1. ABSTRAC T HARMONIC ANALYSIS: A SURVEY 7
and for w £ 5, the w-th component of as is the same as that of a. This allows us
to write
x{o)=nx^'x^5)=n
^(^)=n^^
ves ves v
because Xv{&v) 1 for all v S. This proves the lemma.
As a complementary statement to the above lemma we also have
LEMM A 2. Suppose that \v £ Homc(GVj
Gx)
is given for each v E S with Xv
trivial on Hv for almost all v. Then the product
V
defines a quasi-character in Homc(G,Cx).
Proof, x
is
clearly a homomorphism. To prove continuity we select a set S contain-
ing all v where Xv(Hv) ^ 1. Let s be the cardinality of 5. Given a neighborhood U
of 1 in C x we can select another neighborhood V of 1, possibly smaller, such that
Vs C U, i.e. any product of 5 elements of V is an element in U. For each v choose
a neighborhood Nv of 1 in Gv such that Xv(Nv) C V for all v G S and Nv Hv
for all v ^ S. Then clearly
X([[NV)CVSCU.
V
This proves the lemma.
We observe that a quasi-character x = 11^ Xv of G is a character if and only
if all the local components Xv are also characters. Now let H^ be the subgroup of
the dual group Gv consisting of the characters of Gv which are trivial on Hv. Since
Hv is compact by Pontrjagin duality, its dual group Hv = G/H^ is discrete and
hence H^ is open; similarly, since Hv is open and compact, Gv/Hv is discrete and
its dual group (GV/HV)A = H^ is compact.
We are now able to view the dual group of G as the restricted direct product
of the dual groups Gv with respect to the subgroups H^. In fact the product
decomposition \ EL Xv gives an algebraic isomorphism. We state this formally.
T H E O R E M 7. The factorization x Ylv Xv of a quasi-character gives an
algebraic and topological isomorphism
Hvmc([(Gv,T) ^ Jj'ffomc(Gt,,T),
V V
where the first product Y[v Gv denotes the restricted direct product of the {Gv} with
respect to the subgroups {Hv} and the second is the restricted direct product of the
groups {Homc(Gv,T)} with respect to the subgroups {H^}.
Proof. We need only verify that indeed we have a topological isomorphism. To
say that x Ylv Xv is close to 1 means that x is close to 1 for a large compact
subset B = Yiv By of G where Bv C Gv is compact and Bv = Hv for almost all
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