Suppose that c is an operator on a Hilbert Space H such that the von Neumann algebra N generated by c is finite. Suppose that tau is a faithful normal tracial state on N. Let B denote the spectal scale of c with respect to tau. We show that the boundary of the numerical range of c is exactly the set of radial complex slopes on B at the origin. Further, we show that points on this boundary that lie in the numerical range are visible as line segments in the boundary of B. Also, line segments on the boundary which lie in the numerical range show up as faces of dimension two in the boundary of B. Finally, when c is normal, we prove that the point spectrum of c is exactly the set of complex slopes of 1-dimensional faces of B.

Given an n-tuple {b_1, ..., b_n} of self-adjoint operators in a finite von Neumann algebra M and a faithful, normal tracial state tau on M, we define a map Psi from M to R^{n+1} by Psi(a) = (tau(a), tau(b_1a), ..., tau(b_na)). The image of the positive part of the unit ball under Psi is called the spectral scale of {b_1, .., b_n} relative to tau and is denoted by B. In a previous paper with Nik Weaver we showed that the geometry of B reflects spectral data for real linear combinations of the operators {b_1, .., b_n}. For example, we showed that an exposed face in B is determined by a certain pair of spectral projections of a real linear combination of {b_1, .., b_n}. In the present paper we extend this study to faces that are not exposed. We completely describe the structure of arbitrary faces of B in terms of {b_1, .., b_n} and tau. We also study faces of convex, compact sets that are exposed by more than one hyperplane of support. Although many of the conclusions of this study involve too much notation to fit nicely in an abstract, there are two results that give their flavor very well. Let N be the algebra generated by {b_1, ..., b_n} and the identity. Theorem 6.1: If the set of extreme points of B is countable, then N is abelian. Corollary 5.6: B has a finite number of extreme points if and only if N is abelian and finite dimensional.

The concept of regularity in the meta-topological setting of projections in the double dual of a C*-algebra addresses the interrelations of a projection p with its closure, for instance in the form that such projections act identically, in norm, on elements of the C*-algebra. This concept has been given new actuality with the recent plan of Peligrad and Zsido to find a meaningful notion of Murray-von Neumann type equivalence among open projections. Although automatic in the commutative case, it has been known since the late sixties that regularity fails for many projections. The original investigations, however, did not answer a question such as: "Are all open and dense projections regular in A, when A is simple?" We report here that this and related questions have negative answers. In the other direction, we supply positive results on regularity of large open projections.

In many cases the convexity of the image of a linear map with range is $R^n$ is automatic because of the facial structure of the domain of the map. We develop a four step procedure for proving this kind of ``automatic convexity''. To make this procedure more efficient, we prove two new theorems that identify the facial structure of the intersection of a convex set with a subspace in terms of the facial structure of the original set. Let $K$ be a convex set in a real linear space $X$ and let $H$ be a subspace of X that meets $K$. In Part I we show that the faces of $K\cap H$ have the form $F\cap H$ for a face $F$ of $K$. Then we extend our intersection theorem to the case where $X$ is a locally convex linear topological space, $K$ and $H$ are closed, and $H$ has finite codimension in $X$. In Part II we use our procedure to ``explain'' the convexity of the numerical range (and some of its generalizations) of a complex matrix. In Part III we use the topological version of our intersection theorem to prove a version of Lyapunov's theorem with finitely many linear constraints. We also extend Samet's continuous lifting theorem to the same constrained siuation.

There is a hierarchy of structure conditions for convex sets. In this paper we study a recently defined [3, 8, 9] condition called locally nonconical convexity (abbreviated LNC). Is is easy to show that every strictly convex set is LNC, as are half-spaces and finite intersections of sets of either of these types, but many more sets are LNC. For instance, every zonoid (the range of a nonatomic vector-valued measure) is LNC (Corollary 34). However, there are no infinite-dimensional compact LNC sets (Theorem 23). The LNC concept originated in a search for continuous sections, and the present paper shows how it leads naturally (and constructively) to continuous sections in a variety of situations. Let Q be a compact, convex set in R^n, and let T be a linear map from R^n into R^m. We show (Theorem 1) that Q is LNC if and only if the restriction of any such T to Q is an open map of Q onto T(Q). This implies that if Q is LNC, then any such T has continuous sections (i.e. there are continuous right inverses of T) that map from T(Q) to Q, and in fact it is possible to define continuous sections constructively in various natural ways (Theorem 3, Corollary 4, and Theorem 5). If Q is strictly convex and T is not 1-1, we can construct continuous sections which take values in the boundary of Q (Theorem 6). When we give up compactness it is natural to consider a closed, convex, LNC subset Q of a Hilbert space X which may be infinite-dimensional. In this case we must assume that T is left Fredholm, i.e. a bounded linear map with closed range and finite-dimensional kernel. We can then prove results analogous to those mentioned in the last paragraph (Theorems 16-20). We also prove that T(Q) is LNC (Theorem 25).

