The distance between two operator algebras acting on a Hilbert space H is defined to be the Hausdorff distance between their unit balls. We investigate the structural similarities between two close AW*-algebras A and B acting on a Hilbert space H. In particular, we prove that if A is of type I and separable, then A and B are *-isomorphic.

We construct a suitable representation of a C*-algebra that carries single elements to rank one operators. We also prove an abstract spectral theorem for compact elements in the algebra. This leads naturally to an abstract definition of Cp-classes of compact elements in the algebra.

We give an explicit construction of a continuous trace C*algebra with prescribed Dixmier-Douady class, and with only finite-dimensional irreducible representations. These algebras often have non-trivial automorphisms, and we show how a recent description of the outer automorphism group of a stable continuous trace C*algebra follows easily from our main result. Since our motivation came from work on a new notion of central separable algebras, we explore the connections between this purely algebraic subject and C*a1gebras.

By a theorem of Fell and Tomiyama-Takesaki, an N-homogeneous C*-algebra with spectrum X has the form Γ(E) for some bundle E over X with fibre MN(C), and its isomorphism class is determined by that of E and its pull-backs f*E along homeomorphisms f of X. We describe the homogeneous C*-algebras with spectrum T2 or T3 by classifying the MN-bundles over Tk using elementary homotopy theory. We then use our results to determine the isomorphism classes of a variety of transformation group C*-algebras, twisted group C*-algebras and more general crossed products.

Let G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

A joint spectral theorem for an n-tuple of doubly commuting unbounded normal operators in a Hilbert space is proved by using the techniques of GB*-algebras.

Let Ω be the weakly pseudoconvex domain

and let ∂Ω be its boundary. If ϕ ∈ L∞ (∂Ω), we denote by Tϕ, the Toephtz operator with symbol ϕ acting on the Hardy space H2(∂Ω), and by J(∂Ω) the C*-subalgebra of B(H2(∂Ω)) generated by the Toeplitz operators with continuous symbol. Our main theorem asserts that J(∂Ω) contains the ideal K of all compact operators on H2(∂Ω), and that the symbol map ϕ→Tϕ induces an isomorphism of C(∂Ω) onto the quotient C*-algebra ℑ(∂Ω)/K. Similar results have been established before for other domains, and in particular when Ω is strongly pseudoconvex. The main interest of our results lies in their proofs: ours are elementary, whereas those used in the strongly pseudoconvex case depend heavily on the theory of the tangential Cauchy-Riemann operator.

Let υ be a C*-algebra, α a *-anti-automorphism of order 2, and υα(±1) = {A; A ∈ υ, α(A) = ± A} the spectral subspaces of α. It follows that υα(+ 1) is a Jordan algebra and υα(− 1) is a Lie algebra. We begin the classification of pairs of Jordan and Lie algebras which can occur in this manner by examining υ = ℒ(ℋ), the algebra of bounded operators on a Hilbert space ℋ.