Robert J. Archbold

On factorial states of operator algebras

Results for the factorial state space of a C∗-algebra A which are analogous to results of Glimm (Ann. of Math. 72 (1960), 216–244; 73 (1961), 572–612),Tomiyama and Takesaki (Tohoku Math. J. (2) 13 (1961), 498–523) for the pure state space. It is shown that A is prime if and only if the (type I) factorial states are dense in the state space. It follows that every factorial state is a w∗-limit of type I factorial states. The factorial state space of a von Neumann algebra is determined, and it is shown that if A is unital and acts non-degenerately on a Hilbert space then the factorial state space of the generated von Neumann algebra restricts precisely to the factorial state space of A. It is shown that the set of factorial states is w∗-compact if and only if A is unital, liminal and has Hausdorff primitive ideal space.

Quasi-standard C*-algebras

A necessary and sufficient condition is given for a separable C*-algebra to be *-isomorphic to a maximal full algebra of cross-sections over a base space such that the fibre algebras are primitive throughout a dense subset. The condition is that the relation of inseparability for pairs of points in the primitive ideal space should be an open equivalence relation.

On the norm of an inner derivation of a C*-algebra

Let A be a C *-algebra with centre Z . If a ∈ A , the bounded linear mapping x → + ax – xa ( x ∈ A ) is called the inner derivation of A induced by a , and we denote it by D ( a, A ). A simple application of the triangle inequality shows that where d ( a; Z ) denotes the distance from a to Z in the normed space A .

Ideal Spaces of the Haagerup Tensor Product of C*-Algebras

Following the work of Allen, Sinclair and Smith on the primitive ideal space of the Haagerup tensor product A ⊗ hB of C*-algebras A and B, we investigate the hull-kernel topology and use this to determine various other ideal spaces and their topologies in relation to the corresponding ideal spaces of A and B. We study the semi-continuity of norm functions I → ||x + I||(x ∈ A ⊗h B) on these ideal spaces and identify the separated points of Prim(A ⊗h B). Finally, we exhibit several conditions each of which is equivalent to the quasi-standardness of A ⊗h B.

On the topology of the dual of a nilpotent Lie group

In this paper we investigate separation properties in the dual Ĝ of a connected, simply connected, nilpotent Lie group G . Following [ 4 , 19 ], we are particularly interested in the question of when the group G is quasi-standard, in which case the group C *-algebra C *( G ) may be represented as a continuous bundle of C *-algebras over a locally compact, Hausdorff, space such that the fibres are primitive throughout a dense subset. The same question for other classes of locally compact groups has been considered previously in [ 1 , 5 , 18 ]. Fundamental to the study of quasi-standardness is the relation of inseparability in Ĝ[ratio ]π∼σ in Ĝ if π and σ cannot be separated by disjoint open subsets of Ĝ. Thus we have been led naturally to consider also the set sep (Ĝ) of separated points in Ĝ (a point in a topological space is separated if it can be separated by disjoint open subsets from each point that is not in its closure).

Simply connected nilpotent Lie groups with quasi-standard *-algebras

The problem of when the group C*-algebra of a locally compact group is quasi-standard is investigated for certain simply connected nilpotent Lie groups. The characterization is in terms of the coadjoint orbit structure in the dual of the Lie algebra.

Strength of convergence and multiplicities in the spectrum of a C*-dynamical system

We consider separable

Stable rank and real rank for some classes of group *-algebras

C^*

Norms of inner derivations for multiplier algebras of C⁎-algebras and group C⁎-algebras, II☆

-dynamical systems

The Dixmier property and tracial states for C*-algebras

(A,G,\alpha)