Vol. 15, No. 4, 1965

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Group extension representations and the structure space

Robert James Blattner

Vol. 15 (1965), No. 4, 1101–1113

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 KS. For any two representations V 1 and V 2, (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 xkx1, 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 (UL,UL) is isomorphic to (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.

Mathematical Subject Classification
Primary: 22.60
Received: 30 September 1964
Published: 1 December 1965
Robert James Blattner