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