Gert K. Pedersen

C∗-algebras of real rank zero

Abstract The concept of real rank of a C ∗ -algebra is introduced as a non-commutative analogue of dimension. It is shown that real rank zero is equivalent to the previously defined conditions FS and HP, and that it is invariant under strong Morita equivalence, in particular under stable isomorphism. Real rank zero is also invariant under inductive limits and split extensions, and the class may well be regarded as the conceptual completion of the AF-algebras. In some cases, notably when the algebra is matroid, it is shown that the multiplier algebra also has real rank zero—although that is not true in general. By a result of G. J. Murphy, this implies a Weyl-von Neumann type result for self-adjoint multiplier elements in these cases.

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.

The Radon-Nikodym theorem for von neumann algebras

Let ϕ be a faithful normal semi-finite weight on a von Neumann algebraM. For each normal semi-finite weight ϕ onM, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operatorh, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ(h·). Conversely, each suchh determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result thatM is semi-finite if and only if Σ consists of inner automorphisms.

Jensen's Operator Inequality

Jensens operator inequality and Jensens trace inequality for real functions defined on an interval are established in what might be called their definitive versions. This is accomplished by the introduction of genuine non-commutative convex combinations of operators, as opposed to the contractions considered in earlier versions of the theory by the authors, and by Brown and Kosaki. As a consequence, one no longer needs to impose conditions on the interval of definition. It is shown how this relates to the pinching inequality of Davis, and how Jensens trace inequality generalizes to C*-algebras.

Diagonalizing matrices over C(X)

A complete characterization of those compact Hausdorff spaces is given such that for every n, each normal element in the algebra C(X)⊗Mn of continuous functions from X to Mn can be continuously diagonalized. The conditions are that X be a sub-Stonean space with dim X ⩽ 2 and carries no nontrivial G-bundles over any closed subset, for G a symmetric group or the circle group. In particular, diagonalization is assured on every totally disconnected sub-Stonean space, but also on connected spaces of the form β(Y)/Y, where Y is a simply-connected (noncompact) graph.

Equivalence and traces on C∗-algebras

Abstract We introduce an equivalence relation among the positive elements in a C∗ and show that the algebra is (semi-) finite if and only if there is a separating family of (semi-) finite traces. Concentrating on simple, semi-finite C∗-algebras we relate geometrical properties in the cone of equivalence classes to functional analytic properties of the algebra, such as the number of normalized traces and their possible values on a given element. The paper may be considered as an attempt to extend Murray and von Neumanns type and equivalence theory to C∗-algebras.

Approximating derivations on ideals ofC*-algebras

For each*-derivation δ of a separableC*-algebraA and each ε>0 there is an essential idealI ofA and a self-adjoint multiplierx ofI such that ∥(δ−ad(ix))|I∥

Sub-stonean spaces and corona sets

Mathematics Institute, University of Copenhagen, Universitetsparken 5, DK-2100, Denmark Communicated by A. Cannes Received July 1983 A self-contained’account of the theory of sub-Stonean spaces, and their relations to Stonean spaces and Rickart spaces is given. Of particular interest are the corona sets (of the form p(X)\X) for locally compact, u-compact spaces, because these highly nontrivial sub-Stonean spaces lend themselves to Tech-cohomological considerations. The theory of sub-Stonean spaces is essential for our solution of the diagonalization problem for C(X)@ M,, found in K. Grove and G. K. Pedersen, Diagonalizing matrices over C(X), submitted for publication.

Projectivity, transitivity and AF-telescopes

Continuing our study of projective C∗-algebras, we establish a projective transitivity theorem generalizing the classical Glimm-Kadison result. This leads to a short proof of Glimm’s theorem that every C∗-algebra not of type I contains a C∗-subalgebra which has the Fermion algebra as a quotient. Moreover, we are able to identify this subalgebra as a generalized mapping telescope over the Fermion algebra. We next prove what we call the multiplier realization theorem. This is a technical result, relating projective subalgebras of a multiplier algebra M(A) to subalgebras of M(E), whenever A is a C∗-subalgebra of the corona algebra C(E) = M(E)/E. We developed this to obtain a closure theorem for projective C∗-algebras, but it has other consequences, one of which is that if A is an extension of an MF (matricial field) algebra (in the sense of Blackadar and Kirchberg) by a projective C∗-algebra, then A is MF. The last part of the paper contains a proof of the projectivity of the mapping telescope over any AF (inductive limit of finite-dimensional) C∗-algebra. Translated to generators, this says that in some cases it is possible to lift an infinite sequence of elements, satisfying infinitely many relations, from a quotient of any C∗-algebra.

Compact abelian groups of automorphisms of simpleC*-algebras

If α is a continuous representation of a compact abelian groupG as *-automorphisms of a prime or simpleC*-algebraA we study how algebraic properties of the fixed-point algebraA0 are reflected in properties of the spectrum of α and outerness of α1,t∈G. For example, ifA is prime, Γ(α)=Sp(α) if and only ifA0 is prime, and the same with simple (instead of prime) whenG is finite. In the latter case the relative commutant ofA0 inA is trivial if and only if all α1≠l are outer, and ifG is cyclic of prime order all the conditions above are equivalent. This paper replaces the earlier versions [12] and [9].