Derivations on the Algebra of τ-Compact Operators Affiliated with a Type I von Neumann Algebra

Let M be a type I von Neumann algebra with the center Z, and a faithful normal semi-finite trace τ. Consider the algebra L(M, τ) of all τ-measurable operators with respect to M and let S ₀(M, τ) be the subalgebra of τ-compact operators in L(M, τ). We prove that any Z-linear derivation of S ₀(M, τ) is spatial and generated by an element from L(M, τ).      

HNN extensions of von Neumann algebras

Reduced HNN extensions of von Neumann algebras (as well as C^*-algebras) will be introduced, and their modular theory, factoriality and ultraproducts will be discussed. In several concrete settings, detailed analysis on them will be also carried out.     

The Weyl group and the normalizer of a conditional expectation

We define a discrete groupW(E) associated to a faithful normal conditional expectationE : M N forN M von Neuman algebras. This group shows the relation between the unitary groupU _N and the normalizerN _E ofE, which can be also considered as the isotropy of the action of the unitary groupU _M ofM onE. It is shown thatW(E) is finite if dimZ(N)< and bounded by the index in the factor case. Also sharp bounds of the order ofW(E) are founded.W(E) appears as the fibre of a covering space defined on the orbit ofE by the natural action of the unitary group ofM. W(E) is computed in some basic examples.     

Approximate AF flows

When is a flow on a unital AF algebra A such that there is an increasing sequence (A_n) of finite-dimensional -invariant C*-subalgebras of A with dense union, we call an AF flow. We show that an approximate AF flow is a cocycle perturbation of an AF flow.     

Bimodules in crossed products of von Neumann algebras

In this paper, we study bimodules over a von Neumann algebra M   in the context of an inclusion M⊆M_?αG$M\subseteq M{?}_{\alpha }G$, where G is a discrete group acting on a factor M by outer ?-automorphisms. We characterize the M  -bimodules X⊆M_?αG$X\subseteq M{?}_{\alpha }G$ that are closed in the Bures topology in terms of the subsets of G  . We show that this characterization also holds for w^?$w^{?}$-closed bimodules when G has the approximation property (AP  ), a class of groups that includes all amenable and weakly amenable ones. As an application, we prove a version of Mercer's extension theorem for certain w^?$w^{?}$-continuous surjective isometric maps on X.     

Singly generated planar algebras of small dimension, Part II

We classified in Bisch and Jones (Duke Math. J. 101 (2000) 41) all spherical C∗-planar algebras generated by a non-trivial 2-box subject to the condition that the dimension of N′∩M₂ is ?12. We showed that they are given by the Fuss-Catalan systems discovered in Bisch and Jones (Invent. Math. 128 (1997) 89) and one exceptional planar algebra. In the present paper, we extend these results and show that there is only one spherical C∗-planar algebra generated by a single non-trivial 2-box if the dimension of N′∩M₂ is 13. It is given by the standard invariant of the crossed product subfactor , where D₅ denotes the dihedral group with 10 elements.     

Bundle convergence of sequences of pairwise uncorrelated operators in von Neumann algebras and vectors in their L2-spaces

Let H be a separable complex Hilbert space, A a von Neumann algebra in ?(H),a faithful, normal state on A. We prove that if a sequence (X_n: n ≥ 1) of uncorrelated operators in A is bundle convergent to some operator X in A and Σ^∞_n=1n⁻² Var(X_n) log²(n + 1) < ∞, then X is proportional to the identity operator on H. We also prove an analogous theorem for certain uncorrelated vectors in the completion L₂=L₂(A,φ) of A given by the Gelfand-Naimark-Segal representation theorem. Both theorems were motivated by a recent one proved by Etemadi and Lenzhen in the classical commutative setting.     

