebra C(∂∆)⋊Γ depends only on the rank of Γ and is a Cuntz-Krieger algebra whose structure is explicitly determined. The cros...

A linear mapping φ on an algebra A is called a centralizable mapping at G ∈ A if φ(AB) = φ(A)B = Aφ(B) for each A and B in A with AB = G, and φ is called a derivable mapping at G ∈ A if φ(AB) = φ(A)B + Aφ(B) for each A and B in A with AB = G. A point G in A is called a full-centralizable point (resp. full-derivable point) if every centralizable (resp. derivable) mapping at G is a centralizer (r...

Recently, Takahashi has introduced the James and von Neumann-Jordan type constants. In this paper, we present some sufficient conditions for uniform normal structure and therefore the fixed point property of a Banach space in terms of the James and von Neumann-Jordan type constants and the Ptolemy constant. Our main results of the paper significantly generalize and improve many known results in...

In this work we discuss the notion of observable both quantum and classical from a new point of view. In classical mechanics, an observable is represented as a function (measurable, continuous or smooth), whereas in (von Neumann’s approach to) quantum physics, an observable is represented as a bonded selfadjoint operator on Hilbert space. We will show in part II of this work that there is a com...