SECTION 2: FUNDAMENTAL FAMILIES

Let G be a compact Lie group. We will be interested in

certain full subcategories of the category of separable nuclear

*

C -algebras equipped with a continuous action of G by

•-automorphisms. For simplicity, we'll call such an object a

G-algebra. The morphisms in this category are G-equivariant

•-homomorphisms. A certain G-algebra which plays a vital role is

K, the algebra of compact operators on the representation space

of an infinite-dimensional unitary representation of G, which we

will generally take to be an infinite direct sum of copies of the

regular representation of G. The spectrum A of a G-algebra has

the structure of a G-space in a natural manner. When we regard

the spectrum as a G-space with its canonical structure, we use

the term G-spectrum of A. Open G-invariant subsets correspond to

G-invariant ideals, and closed G-invariant subsets correspond to

G-lnvariant quotients. We remind the reader that the G-structure

on A does not determine the G-action on A in general. This is a

difficult point which enormously complicates the study of

G-algebras. We shall return to this matter later in this

section.

Occasionally we shall wish to vary the group G. Therefore

it's useful to record for future reference that if A is a

G-algebra and H is a closed subgroup of G, then A may also be

viewed by restriction as an H-algebra. In the other direction, if

B is an H-algebra, we may define the induced G-algebra (cf.

[RR,%3]) to be

IndHTGA = C(GXHA) = {f€C(G,A):f(gh) = h"1-f(g), for g€G, h€H .

Here G operates by left translation.

Unless stated otherwise, spaces are to be locally compact,

second countable, and equipped with continuous G-actions. (If X

is such a space, then C (X) is a typical abelian G-algebra.) AQ

denotes the category of abelian G-algebras. CQ denotes the

smallest subcategory of G-algebras containing the separable Type

I G-algebras, and closed under G-kernels, G-quotients,

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