T heorem . A n LCA group G which is not a singleton is not anti-self dual.

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X IX (1976) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE X IX (1976)

A. O lubummo and M. B ajagopalan (Memphis) Anti-selî dual groups

Let G be a commutative topological group and G its dual (the group of all characters of G with compact open topology). G is called anti-self dual if the only continuous homomorphism of G into G is the constant homomorphism and conversely. The main theorem of Armacoast [1] is th a t each LCA group G is not anti-self dual unless G is a singleton. We give here an easy proof of this result.

T heorem . A n LCA group G which is not a singleton is not anti-self dual.

P ro o f. Let the dual of G be G. Let 0 denote the О-elements of G and G and also the constant homomorphisms between any two groups.

If G = Gx x R n with n Ф 0 and Gx LCA group, then G = Gxx B n and so the map 0 x id: Gx x B n->GX x B n is a non-zero continuous homomorphism of G into G. So we can assume th a t G contains a compact open subgroup H.

Let G

be connected component of 0 in G and L — G q c G. Let G

Ф ( 0 ).

Then A xQe G of infinite order and G0[L contains an element y

of oo order. So there is an isomorphism 99 : [#0]-*-[^0], where [a] denotes the cyclic group generated by 'a'. Since G

is divisible <p extends to a homo­

morphism (px: G0/L ->G0. S o 9^0 A: G ->G is a non-zero continuous homo­

morphism from G -+G, where A: G ->G/L is canonical map. So the problem reduces to the case when both G and G are totally disconnected. Let G have an element % of finite order n Ф 1 . Let H c G be the kernel of %.

Then there is an isomorphism T : (r/H -»[#]. So To A: G ->G is a non-zero continuous homomorphism, where A: G-^ G/H is the canonical map. So the problem reduces to the case when both G and G are totally disconnected and torsion free. Then G = J p x G x for some LCA group Gx and for some prime p, where J p denotes the LCA group of p-adic numbers (see Theorem 12 of [3] and Lemma 7 of [4]). Then û = J p x û x and so the map id xO:

J‘ rp x G x ^ А уз is a non-zero continuous homomorphism. This proves

the theorem.

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A. O l u b u m m o and M. R a j a g o p a l a n

References

[1] Arma Coast, Can an LCA group be anti-self-dual ?, Proc. Amer. Math. Soc. 27 (1971), p. 186-188.

[2] E. H ew itt and K. A. Ross, Abstract harmonie analysis, Vol. 1., Springer, Berlin 1963.

[3] M. R ajagop alan , Structure of monogenie groups, III. J. Math. 12 (1968), p.

205-214.

[4] — and T. S ou n d ararajan , Structure of self-dual torsion-free metric LCA groups, Fund. Math. (1969), p. 209-216.

[5] W. Rudin, Fourier analysis on groups, John Wiley, New York 1962.

DEPARTMENT OF MATHEMATICS UNIVERSITY OF IBADAN IBADAN (NIGERIA) and

DEPARTMENT OF MATHEMATICS MADURAI UNIVERSITY

MADURAI-2 (INDIA)

MEMPHIS STATE UNIVERSITY MEMPHIS (TENN).