Douglas W. B. Somerset

On the topology of the dual of a nilpotent Lie group

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

COMPLETELY BOUNDED MAPPINGS AND SIMPLICIAL COMPLEX STRUCTURE IN THE PRIMITIVE IDEAL SPACE OF A C*-ALGEBRA

We consider the natural contraction from the central Haagerup tensor product of a C*-algebra A with itself to the space of completely bounded maps CB(A) on A and investigate those A where there exists an inverse map with finite norm L(A). We show that a stabilised version L(A) = sup n L(M n (A)) depends only on the primitive ideal space Prim(A). The dependence is via simplicial complex structures (defined from primal intersections) on finite sets of primitive ideals that contain a Glimm ideal of A. Moreover L(A) = L(A ⊗ K(H)), with K(H) the compact operators, which requires us to develop the theory in the context of C*-algebras that are not necessarily unital.

A NOTE ON IDEAL SPACES OF BANACH ALGEBRAS

In a previous paper the second author introduced a compact topology on the space of closed ideals of a unital Banach algebra A. If A is separable then this topology is either metrizable or else neither Hausdorff nor first countable. Here it is shown that this topology is Hausdorff if A is the algebra of once continuously differentiable functions on an interval, but that if A is a uniform algebra then this topology is Hausdorff if and only if A has spectral synthesis. An example is given of a strongly regular, uniform algebra for which every maximal ideal has a bounded approximate identity, but which does not have spectral synthesis.

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

Abstract The derivation constant K ( A ) ≥ 1 2 has been extensively studied for unital non-commutative C ⁎ -algebras. In this paper, we investigate properties of K ( M ( A ) ) where M ( A ) is the multiplier algebra of a non-unital C ⁎ -algebra A. A number of general results are obtained which are then applied to the group C ⁎ -algebras A = C ⁎ ( G N ) where G N is the motion group R N ⋊ SO ( N ) . Utilizing the rich topological structure of the unitary dual G N ˆ , it is shown that, for N ≥ 3 , K ( M ( C ⁎ ( G N ) ) ) = 1 2 ⌈ N 2 ⌉ .

The Inner Corona Algebra of a C0(X)-Algebra

Let A = C(X) ⊗K(H), where X is a compact Hausdorff space and K(H) is the algebra of compact operators on a separable, infinite-dimensional Hilbert space. Let A be the algebra of strong∗-continuous functions from X to K(H). Then A/A is the inner corona algebra of A. We show that if X has no isolated points then A/A is an essential ideal of the corona algebra of A, and Prim(A/A), the primitive ideal space of A/A, is not weakly Lindelof. If X is also first countable then there is a natural injection from the power set of X to the lattice of closed ideals of A/A. If X = βN N and (CH) is assumed then the corona algebra of A is a proper subalgebra of the multiplier algebra of A/A. Several of the results are obtained in the more general setting of C0(X)-algebras. 2010 Mathematics Subject Classification: 46L05, 46L08, 46L45 (primary); 46E25, 46J10, 54A35, 54C35, 54D15, 54D35, 54G10 (secondary).