Charles A. Akemann

Multipliers of C∗-algebras☆

Abstract For a C ∗ -algebra A let M ( A ) denote the two-sided multipliers of A in its enveloping von Neumann algebra. A complete description of M ( A ) is given in the case where the spectrum of A is Hausdorff. The formula M ( A ⊗ α B ) = M ( A ) ⊗ α M ( B ) is discussed and examples are given where M(A) A is non-simple even though A is simple and separable. As a generalization of Tietzes Extension Theorem it is shown that a multiplier of a quotient of A is the image of an element from M ( A ), if A is separable. Finally, deriving algebras and thin operators and their relations to multipliers are discussed.

Left ideal structure of C∗-algebras

An abelian C∗-algebra is known to be isomorphic to the algebra of all complex continuous functions vanishing at infinity on its maximal ideal space. In an earlier paper the author defined a structure on the closed left ideals of an arbitrary C∗-algebra which was analogous to the structure topology on the maximal ideals which exists for the abelian case. This paper carries that work forward. The first section expands the technical base of the theory by showing that more of the desirable properties of a compact Hausdorff space carry over to the new structure. The following section contains applications of various sorts. For example, Theorem I.1 gives a new characterization of AW∗-algebras. The last section characterizes the maximal and minimal (closed) left ideals of any C∗-algebra.

Weak compactness in the dual space of a C∗-algebra

Abstract Several results in noncommutative measure theory for C ∗ -algebras are proved. A bounded linear map from a C ∗ -algebra to a weakly sequentially complete Banach space is weakly compact (Theorem 4.2). This was a conjecture of Sakai. This result is a consequence of a recent theorem of Pedersen. A theorem of the Vitali-Hahn-Saks type states that a sequence {ƒ i } of states on a W ∗ -algebra converges weakly if it converges weak ∗ (Corollary 3.3).

Geometric characterizations of some classes of operators in C*-algebras and von Neumann algebras

We present geometric characterizations of the partial isometries, unitaries, and invertible operators in C*-algebras and von Neumann algebras.

A Lyapunov‐type theorem from Kadison–Singer

Marcus, Spielman, and Srivastava recently solved the Kadison-Singer problem by showing that if u_1, ..., u_m are column vectors in C^d such that \sum u_iu_i^* = I, then a set of indices S \subseteq {1, ..., m} can be chosen so that \sum_{i \in S} u_iu_i^* is approximately (1/2)I, with the approximation good in operator norm to order \epsilon^{1/2} where \epsilon = \max \|u_i\|^2. We extend their result to show that every linear combination of the matrices u_iu_i^* with coefficients in [0,1] can be approximated in operator norm to order \epsilon^{1/8} by a matrix of the form \sum_{i \in S} u_iu_i^*.

B(H) has a pure state that is not multiplicative on any masa.

Assuming the continuum hypothesis, we prove that ℬ(H) has a pure state whose restriction to any masa is not pure. This resolves negatively old conjectures of Kadison and Singer and of Anderson.

Conditional Correlation Phenomena with Applications to University Admission Strategies

This paper considers the situation (as in college admissions) where one is given two attributes,X and Y, which one uses to predict a third attribute, Z, by some function Ẑ ofX andY. However, one only retains values ofX, Y, andZ for which Ẑ is large. A thorough discussion, under fairly general conditions on the distributions, is given of how the correlation coefficients of X, Y, andZ are affected by this restriction of the range of values. In the case of the normal distribution, where linear prediction is optimal, the role of suppressor variables is discussed.

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.

Derivations of non-separable C∗-algebras

Abstract The question of which C ∗ -algebras have only inner derivations has been considered by a number of authors for 25 years. The separable case is completely solved, so this paper deals only with the non-separable case. In particular, we show that the C ∗ -tensor product of a von Neumann algebra and an abelian C ∗ -algebra has only inner derivations. Other special types of C ∗ -algebras are shown to have only inner derivations as well such as the C ∗ -tensor product of L ( H ) (all bounded operators on separable Hilbert space) and any separable C ∗ -algebra having only inner derivations. Derivations from a smaller C ∗ -algebra into a larger one are also considered, and this concept is generalized to include derivations between C ∗ -algebras connected by a ∗ -homomorphism. Finally, we consider the general problem of a sequence of linear functionals on a C ∗ -algebra which converges to zero (in norm) when restricted to any abelian C ∗ -subalgebra. Does such a sequence converge to zero in norm? The answer is “yes” for normal functionals on L ( H ), but unknown in general.

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.