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## Über dieses Buch

This volume consists of articles contributed by participants at the fourth Ja­ pan-U.S. Joint Seminar on Operator Algebras. The seminar took place at the University of Pennsylvania from May 23 through May 27, 1988 under the auspices of the Mathematics Department. It was sponsored and supported by the Japan Society for the Promotion of Science and the National Science Foundation (USA). This sponsorship and support is acknowledged with gratitude. The seminar was devoted to discussions and lectures on results and prob­ lems concerning mappings of operator algebras (C*-and von Neumann alge­ bras). Among the articles contained in these proceedings, there are papers dealing with actions of groups on C* algebras, completely bounded mappings, index and subfactor theory, and derivations of operator algebras. The seminar was held in honor of the sixtieth birthday of Sh6ichir6 Sakai, one of the great leaders of Functional Analysis for many decades. This vol­ ume is dedicated to Professor Sakai, on the occasion of that birthday, with the respect and admiration of all the contributors and the participants at the seminar. H. Araki Kyoto, Japan R. Kadison Philadelphia, Pennsylvania, USA Contents Preface.... ..... ....... ........... ...... ......... ................ ...... ............... ... vii On Convex Combinations of Unitary Operators in C*-Algebras UFFE HAAGERUP ......................................................................... .

## Inhaltsverzeichnis

### On Convex Combinations of Unitary Operators in C*-Algebras

Abstract
Let A be a unital C*-algebra.
Uffe Haagerup

### Approximately Inner Derivations, Decompositions and Vector Fields of Simple C*-Algebras

Abstract
Given a mathematical structure, one of the basic associated objects is its automorphism group. In differential geometry, manifolds are equipped with a differ-entiable structure and the objects of study are the diffeomorphism group and its subgroups. In quantum mechanics, the observables form a non-Abelian algebra. In some of the more accessible cases, this algebra is a simple C*-algebra, and the objects of study are the associated groups of automorphisms. These are the quantum mechanical symmetry groups.
Palle Jorgensen

### Derivations in Commutative C*-Algebras

Abstract
Prof. S. Sakai began the systematic study of unbounded *-derivations in C*-algebras after his work on bounded *-derivations. For the theory of the unbounded *-derivations, he posed many questions in his lecture notes and his talks ([SI, S2]). In the case of commutative C*-algebras, several authors have developed the theory by trying to solve his problems ([Ba], [G]). In consequence, roughly speaking, now we may say that the structure of closed *-derivations has been almost clarified when the underlying space of the commutative C*-algebra is of 0- or 1-dimension, though a few problems have been left unsolved. Furthermore the structure of *-derivations commuting with group actions has rapidly become clear in the last decade. In this note, we shall not mention these structures, using as our references [Bra] and [T].
H. Kurose

### Representation of Quantum Groups

Abstract
The concept of quantum groups is important for the study of the quantum Yang-Baxter equations, Drinfeld [2], Jimbo [4], Manin [5] and others. On the other hand, Woronowicz [10] introduced the concept of compact matrix pseudogroups through the study of the dual object of groups. As pointed out by Rosso in [8], these two concepts are related to each other as quantum Lie algebras and quantum Lie groups. In this talk we want to indicate that the ideas of Kac algebras studied by Takesaki [9] and Enock and Schwartz [3] et al. are helpful for the study of quantum groups. As a result we can give a geometric interpretation for a q-analogue of a certain class of special functions, which has been a long standing problem of q- analogues.
Tetsuya Masuda, Katsuhisa Mimachi, Yoshiomi Nakagami, Masatoshi Noumi, Kimio Ueno

### Automorphism Groups and Covariant Irreducible Representations

Abstract
Given a C*-dynamical system (A,G,α), I would like to consider the problem of analyzing (Â,G,α*), where A is a C*-algebra with its dual Â, G is a locally compact group, and α is a continuous action of G on A by automorphisms with α * being the transposed action on Â. In other words, I would like to interpret the non-commutative system (A,G,α) in terms of the commutative-like system (Â,G,α*). As this is still too general a problem, my main concern will be with ‘type I orbits’ (which will be defined soon) in more restricted situations; especially with covariant irreducible representations. The latter was also the subject of my talk at the previous US-Japan seminar (cf. [10]).
Akitaka Kishimoto

### Proper Actions of Groups on C*-Algebras

Abstract
Recently I have been attempting to formulate a suitable C*-algebraic framework for the subject of deformation quantization of Poisson manifolds [1,13]. Some of the main examples which I have constructed within this framework [27] involve “proper” actions of groups on C*-algebras, where “proper” actions are to be defined as a generalization of proper actions of groups on locally compact spaces. Much of the material on proper actions which I have developed for this purpose is of a general nature which may be useful in other situations, as it has seemed appropriate to give a separate exposition of it, in the present article.
Marc A. Rieffel

### On The Baum-Connes Conjecture

Abstract
Recently, Connes [3] initiated a new index theory for both dynamical systems and foliated manifolds, which is quite useful to cases with pathological ambient spaces for which the Atiyah-Singer index theory is no longer applicable. The main ideas of his theory are based on both K-theory and cyclic cohomology of algebras, whose validity is illustrated in many papers.
Hiroshi Takai

