Joel Anderson

Extreme points in sets of positive linear maps on B(H)

Three main results are obtained: (1) If D is an atomic maximal Abelian subalgebra of B(H), P is the projection of B(H) onto D and h is a complex homomorphism on D, then h ∘ P is a pure state on B(H). (2) If {Pn} is a sequence of mutually orthogonal projections with rank(Pn) = n and ∑ Pn = I, P is the projection of B(H) onto {Pn}″ given by P(T)=∑tracen(T)Pn and h is a homomorphism on {Pn}″ such that h(Pn) = 0 for all n then h ∘ P induces a type II∞ factor representation of the Calkin algebra. (3) If M is a nonatomic maximal Abelian subalgebra of B(H) then there is an atomic maximal Abelian subalgebra D of B(H) and a large family {Φα} of ∗-homomorphisms from D onto M such that for each α, Φα ∘ P is an extreme point in the set of projections from B(H) onto M. (Here P denotes the projection of B(H) onto D.)

A secular equation for the eigenvalues of a diagonal matrix perturbation

Let D denote a diagonal n × n complex matrix, and suppose x1, …, xr and w1, …, wr are complex n-vectors. It is shown that there is a rational function F such that if λ is not an eigenvalue for D, then λ is an eigenvalue for P = D + x∗1w1 + … + x∗rwr if and only if F(λ) = 0. This generalizes a well-known result for the eigenvalues of a rank one self-adjoint perturbation. An immediate corollary in the rank one self-adjoint case is that the eigenvalues of P and D must interlace if the eigenvalues of D are distinct and the perturbation matrix is irreducible. It is shown that in the general case the function F also carries information about the eigenvalues of P. For example, λ is an eigenvalue of multiplicity m > 0 for P if and only if F(λ) = F′(λ) = … = F(m − 1)(λ) = 0 and F(m)(λ) ≠ 0. In the self-adjoint case, a necessary and sufficient condition for the eigenvalues of P and D to interlace is given, and the problem of determining the multiplicities of the eigenvalues of D as eigenvalues of P is studied. The formula yields a simple algorithm for determining the characteristic polynomial of a tridiagonal matrix.

A Conjecture Concerning the Pure States of B(H) and a Related Theorem

The purpose of this note is twofold. First to present a conjecture concerning the form of the pure states on B(H) and second to prove a theorem related to this conjecture.

The Stone-Weierstrass Problem for C*-Algebras

Suppose A is a C*-algebra and B is a C*-subalgebra of A. Theorems of the Stone-Weierstrass type assert that, if some additional conditions are met, then B=A. If M is the set of maximal modular left ideals of A together with A itself, then the original Stone-Weierstrass theorem can be stated as follows. If A is abelian and B separates M (i.e. for I, JeM, I=J if and only if I⋂B=J⋂B), then B=A. The general Stone-Weierstrass problem, which remains unsolved, is whether the assumption that A is abelian can be dropped. In this paper we shall review the historical development of the Stone-Weierstrass problem, the partial results, solutions for special cases, solutions with stronger hypotheses and variants. Then we shall discuss how some very recent work of Sorin Popa can be used to reduce the problem further. Full details will be given only in the last section since earlier proofs have already been published.

THE SPECTRAL SCALE AND THE

Suppose that c is a linear operator acting on an n-dimensional complex Hilbert Space H, and let tau denote the normalized trace on B(H). Set b_1 = (c+c*)/2 and b_2 = (c-c*)/2i, and write B for the the spectral scale of {b_1, b_2} with respect to tau. We show that B contains full information about (W_k)(c), the k-numerical range of c for each k =1,...,n. We then use our previous work on spectral scales to prove several new facts about (W_k)(c). For example, we show in Theorem 3.4 that the point lambda is a singular point on the boundary of (W_k)(c) if and only if lambda is an isolated extreme point of (W_k)(c). In this case lambda = (n/k)tau(cz), where z is a central projection in in the algebra generated by b_1, b_2 and the identity. We show in Theorem 3.5, that c is normal if and only if (W_k)(c) is a polygon for each k. Finally, it is shown in Theorem 5.4 that the boundary of (W_k)(c) is the finite union of line segments and curved real analytic arcs.

