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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 EV 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 EV -thick if EV (S1) = 0 for every Borel S1 ⊆ 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 kx = 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 EL|K-thick. Then ℛ(UL,UL) is isomorphic to
ℛ(L,L) and {z}G is EUL|K-thick.
Theorem 2. Let M be a representation of G such that {z}G is EM|K-thick for
some z ∈ K∗. Let H be the stability subgroup of G at z. Suppose G∕H is σ-compact.
Then there is a representation L of H such that {z} is EL|K-thick and such that
M ≃ UL.
In the above, UL 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 L|K of these
theorems is a multiple of the (essentially unique) irreducible representation L0 of K
such that Ker(L0) = 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.
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