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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