Ken Dykema

Commutator structure of operator ideals

Commutator structure of operator ideals Ken Dykema, ,1 Tadeusz Figiel, Gary Weiss, and Mariusz Wodzicki Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA b Instytut Matematyczny Polskiej Akademii Nauk, 81-825 Sopot, Poland Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

EXACTNESS OF CUNTZ–PIMSNER C*-ALGEBRAS

Let H be a full Hilbert bimodule over a C*-algebra A. We show that the Cuntz-Pimsner C*-algebra associated to H is exact if and only if A is exact. Using this result, we give alternative proofs for exactness of reduced amalgamated free products of exact C*-algebras. In the case that A is a finite dimensional C*-algebra, we also show that the Brown-Voiculescu topological entropy of Bogljubov automorphisms of the Cuntz-Pimsner algebra associated to an A,A Hilbert bimodule is zero.

Free entropy dimension in amalgamated free products

We calculate the microstates free entropy dimension of natural generators in an amalgamated free product of certain von Neumann algebras, with amalgamation over a hyperfinite subalgebra. In particular, some ‘exotic’ Popa algebra generators of free group factors are shown to have the expected free entropy dimension. We also show that microstates and non-microstates free entropy dimension agree for generating sets of many groups. In the appendix, the first L 2 -Betti number for certain amalgamated free products of groups is calculated.

The stable rank of some free product

It is proved that the reduced group C*-algebra C*_{red}(G) has stable rank one (i.e. its group of invertible elements is a dense subset) if G is a discrete group arising as a free product G_1*G_2 where |G_1|>=2 and |G_2|>=3. This follows from a more general result where it is proved that if (A,tau) is the reduced free product of a family (A_i,tau_i), i\in I, of unital C*-algebras A_i with normalized faithful traces tau_i, and if the family satisfies the Avitzour condition (i.e. the traces, tau_i, are not too lumpy in a specific sense), then A has stable rank one.

C^\ast

Abstract In [4] we introduced the class of DT-operators, which are modeled by certain upper triangular random matrices, and showed that if the spectrum of a DT-operator is not reduced to a single point, then it has a nontrivial, closed, hyperinvariant subspace. In this paper, we prove that also every DT-operator whose spectrum is concentrated on a single point has a nontrivial, closed, hyperinvariant subspace. In fact, each such operator has a one-parameter family of them. It follows that every DT-operator generates the von Neumann algebra L( F 2 ) of the free group on two generators.

-algebras

A necessary and sufficient condition for the simplicity of the C*-algebra reduced free product of finite dimensional abelian algebras is found, and it is proved that the stable rank of every such free product is 1. Related results about other reduced free products of C*-algebras are proved.

Invariant subspaces of the quasinilpotent DT-operator

The intersection ring of a complex Grassmann manifold is generated by Schubert varieties, and its structure is governed by the Littlewood–Richardson rule. Given three Schubert varieties S1, S2, S3 with intersection number equal to one, we show how to construct an explicit element in their intersection. This element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags, and is therefore well defined over an arbitrary field of scalars. Moreover, this result also applies to appropriately defined analogues of Schubert varieties in the Grassmann manifolds associated with a finite von Neumann algebra. The arguments require the combinatorial structure of honeycombs, particularly the structure of the rigid extremal honeycombs. It is known that the eigenvalue distributions of self-adjoint elements a,b,c with a+b+c=0 in the factor Rω are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. In particular, if x,y,z are self-adjoint elements of such a factor and x+y+z=0, then there exist self-adjoint a,b,c∈Rω such that a+b+c=0 and a (respectively, b,c) has the same eigenvalue distribution as x (respectively, y,z). A (‘complete’) matricial form of this result is known to imply an affirmative answer to an embedding question formulated by Connes. The critical point in the proof of this result is the production of elements in the intersection of three Schubert varieties. When the factor under consideration is the algebra of n×n complex matrices, our arguments provide new and elementary proofs of the Horn inequalities, which do not require knowledge of the structure of the cohomology of the Grassmann manifolds.

Simplicity and the stable rank of some free product C*-algebras

Using free probability constructions involving Cuntz-Pimsner C*-algebras we show that the topological entropy of the free product of two automorphisms is equal to the maximum of the individual entropies. As applications we show that general free shifts have entropy zero. We show that any nuclear C*-dynamical system admits an entropy preserving covariant embedding into the Cuntz algebra on infinitely many generators. It follows that any simple nuclear purely infinite C*-algebra admits an automorphism with any given value of entropy. As a final application we show that if two automorphisms satisfy a CNT-variational principle then so does their free product.

Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor

Abstract. A reduction formula for compressions of von Neumann algebra II

Topological entropy of free product automorphisms

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