k

Suppose that c is a linear operator acting on an n-dimensional complex Hilbert Space H, and let tau denote the normalized trace on B(H). Set b_1 = (c+c*)/2 and b_2 = (c-c*)/2i, and write B for the the spectral scale of {b_1, b_2} with respect to tau. We show that B contains full information about (W_k)(c), the k-numerical range of c for each k =1,...,n. We then use our previous work on spectral scales to prove several new facts about (W_k)(c). For example, we show in Theorem 3.4 that the point lambda is a singular point on the boundary of (W_k)(c) if and only if lambda is an isolated extreme point of (W_k)(c). In this case lambda = (n/k)tau(cz), where z is a central projection in in the algebra generated by b_1, b_2 and the identity. We show in Theorem 3.5, that c is normal if and only if (W_k)(c) is a polygon for each k. Finally, it is shown in Theorem 5.4 that the boundary of (W_k)(c) is the finite union of line segments and curved real analytic arcs.

NUMERICAL RANGE

The concept of a diffuse sequence in a C*-algebra is introduced and exploited to complete the classification of separable, perfect C*-algebras. A C*-algebra is separable and perfect exactly when the closure of the pure state space consists entirely of atomic states. 0. Introduction. In [3] the first author and F. Shultz introduced the notion of a perfect C*-algebra (see ?1 for the definition) and characterized separable, type I, perfect C*-algebras. In Theorem 3.11 we complete this classification by showing that no separable, nontype I C*-algebra is perfect. Our approach requires the introduction of the concept of a diffuse sequence {bn} of operators in a C*-algebra (see Definition 2.1). Such a sequence is considered to be trivial if lim IIbn = 0. In ?2 we develop the basic facts about diffuse sequences. As applications we show that the existence of nontrivial diffuse sequences is largely a phenomenon of separable C*-algebras with very non-Hausdorff spectrum. In particular, we show that neither separably represented von Neumann algebras nor corona algebras (M(A)/A, A aunital) can have nontrivial diffuse sequences. In ?3 we are aiming for the characterization of separable, perfect C*-algebras in Theorem 3.11, but our route takes us deeply into the Fermion algebra. A somewhat generalized theorem of Glimm [9, 6.7.3] allows us to do most of our specific construction in the Fermion algebra because we can use our lifting result, Proposition 2.11, to lift the constructed sequences to an arbitrary, separable, nontype I C*-algebra. 1. Notation and preliminaries. Generally we follow the notation of [9]. The letters A and B will always denote C*-algebras with elements a, b, c, d, e, p, q, r, s, u, v, w, x, y. The letters f, g, h will denote generic elements of A*, the dual space of A (with j, , and o used for some special elements). We shall frequently consider A as canonically embedded in its double dual A**, identified with the weak closure of A in its universal representation (see [9, p. 60]). For any elements a, b, c C A and f c A* define (afb) c A* by (afb)(c) = f(acb). Let S(A) denote the state space of A, Q(A) the quasi-state space of A, and P(A) the pure state space of A. Convergence in A* will default to weak* convergence, while the default convergence in A** is strong*. The letter z will be reserved for the central projection in A** covering the reduced atomic representation of A (see [9, p. 103]). Any g c Q(A) with g(z) = g(l) is called atomic while any f c Q(A) with f(z) = 0 is called diffuse, and, by [1, Lemma 1.3], Ilf gll = Ilf + gl. Received by the editors August 28, 1985 and, in revised form, January 10, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46Lxx.

THE SPECTRAL SCALE AND THE NUMERICAL RANGE

Abstract 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 2 k . 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 2 k ’ 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 Fialas 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.