§1. C*-ALGEBRAS AS ALGEBRAS OF EQUIVARIANT FUNCTIONS

Throughout this section, A will be a C*-algebra and H

will be a Hilbert space of 'sufficiently large dimension', by

which we mean that the dimension of H exceeds the dimension of

every cyclic representation of A and is infinite. (We need H

large enough so that a unitary equivalence of cyclic

representations of A on H can always be implemented by a unitary

on H.) We let Rep(A) denote the space of *-homomorphisms from A

into B(H), with the topology of pointwise strong convergence.

(This is known to be equivalent to pointwise weak, j-weak,

a-strong, or cr-*-strong convergence, cf. [17]).

We will be working with certain subspaces of Rep(A) with

the induced topology. If n is in Rep(A), let H(TT) be the

essential subspace of ix, i.e.

H(7T)

=

(TT(A)H)"~.

We will let

Repc(A) denote those n in Rep(A) which admit a cyclic vector when

7i(A) is restricted to the essential subspace H(JI). Similarly,

Irre(A) will denote those non-zero n in Rep(A) such that n(A) is

irreducible when restricted to H(TT)? clearly Irre(A) is contained

in Repc(A). We will let Irre(A)~ denote the closure of Irre(A) in

Repc(A) (not in Rep(A)).

For n in Rep(A), we let p(n) denote the projection on the

subspace

H(TT).

If X is a subset of Rep(A), a map y from X into

1