On the Characters and the Plancherel Formula of Nilpotent Groups ...

STRUCTURE SPACES AND EXTENSIONS OF C*-ALGEBRAS 377
function on Prim (B), we see that D, and 0s have disjoint neigh-
borhoods in Prim (B) and so Prim (B) is HausdorE
Conversely, suppose Prim (I?) is Hausdorff. Condition (1) is then
clear and (2) follows from Theorem 3.3. Now denote r-l(C,(O(A)) by
K, and consider K n C(C). This is easily seen to be a C*-algebra
of continuous functions on Prim (C) (the usual identifications being
in force). This algebra contains constants since
r(l) = lo(a) ~~,W)).
Prim (B) is Hausdortf, and so for any @ = S-l(Qn,), i = 1, 2,
L$ E Prim (C) there must be an element (m, c) E C(B) with (m, c) E DI
and (m,c)$ On,. Thus there is a c E C(C) such that c E L?, and
c 4 Q, . This property together with the properties of containing
constants and being a C*-algebra, imply in the usual way using Stone-
Weierstrass Theorem, that K n C(C) = C(C) and condition (3) holds.
This completes the proof.
RRFERRNCR.S
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