On the asymptotic tensor C*-norm

We introduce a new asymptotic one-sided and symmetric tensor norm, the latter of which can be considered as the minimal tensor norm on the category of separable C^*-algebras with homotopy classes of asymptotic homomorphisms as morphisms. We show that the one-sided asymptotic tensor norm differs in general from both the minimal and the maximal tensor norms and discuss its relation to semi-invertibility of C^*-extensions. Received: 23 September 2004; revised: 30 May 2005     

Amenability and uniqueness

The main result of this paper is a characterization of properly infinite injective von Neumann algebras and of nuclear C^∗$C^{\ast }$-algebras by using a uniqueness theorem, based on generalizations of Voiculescu’s famous Weyl–von Neumann theorem.     

Entropy of automorphisms arising from dynamical systems through discrete groups with amenable actions

Let G⊂Aut(A) be a discrete group which is exact, that is, admits an amenable action on some compact space. Then the entropy of an automorphism of the algebra A does not change by the canonical extension to the crossed product A×G. This is shown for the topological entropy of an exact C∗-algebra A and for the dynamical entropy of an AFD von Neumann algebra A. These have applications to the case of transformations on Lebesgue spaces.     

Remarks on ghost projections and ideals in the Roe algebras of expander sequences

We use repeating sequences of expander graphs or small perturbations of expanders to present examples of ideals in the Roe algebras of bounded geometry discrete metric spaces which cannot be expressed as the sum of a ghost ideal and an ideal in which finite propagation operators are dense. This gives a negative answer to a question in [1, 3]. Received: 9 December 2006, Revised: 18 April 2007     

On ergodic theorems for free group actions on noncommutative spaces

We extend in a noncommutative setting the individual ergodic theorem of Nevo and Stein concerning measure preserving actions of free groups and averages on spheres s_2n of even radius. Here we study state preserving actions of free groups on a von Neumann algebra A and the behaviour of (s_2n(x)) for x in noncommutative spaces L^p(A). For the Cesàro means this problem was solved by Walker. Our approach is based on ideas of Bufetov. We prove a noncommutative version of Rota ``Alternierende Verfahren' theorem. To this end, we introduce specific dilations of the powers of some noncommutative Markov operators.     

Crossed products by finite group actions with certain tracial Rokhlin property

We introduce a special tracial Rokhlin property for unital C~*-algebras. Let A be a unital tracial rank zero C~*-algebra(or tracial rank no more than one C~*-algebra). Suppose that α : G → Aut(A) is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C~*-algebra. Then, the crossed product C~*-algebra C~*(G, A, α) has tracia rank zero(or has tracial rank no more than one). In fact,we get a more general results.     

The Rokhlin property and the tracial topological rank

Let A be a unital separable simple C∗-algebra  with TR(A)?1 and α be an automorphism. We show that if α satisfies the tracially cyclic Rokhlin property then . We also show that whenever A has a unique tracial state and α^m is uniformly outer for each m(≠0) and α^r is approximately inner for some r>0, α satisfies the tracial cyclic Rokhlin property. By applying the classification theory of nuclear C∗-algebras, we use the above result to prove a conjecture of Kishimoto: if A is a unital simple -algebra of real rank zero and α∈Aut(A) which is approximately inner and if α satisfies some Rokhlin property, then the crossed product is again an -algebra of real rank zero. As a by-product, we find that one can construct a large class of simple C∗-algebras with tracial rank one (and zero) from crossed products.     

On a conjecture by Ghahramani-Lau and related problems concerning topological centres

Let A be a Banach algebra, and consider A^** equipped with the first Arens product. We establish a general criterion which ensures that A is left strongly Arens irregular, i.e., the first topological centre of A^** is reduced to A itself. Using this criterion, we prove that for a very large class of locally compact groups, Ghahramani-Lau's conjecture (cf. [Lau 94] and [Gha-Lau 95]) stating the left strong Arens irregularity of the measure algebra M(G), holds true. (Our methods obviously yield as well the right strong Arens irregularity in the situation considered.)Furthermore, the same condition used above implies that every linear left A^**-module homomorphism on A^* is automatically bounded and w^*-continuous. We finally show that our criterion also yields a partial answer to a question raised by Lau-Ülger (Trans. Amer. Math. Soc. 348 (3) (1996) 1191) on the topological centre of the algebra (A^*⊙A)^*, for A having a right approximate identity bounded by 1.