MINIMUM MODULI IN VON NEUMANN ALGEBRAS

Abstract

In this paper we answer a question raised in [12] in the affirmative, namely that the essential minimum modulus of an element in a von Neumann algebra, relative to any norm closed two-sided ideal, is equal to the minimum modulus of the element perturbed by an element from the ideal. As a corollary of this result, we extend some basic perturbation results on semi-Fredholm elements to a von Neumann algebra setting. We then characterize the semi-Fredholm elements in terms of the points of continuity of the essential minimum modulus function.     

C*-Bundles for Certain Liminal C*-Algebras

It is shown that certain liminal C*-algebras whose limit sets in their primitive ideal space are discrete can be described as algebras of continuous sections of a C*-bundle associated with them. Their multiplier algebras are also described in a similar manner. The class of C*-algebras under discussion includes all the liminal C*-algebras with Hausdorff primitive ideal spaces but also many other liminal algebras. A large sub-class of examples is examined in detail.      

Ghost ideals in uniform Roe algebras of coarse spaces

Let be the uniform Roe algebra of a coarse space X with uniformly locally finite coarse structure. We show that an operator G in is a ghost element if and only if the finite propagation operators in the principal ideal G are all compact operators. In contrast, if X is a discrete metric space with Yus property (A), then any ideal in is the closure of the finite propagation operators in the ideal.Received: 9 June 2004     

Matrix-valued interpolation and hyperconvex sets

We prove that it is possible for two uniform algebras to have the same scalar interpolating sets, yet still have different matrix-valued interpolating sets.We prove a result for tensor products of uniform algebras that extends Agler's interpolation formula for the bidisk to more general product domains. This is accomplished by introducing a dual object for interpolation problems, which we call a Schur ideal, and proving that the Schur ideal for a tensor product is the intersection of the corresponding Schur ideals.Research supported in part by a grant from the NSF     

Graph C *-Algebras and Their Ideals Defined by Cuntz-Krieger Family of Possibly Row-Infinite Directed Graphs

Let E be a possibly row-infinite directed graph. In this paper, first we prove the existence of the universal C^*-algebra C^*(E) of E which is generated by a Cuntz-Krieger E-family {s_e, p_v}, and the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for the ideal of C^*(E). Then we get our main results about the ideal structure of Finally the simplicity and the pure infiniteness of is discussed.     

THE UNIQUENESS OF OPERATIONAL QUANTITIES IN VON NEUMANN ALGEBRAS

Abstract

Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N].     

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     

Ideal structure of uniform Roe algebras of coarse spaces

Let be the uniform Roe algebra of a coarse space with uniformly locally finite coarse structure. By a controlled truncation technique, we show that the controlled propagation operators in an ideal I of are exactly the controlled truncations of elements in I. It follows that the lattice of the ideals of the uniform Roe algebra in which controlled propagation operators are dense, the lattice of the invariant open subsets in the unit space of the groupoid G(X) introduced by Skandalis, Tu and Yu, the lattice of the ideals of the coarse structure , and the lattice of the ideals of the coarse space X are mutually isomorphic. These lattices also give rise to a type of classification for the ideals of .     

Ideals of operators and the metric approximation property

We prove that a Banach space X has the metric approximation property if and only if , the space of all finite rank operators, is an ideal in , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if is an ideal in for every Banach space Y.Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.     

Topological and bornological characterisations of ideals in von Neumann algebras: II

SupposeM is a von Neumann algebra on a Hilbert spaceH andI is any norm closed ideal inM. We extend to this setting the well known fact that the compact operators on a Hilbert space are precisely those whose restrictions to the closed unit ball are weak to norm continuous.     

Classification of sequentially weakly continuous $JB^*$-triples

Let D be the open unit ball of a -triple A and let Aut(D) be the group of all biholomorphic automorphisms of D. It is shown that every element of Aut(D) is sequentially weakly continuous if and only if every primitive ideal of A is a maximal closed ideal and is a type I -triple without infinite-spin part. Implications for general structure theory are explored. In particular, it is deduced that every -triple A contains a smallest ideal J for which the sequentially weakly continuous biholomorphic automorphisms of the open unit ball of A/J are all linear. Received August 27, 1998; in final form February 10, 1999     

On<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-algebras whose Glimm ideals are primitive

Every Glimm ideal of anAW ^*-algebra is a (minimal) primitive ideal.     

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 some subalgebras of von Neumann algebras with analyticity

Let (M,α,G) be a covariant system on a locally compact Abelian group G with the totally ordered dual group which admits the positive semigroup . Let H^∞(α) be the associated analytic subalgebra of M; i.e. . Let be the analytic crossed product determined by a covariant system . We give the necessary and sufficient condition that an analytic subalgebra H^∞(α) is isomorphic to an analytic crossed product related to Landstad's theorem. We also investigate the structure of σ-weakly closed subalgebra of a continuous crossed product N?_θR which contains N?_θR₊. We show that there exists a proper σ-weakly closed subalgebra of N?_θR which contains N?_θR₊ and is not an analytic crossed product. Moreover we give an example that an analytic subalgebra is not a continuous analytic crossed product using the continuous decomposition of a factor of type III_λ(0?λ<1).     

KMS states of quasi-free dynamics on Pimsner algebras

A continuous one-parameter group of unitary isometries of a right-Hilbert -bimodule induces a quasi-free dynamics on the Cuntz-Pimsner -algebra of the bimodule and on its Toeplitz extension. The restriction of such a dynamics to the algebra of coefficients of the bimodule is trivial, and the corresponding KMS states of the Toeplitz-Cuntz-Pimsner and Cuntz-Pimsner -algebras are characterized in terms of traces on the algebra of coefficients. This generalizes and sheds light onto various earlier results about KMS states of the gauge actions on Cuntz algebras, Cuntz-Krieger algebras, and crossed products by endomorphisms. We also obtain a more general characterization, in terms of KMS weights, for the case in which the inducing isometries are not unitary, and accordingly, the restriction of the quasi-free dynamics to the algebra of coefficients is nontrivial.     

The central kernel of the Peirce-one space

Further investigation into the properties of the Peirce-one space J₁ corresponding to a weak^*-closed inner ideal J in a JBW^*-triple A is carried out, and, in particular, it is shown that J₁ contains no non-trivial weak^*-closed ideals.Received: 12 June 2002     

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 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.