The corona algebra M (A)/A contains essential information on the global structure of A, as demonstrated for instance by Busby theory.It is an interesting and surprisingly difficult task to determine the ideal structure of M (A)/A by means of the internal structure of A.Toward this end, we generalize Freudenthal's classical theory of ends of topological spaces to a large class of C * -algebras.However, mirroring requirements necessary already in the commutative case, we must restrict attention to C * -algebras A which are σ-unital and have connected and locally connected spectra.Furthermore, we must study separately a certain pathological behavior which occurs in neither commutative nor stable C * -algebras.We introduce a notion of sequences determining ends in such a C * -algebra A and pass to a set of equivalence classes of such sequences, the ends of A. We show that ends are in a natural 1-1 correspondence with the set of components of M (A)/A, hence giving a complete description of the complemented ideals of such corona algebras.As an application we show that corona algebras of primitive σ-unital C * -algebras are prime.Furthermore, we employ the methods developed to show that, for a large class of C *algebras, the end theory of a tensor product of two nonunital C * -algebras is always trivial. Introduction.The corona algebra M (A)/A ([34]) of a non-unital C * -algebra A contains essential information on the global structure "at infinity" of A. An important instance of this is the bijective correspondence between * -homomorphisms from B to M (A)/A and equivalence classes of extensions of C * -algebras 0 -→ A -→ X -→ B -→ 0 noted by Busby ([13]).This observation is fundamental in BDF-theory ([11]) and its generalizations, which apply to describe the set of extensions by K-theory in certain cases.The objective of the present paper is to develop and then apply a generalized form of end theory to describe the ideal structure of a corona algebra

Let X denote a finite set, k and n denote natural numbers and S-1,...,S-n denote subsets of X. Assume that no point of X lies in more than k of these subsets. In 1981 Beck and Fiala proved that there is a 2-coloring of X such that each of the subsets has discrepancy less than 2k. This result has an interpretation as a theorem about incidence matrices and its generalization to real matrices (with essentially the same proof) is called the continuous Beck-Fiala theorem. We investigate the continuous version of the conjecture of Beck and Fiala that 'less than 2k' could be replaced by 'less than or equal to k'. For matrices with nonnegative entries, we show that the answer to the corresponding continuous problem is 'no', so the continuous Beck-Fiala theorem is optimal in this case. However our methods do not provide a counterexample to Beck and Fiala's original conjecture. On the other hand we show that the answer to the corresponding continuous problem is 'yes' when the dimension of the matrix that corresponds to n is 1, 2 or 3.

We give a complete description of the closed faces in twelve kinds of convex sets that appear in operator algebra theory. These consist of positive parts of unit balls for C*-algebras and their dual spaces, and for von Neumann algebras and their pre-duals; of self-adjoint parts of unit balls in the same four classes and finally of general unit balls in the four classes. All these faces are shown to be semi-exposed and naturally paired with a 'polar' face in the dual (or pre-dual) space. We establish a density theorem (a la Kap) by showing that every face in the unit ball of a C*-algebra is weakly dense in its 'bi-polar' face; and we conclude with results about faces in unit balls for certain one-sided ideals and their quotients.

In 1940 Lyapunov proved that the range of a nonatomic vector-valued measure is compact and convex. This theorem translates into the language of operator algebras as follows.Lyapunov's Theorem: If PSI is a weak* continuous linear map from an abelian, nonatomic von Neumann algebra M to a finite dimensional space, then PSI(P) = PSI((M+)1), where P denotes the set of projections in M and (M+)1 denotes the positive portion of the unit ball of M.

The concept of a diffuse sequence in a C ∗ {C^{\ast }} -algebra is introduced and exploited to complete the classification of separable, perfect C ∗ {C^{\ast }} -algebras. A C ∗ {C^{\ast }} -algebra is separable and perfect exactly when the closure of the pure state space consists entirely of atomic states.

Negative definite functions (all definitions are given in § 1 below) on a locally compact, σ-compact group G have been used in several different contexts recently [2, 5, 7, 11]. In this paper we show how such functions relate to other properties such a group may have. Here are six properties which G might have. They are grouped into three pairs with one property of each pair involving negative definite functions. We show that the paired properties are equivalent and, where possible, give counter-examples to other equivalences. We assume throughout that G is not compact.(1A) G does not have property T.(IB) There is a continuous, negative definite function on G which is unbounded.(2A) G has the (weak and/or strong) dual R-L property.(2B) For every closed, non-compact set Q ⊂ G there is a continuous, negative definite function on G which is unbounded on Q.

Every discrete group G generates a C*-algebra 9Q1(G) of operators on the Hilbert space L2(G) of square summable complex valued functions on G. 9Q1(G), in which elements are written as formal (generally infinite) complex linear combinations ExE -GaXx, is the closure in operator norm of the group algebra L(G) of finite linear combinations, which acts on L2(G) by left multiplication (the left regular representation of G). Our main result is a formula for the explicit calculation of the norm of certain operators in L(G). We show that, if xl, . . ., x, satisfy a certain freedom condition (in particular if they freely generate a free subgroup of G), then for arbitrary complex scalars al,,... ,an5 a~ ~ ~ ~ ~~~Z Z21 i1 _Z JZ>A7