### On Primitive Ideal Spaces of C*-Algebras over Certain Locally Compact Groupoids

Abstract
Let F be a locally compact Hausdorff second countable groupoid with a left Haar system {v x } xX in the sense of [9] (X = the unit space of Γ). By analogy with Fell’s algebraic bundles over groups, we define the notion of C*-algebras over F and, given a C*-algebra A over Γ, we can form a C*-algebra C*(Γ, A) as the completion of the cross sectional algebra of A. In this note, under some stringent assumptions on Γ, we present a concrete realization of the primitive ideal space of C*(Γ, A). This is a C*-version of [12].
Shigeru Yamagami

### On Sequences of Jones’ Projections

Abstract
In the index theory for finite factors introduced by Jones [3], the following sequence {e i ; i = 1,2,…} of projections plays an important role:
Marie Choda

### The Powers’ Binary Shifts on the Hyperfinite Factor of Type II1

Abstract
A unit preserving *-endomorphism σ on the hyperfinite II1 factor R is called a shift if $$\cap _{{i = 0}}^{\infty }{{\sigma }^{k}}(R) = \{ \lambda I;\lambda \in \mathbb{C}\}$$. A shift σ is called a Powers binary shift if there is a self adjoint unitary u such that R = {σ n (u); n ∈ IN ∪ {0}}″ and σ k (u)u = ±uσ k (u) for k ∈ IN ∪ {0}. Let q(σ) be the number min{k ∈ IN; σ k (R)∲ ∩R = ℂI}. It is shown that the number q(σ) is not the complete outer conjugacy invariant for a Powers’ binary shift.
Masatoshi Enomoto

### Index Theory for Type III Factors

Abstract
We describe the structure of (finite-index) inclusion of type III factors based on analysis of involved flows of weights. Roughly speaking, a type HI index theory splits into a “purely type III” index theory and an (essentially) type II index theory. The factor flows constructed in [1] serve as the complete invariant for the former in the AFD case while the latter can be analyzed by paragroups or quantized groups (as announced in [7]). Therefore, classification of subfactors in an AFD type III factor reduces to classification of factor flows and an “equivariant” paragroup theory.
Hideki Kosaki

### Relative Entropy of a Fixed Point Algebra

Abstract
The relative entropy H(MN) for a pair NM of finite von Neumann algebras was introduced and studied by M. Pimsner and S.Popa in [7]. One of their important results was to clarify the relationship between H(MN) and the Jones index [M : N] for a pair of finite factors ([2]). On the other hand, in [1], V. Jones succeeded in classifying actions of a finite group G on the hyperfinite type II1 factor R, up to conjugacy, associated with normal subgroups of G, characteristic invariants and inner invariants.
Satoshi Kawakami

### Jones Index Theory for C*-Algebras

Abstract
The notion of index [M : N] was introduced by Jones [13] as an invariant for subfactors N of a factor M of type II1. Subsequently Kosaki [18] defined an index E for a conditional expectation E of an arbitrary factor M onto a subfactor N using the spatial theory of Connes [6] and the theory of operator-valued weights of Haagerup [9]. We shall define an index E for a conditional expectation E on a C*-algebra. This index theory for C*-algebras is a mixture of the index theory by Jones and the theory of Morita equivalence by Rieffel [24], [25]. We establish the link between transfer in K -theory and a multiplication by Index E.
Yasuo Watatani

### Three Tensor Norms for Operator Spaces

Abstract
The purpose of these notes is to give a survey of some of the results, questions, and applications surrounding three tensor norms which occur in the non-self adjoint theory of operator algebras. The first norm we will focus on is the Haagerup norm on tensor products of subspaces of C* -algebras (which we call operator spaces). This portion of our talk includes some joint work with R.R. Smith [18]. The other tensor norms are the min and max norms on tensor products of non-selfadjoint operator algebras, which includes some joint work with S.C. Power ([16] and [17]).
Vern I. Paulsen

### Extension Problems for Maps on Operator Systems

Abstract
Given operator systems EF and a C*-algebra B the extension problem considers a map φ:EB and asks for an extension ψ:FB which makes the following diagram commute:
R. R. Smith

### Multivariable Toeplitz Operators and Index Theory

Abstract
In these notes we describe the recent progress made in the study of multivariable Toeplitz operators on domains in ℂ n , the C*-algebras generated by these operators and the index theory associated with C*-algebra extensions of Toeplitz type. These results are important for a better understanding of multivariable complex analysis and also connect Toeplitz operators with interesting C*-algebras not of type I, namely foliation C*-algebras and irrational rotation algebras.
Harald Upmeier

### On Maximality of Analytic Subalgebras Associated with Flow in von Neumann Algebras

Abstract
Let M be a von Neumann algebra on a Hilbert space H and let {α t } tIR be a σ-weakly continuous flow on M; i.e. suppose that {α t } tIR be a one-parameter group of *-automorphisms of M such that, for each p in the predual, M*, of M and for each xM, the function of t, p t (x)), is continuous on IR.
Kichi-Suke Saito

### Reflections Relating a von Neumann Algebra and Its Commutant

Abstract
The initial development of the theory of von Neumann algebras, proposed by von Neumann [12] and carried out by him in collaboration with F.J. Murray [9,10,11,13] can be viewed as consisting of two parts, an “algebraic theory” and a “spatial theory.” In the algebraic theory, the results refer to the von Neumann algebra R and make no reference to the corn-mutant; in the spatial theory, the results involve the commutant either explicitly or implicitly. Recognizing this mathematical dichotomy, Kaplan-sky [7,8] studied the algebraic structure of von Neumann algebras, without reference to their action on a space, isolating and putting in sharp focus many of the natural techniques that are basic to our subject. Of course, Murray and von Neumann had taken the algebraic theory to an advanced stage in their own way [9,10,11,13].