The canonical mapping on the product of a LCA group with its dual is shown to extend uniquely in a homomorphic and analytic way to the product of the respective complexifications. According to the Pontryagin-Van Kampen theory, locally compact Abelian groups exist in pairs. If (G, ) is one of these pairs, one can either view G as the character group of 9, or 9 the character group of G. For x E G and y E 9, we shall write the value of x at y (or the value of y at x, depending on one's point of view) as . Let r(G) be the group of all continuous, nonzero complex-valued, homomorphisms (relative to multiplication in C). Then, for each f in r(G), 0 M f(g ) and f is the (unique) product of a homomorphism If I I9 -R + and a character f/lIf . Thus G may be regarded as a direct summand of the group r(G), such that the complementary summand 11(G) is isomorphic with the group of continuous homomorphisms h I R R. The canonical map on G x 9 extends in a trivial way to maps (which we denote also by ) r(G)x C and GxFr(g)-C. When G is n-dimensional Euclidean space E, then 9 = E as well and = eiXY (Vx,y EE ) is the canonical mapping. Here r(G) may be identified with n-dimensional unitary space U (in which E is viewed as a real form) and the extension of the canonical map is = eizy (Vz E U, y E E). In this particular case, however, the extension proceeds even further to a map on r(G) x r(g), U x U 3 (a, b) eia b (where is the inner produce on U). It is the business of this paper to show that an analogous extension obtains for the general LCA group G. The principal tool employed in studying the extension is a space Z(G) of test functions on G. This space was introduced in [4] for the purpose of finding as large Received by the editors December 9, 1979 and, in revised form, May 9, 1980. 1980 Mathematics Subject Classification. Primary 43A40; Secondary 43A25. C 1981 American Mathematical Society 0002-9939/81 /0000-0468/$03.00 307 This content downloaded from 207.46.13.169 on Sat, 01 Oct 2016 04:25:10 UTC All use subject to http://about.jstor.org/terms 308 DAVID NOVAK AND KELLY MCKENNON a space of distributions as possible on which the Fourier transform was an isomorphism. It is of intrinsic interest itself and will be the subject of a subsequent paper. If a E r(G) and b E r(),we shall write a and b for the homomorphisms ql3 3y- , blG 3 x- ; we write lal and IbI for the elements in H(G) and Hl( ), respectively, such that lal 1 y -3 Jq(y)l and lbl IG i3 x -Ib(x)l. By convention, we use additive notation for the operations on the Abelian groups G and 9. 1. The space of test functions. Let W(G) be the set of all f e 21(G) such that f e 21(o). As is well known, W(G) is a dense ideal of the Banach (convolution) algebra 21(G). By Z(G), we shall mean the set of all functionsfi G -* C such that (i) bf E W(G) for all b e r(g) and (ii) a(bf)-E%(9) for all (a, b) e r(G) x r(o) (where ^ denotes the Fourier transformation). For complementary subgroups SI, S2 of G and functions f1ISI -* C and f2jS2 C, we definefC 0 f21 G -* C by fl 03 f2(SI + s2) = fl(sI) .f2(s2) (Vs1 + 52 E S1 3 = G). LEMMA 1. Let G be a direct sum SI e S2 of closed subgroups and suppose fj E Z(Sj) (j = 1, 2). Then we have f1 O f2 E Z(G). PROOF. We have 9 = SI E 52 where 91 and 52 are the annihilators of S2 and SI, respectively, [2, 23.18]. Furthermore, 5, and 52 are the duals of SI and S2, respectively. Recall that the Haar measure of a product is the product of the Haar measures [2, 15.29]. Thus (fl (0 f2)^= f1 0/ 2If a E r(G) and b e r(() are arbitrary, then a = a1 0 a2 and b = b I 0 b2 for aj e r(Sj) and