The local index formula in semifinite von Neumann algebras II: The even case

We generalise the even local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra A of a general semifinite von Neumann algebra. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula.     

The Hochschild class of the Chern character for semifinite spectral triples

We provide a proof of Connes’ formula for a representative of the Hochschild class of the Chern character for (p,∞)-summable spectral triples. Our proof is valid for all semifinite von Neumann algebras, and all integral p?1. We employ the minimum possible hypotheses on the spectral triples.     

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.     

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.     

Duality and operator algebras: automatic weak* continuity and applications

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras, and give several applications of the surprising fact that certain maps are always weak^*-continuous on dual spaces. In particular, if X is a subspace of a C^*-algebra A, and if a∈A satisfies aX⊂X, then we show that the function x?ax on X is automatically weak^* continuous if either (a) X is a dual operator space, or (b) a^*X⊂X and X is a dual Banach space. These results hinge on a generalization to Banach modules of Tomiyama's famous theorem on contractive projections onto a C^*-subalgebra. Applications include a new characterization of the σ-weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and a generalization of the theory of W^*-modules to the framework of modules over such algebras. We also give a Banach module characterization of σ-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.     

Duality and normal parts of operator modules

For an operator bimodule X over von Neumann algebras A⊆B(H) and B⊆B(K), the space of all completely bounded A,B-bimodule maps from X into B(K,H), is the bimodule dual of X. Basic duality theory is developed with a particular attention to the Haagerup tensor product over von Neumann algebras. To X a normal operator bimodule X_n is associated so that completely bounded A,B-bimodule maps from X into normal operator bimodules factorize uniquely through X_n. A construction of X_n in terms of biduals of X, A and B is presented. Various operator bimodule structures are considered on a Banach bimodule admitting a normal such structure.     

The Krein Spectral Shift Function in Semifinite von Neumann Algebras

We show the existence of a spectral shift function in the sense of Krein for bounded trace class perturbations of a self-adjoint operator affiliated with a semifinite von Neumann algebra     

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.     

Operator spaces with prescribed sets of completely bounded maps

Suppose A is a dual Banach algebra, and a representation π:A→B(?₂) is unital, weak* continuous, and contractive. We use a “Hilbert-Schmidt version” of Arveson distance formula to construct an operator space X, isometric to ?₂⊗?₂, such that the space of completely bounded maps on X consists of Hilbert-Schmidt perturbations of π(A)⊗I_?2. This allows us to establish the existence of operator spaces with various interesting properties. For instance, we construct an operator space X for which the group K₁(CB(X)) contains Z₂ as a subgroup, and a completely indecomposable operator space containing an infinite dimensional homogeneous Hilbertian subspace.     

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.     

Quasi-multipliers and algebrizations of an operator space

Let X be an operator space, let φ be a product on X, and let (X,φ) denote the algebra that one obtains. We give necessary and sufficient conditions on the bilinear mapping φ for the algebra (X,φ) to have a completely isometric representation as an algebra of operators on some Hilbert space. In particular, we give an elegant geometrical characterization of such products by using the Haagerup tensor product. Our result makes no assumptions about identities or approximate identities. Our proof is independent of the earlier result of Blecher, Ruan and Sinclair [D.P. Blecher, Z.-J. Ruan, A.M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1) (1990) 188-201] which solved the case when the bilinear mapping has an identity of norm one, and our result is used to give a simple direct proof of this earlier result. We also develop further the connections between quasi-multipliers of operator spaces and their representations on a Hilbert space or their embeddings in the second dual, and show that the quasi-multipliers of operator spaces defined in [M. Kaneda, V.I. Paulsen, Quasi-multipliers of operator spaces, J. Funct. Anal. 217 (2) (2004) 347-365] coincide with their C^∗-algebraic counterparts.     

Characterizations of Noncommutative H ∞

We transfer a large part of the circle of theorems characterizing the generalization of classical H^∞ known as ‘weak* Dirichlet algebras’, to Arveson’s very general noncommutative setting of subalgebras of finite von Neumann algebras. This solves the long-standing open question of the equivalence of principles such as Szeg?’s theorem, the weak* density of A + A*, and so on, within the noncommutative setting. The techniques should also be useful in future developments in noncommutative H^p theory.     

Preduals of von Neumann Algebras

Proofs of two assertions are sketched. 1) If the Banach space of a von Neumann algebra A is the third dual of some Banach space, then the space A is isometrically isomorphic to the second dual of some von Neumann algebra A and the von Neumann algebra A is uniquely determined by its enveloping von Neumann algebra (up to von Neumann algebra isomorphism) and is the unique second predual of A (up to isometric isomorphism of Banach spaces). 2) An infinite-dimensional von Neumann algebra cannot have preduals of all orders.     

Compressions of free products of von Neumann algebras

A reduction formula for compressions of von Neumann algebra II–factors arising as free products is proved. This shows that the fundamental group is for some such algebras. Additionally, by taking a sort of free product with an unbounded semicircular element, continuous one parameter groups of trace scaling automorphisms on II–factors are constructed; this produces type III factors with core , where can be a full II–factor without the Haagerup approximation property. Received: 26 October 1998 / in final form 18 March 1999     

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.     

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.     

Fully symmetric operator spaces

It is shown that certain interpolation theorems for non-commutative symmetric operator spaces can be deduced from their commutative versions. A principal tool is a refinement of the notion of Schmidt decomposition of a measurable operator affiliated with a given semi-finite von Neumann algebra.Research supported by A.R.C.     

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].     

Spectral flow and Dixmier traces

We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd -summable Breuer-Fredholm module in terms of a Hochschild 1-cycle. We explain how to derive a Wodzicki residue for pseudo-differential operators along the orbits of an ergodic action on a compact space X. Finally, we give a short proof of an index theorem of Lesch for generalised Toeplitz operators.     

Operator biflatness of the Fourier algebra and approximate indicators for subgroups

We investigate if, for a locally compact group G, the Fourier algebra A(G) is biflat in the sense of quantized Banach homology. A central rôle in our investigation is played by the notion of an approximate indicator of a closed subgroup of G: The Fourier algebra is operator biflat whenever the diagonal in G×G has an approximate indicator. Although we have been unable to settle the question of whether A(G) is always operator biflat, we show that, for , the diagonal in G×G fails to have an approximate indicator.