Let K be a locally compact
group. K^{∗} will denote the Jacobson structure space of C^{∗}(K), the group C^{∗}algebra
of K. For any unitary representation V of K on a Hilbert space, let E_{V } denote the
projection valued measure on the Borel sets of K^{∗} defined by Glimm (Pacific
J. Math. 12 (1962), 885–911; Theorem 1.9). A (not necessarily BoreI) subset S of
K^{∗} is called E_{V }thick if E_{V }(S_{1}) = 0 for every Borel S_{1} ⊆ K^{∗}∼ S. For any two
representations V _{1} and V _{2}, ℛ(V _{1},V _{2}) will denote the space of operators intertwining
V _{1} and V _{2}.
Suppose K is a closed normal subgroup of the locally compact group G. If V is a
representation of K and x ∈ G, V ^{x} is defined by V _{k}^{x} = V _{xkx−1}, k ∈ K.
If z ∈ K^{∗}, zx = Ker(V ^{x}), where V is any irreducible repesentation such
that z = Ker(V ). (By Ker we mean the kernel in the group C^{∗}algebra.)
This composition turns (K^{∗},G) into a topological transformation group
(Glimm, op. cit., Lemma 1.3). The present paper first shows that the stability
subgroups of G at points z ∈ K^{∗} are closed. Then the following two theorems are
proved:
Theorem 1. Let z ∈ K^{∗} and let H be the stability subgroup of G at z. Let L be a
representation of H such that {z} is E_{LK}thick. Then ℛ(U^{L},U^{L}) is isomorphic to
ℛ(L,L) and {z}G is E_{ULK}thick.
Theorem 2. Let M be a representation of G such that {z}G is E_{MK}thick for
some z ∈ K^{∗}. Let H be the stability subgroup of G at z. Suppose G∕H is σcompact.
Then there is a representation L of H such that {z} is E_{LK}thick and such that
M ≃ U^{L}.
In the above, U^{L} denotes the representation of G induced by L.
It is shown further that if C^{∗}(K)z contains an ideal isomorphic to the algebra of
all compact operators on some Hilbert space, then the representation LK of these
theorems is a multiple of the (essentially unique) irreducible representation L^{0} of K
such that Ker(L^{0}) = z. Finally, it is shown that if M is primary and if K^{∗}∕G
is almost Hausdorff (i.e., every nonvoid closed subset contains a nonvoid
relatively open Hausdorff subset), then M satisfies the hypothesis of Theorem
2.