bj E r(sj),j = 1, 2. We have b(f1 O f2) = b1 fi 0 b2f2, a(b(fi 0f2))^= a1(b1f1)'^0a2(b2f2)', and (a(b(fi 0gf2))^)Y = (q1(b 1f1)Y) ) (a2(b2f2) ) Thatf1 0 f2 is in Z(G) is now evident. Q.E.D. Ford E G andflG -* C, we define dfIG -* C by df(x) = f(x d) (Vx E G). THEOREM 1. There exists a minimal approximate identity for V1(G) in Z(G). PROOF. The structure theory [2, 24.30] provides a decomposition G = E D H in which E may be taken as n-dimensional Euclidean space and H a group with compact open subgroup K. Consequently 9 = & E 'K where 6_ {y E : = 1 (Vx E H)) This content downloaded from 207.46.13.169 on Sat, 01 Oct 2016 04:25:10 UTC All use subject to http://about.jstor.org/terms LOCALLY COMPACT ABELIAN GROUPS 309 and SC--{y E g: = 1 (Vx E E)). Furthermore [2, 23.18], S; and SC are the duals of E and H, respectively-thus & may also be taken as n-dimensional Euclidean space, and the annihilator i = {y E 'S: = 1 (Vx E K)) of K in 'S is a compact, open subgroup of 'S [2, 23.29]. The duality theory [2, 24.11] implies that 'S/$ is topologically isomorphic to the dual of K. The Banach algebra 21(K) admits a minimal approximate identity {ha) consisting of nonnegative, positive-definite functions in 9(K) such that the support of each ha is a finite set (1) ([3, 33.12] and [2, 23.17]). For each index a, let g IH -* R be equal to ha on K and vanish on every other coset of K in H. Let a E r(H) and b E r(9C) be arbitrary, and let a be an arbitrary index for the net { ga}. The support of bsa being a subset of the compact set K, evidently bg, E 2 (H). (2) Since K is compact, bIK is a character and therefore the Extension Theorem [2, 24.12] provides an element d of '1 such that dIK = NIK. For ally E 'JC we have ( gY)'(y) = b(x)g.(x) y(x) dx = "d(x) g(x)y(x) dx = Ra(y d). (3) From (1) follows that there exists a finite subset 13 of '1 such that the cosets y + i, y E C5, are pairwise disjoint and the support supp(ga,,f) of g, is a subset of 5 + i. Thus (3) implies that supp((bg,)^) c d + f + %. Since % is compact, so is d + Y + % and (bg,f) is in 21('X). This, with (2), yields b-g, E 9(H)(4) The compactness of the support of (bgJ)' also implies that a (pg.)' E 2 1 (5C). (5) Since al$ is a character on i, the Extension Theorem provides an element r of H such that rl$ = aIl. From (3) we have (where (x denotes the characteristic function of XforanyX c 9) (b9ga=d(ga) =d 9 y+) =d(ga) E (d+y)(($) YE y C-6 which yields a (bg)= =d(ga) E r(d + y) r (+)i (6) co (d +y)Since . is continuous and positive-definite [3, 32.43], it follows from (6) and [3, 32.10] that a(bga,) is in the Fourier-Stieltjes algebra B(G) of G. Thus, by a This content downloaded from 207.46.13.169 on Sat, 01 Oct 2016 04:25:10 UTC All use subject to http://about.jstor.org/terms 310 DAVID NOVAK AND KELLY MCKENNON corollary of Bochner's Theorem [3, 33.10], (a(bgj)Y is in VI(H). This, with (5), implies that a(bg,)^ is in W(1K). Hence, in view of (4), g, EE Z(H). (7) It is shown in [4, Lemma 4] that there exists a minimal approximate identity {v}) for 21(E) in 5Z(E). Thus, by Lemma 1, {v , 0 g,} is a minimal approximate identity for 21(G) in 5Z(G). Q.E.D. LEMMA 2. Let L be a complex regular Borel measure on G such that JG b djl pL < x for all b E H(g ), and let f be in 5Z(G). Then L * f is in Z(G) as well. PROOF. Let a E r(G) and b E r(g) be arbitrary. A direct calculation shows that b( * f) = * bJ (where bjL is the measure whose value at any Borel set B is fB b dfu). Since bjL is by hypothesis a bounded measure, and since bf is in 21(G) (f being in 5Z(G)), we have b( * f) = by * bf E 21(G). (8)