The translation invariant uniform approximation property for compact groups
We show that, unlike the abelian case, the translation invariant uniform approximation property fails in a strong way for l) of a compact connected semisimple Lie group. 1. We recall (see [5, 4 and 1]) that a Banach space X is said to have the uniform approximation property if there are k s* 1 and a sequence q(n) of positive numbers such that for any w-dimensional subspace E GX there exists an operator T: X -» X for which Tx — x for x G E, \\T\\ < k, dim TX < q(m). As an important example we recall (see [4]) that the reflexive Orlicz spaces have the uniform approximation property. The purpose of this paper is to show that the situation is strongly different when we consider a Banach function space on a compact group and the translation invariant analogue of the uniform approximation property, i.e. if we modify the above definition assuming that X, E and T are translation invariant. This notion was introduced by Bozeiko and Pelcziñski in [1], where the case of a compact abelian group was considered; they proved the theorem below and they obtained, as a consequence, a general result on the translation invariant analogue of the uniform approximation property for translation invariant function Banach spaces on G. Theorem 1 (BoZeiko and Pelcziñski). Let G be a compact abelian group with dual group T. For every k > 1 there exists a positive sequence qk( n ) such that for every finitesetM C Y there exists a trigonometric polynomial g such that g\M = 1, llgll) < k, card(supp(g)) < ^(card M). The above theorem is no longer true for arbitrary compact groups. We shall prove the following Theorem 2. Let G be a compact connected semisimple Lie group with dual object G (a maximal set of pairwise inequivalent unitary irreducible representations of G). Let m > 1, let [Mh}^x be a sequence of pairwise disjoint subsets of G of cardinality at most m and let {Ph}TM=x be a sequence of trigonometric polynomials such that Ph(o) = Ia for all a G Mh and card(supp(PA)) < constm, then || Ph ||, -» oo (as h -* oo). The author wishes to thank P. M. Soardi for raising the question considered here. In the next section we shall fix the notation. Received by the editors August 4, 1981. 1980 Mathematics Subject Classification. Primary 43A15; Secondary 22E46. ©1982 American Mathematical Society 0O02-9939/82/0O0O-0223/$01.75 410 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use UNIFORM APPROXIMATION PROPERTY 411 2. Let G denote a compact connected semisimple Lie group with Lie algebra g and T a maximal torus with Lie algebra t. The complexification tc is a Cartan subalgebra of gc and we denote by A the set of roots of (gc, tc). We choose in A a system P of positive roots and the associated system of simple roots. We set ß = ^1aePa. The weights of G are ordered by letting ^-i =^A2 ifX2 — X, is a sum (possibly empty) of simple roots [6, p. 314]. We write Xx < X2 if X, < X2 and X, ¥= X2. If £ is a set of weights of G, we say that X is a minimal (maximal) weight in E if no weight y in E satisfies y < X (y > X). For every weight X the symmetric sum S(X) is defined on T by S(X)(t) — S^expu^w), where t = exp u (u G t) and the summation runs over all ¡u in the orbit of X under the action of the Weyl group W; we denote by S(X) the unique central continuous extension of S(X) to the whole of G. We recall that the dual object G may be identified with the semilattice 2 of the dominant weights of G; throughout this paper we shall use the same symbol to denote a representation in G and his highest weight in 2. Finally, for every X in G, x\ and dx denote the character and the dimension of X respectively, while Ix is the identity operator on the Hilbert space Hdx. 3. We need a lemma, whose proof is contained in [2]. Lemma. For any X in 2 the function S(X + 2ß) has the following Fourier expansion (on G): S(X + 2ß) = Xx+iß + 2 myxy + rxx, where t = ±1, the my are relative integers and X < y < X + 2ß for any y in the above summation. Proof of Theorem 2. We choose a maximal weight Xh in Mh and we write Fx = supp^X,, + 2ß)) n supp(PA); we have two cases: (¿)Fx = {Xh}, (W)Fx = {Xh,y¡,...,yir}(r>l). In the first case we obtain