Peak Points for Algebras on Circled Sets

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Then L(F(0) = 5 + (t-s) Re(0 hes on the interval between 5 and t, so that F{QeK for (g D Since F is analytic and locally one-to-one, each point of the image of F lies on an analytic disc in К Choosing the parameter a so that F(l/2) = w, we find that w hes on an analytic disc in К Hence (111) implies (11)

That (1) implies (m) is trivial It suffices to estabhsh that (11) implies (1) For this, we need the following lemma

Lemma 1. Let К be a compact, connected, circled subset of (£" such that r{K) = K Suppose that К meets the /^ coordinate axis C^ Then (z^, ,^j-i, Az^, ^j+v ^^r,)^^ whenever (z^, ,zJeK and |Я|^1

Proof We may assume that j = l and that 0<|Я|<1 Choose q = {0,q2, ,q„)GKnC^ Let {KJ be a sequence of compact, connected, circled subsets of C" such that г{К^) = К^, K^DK^^^, and K^-^K Let yeK^^Q and let peK\C be near q Since K^ is circled and logarithmically convex, {\Pif\yi\'~\ , WW"0 belongs to K^ for 0<t<l Letting p-^q, then letting r^O, we find that (0,|);2l, Ay„\)eK, Smœ К^СК^.ЛЩ'МУА Ау„\)еК^.,(от s>0 sufficiently small Also (|3;J, ,\yJ)eK^_i Since K^_^ is logarithmically convex, (|A||};J^~^ \у21 АУп\)^^1-1 Letting 5->0, and then y-^z, we obtain (|Я| \z^I, IZ2I, , |z„|)gK^_ 1 Since this is true for all i, and since К is circled, we obtain (Azi,Z2, ,zJeK This proves the lemma

Now we return to the proof that (11) implies (1) The proof is by induction on the dimension of the convex set L{K\C) If its dimension is zero, then L(K\C) is a singleton, and К is a multitorus In this case, H(K) = C(K\ and the theorem is trivial

Suppose then that ЦК\С) is not a singleton Since L{w) is assumed to be an extreme point of L(K\CX there is a hnear functional \p on IR" such that \p is not constant on K\C, and xp attains its maximum value over K\C at L(w) Set

& = V;(L(w)) = max{ip(L(z)) zeK\C} Define

J , = {zeK\C xp{L{z)) = b},

so that L{Jq) is a closed proper face of L{K\C) Set

J = r(J^)

Then J IS a compact, connected, circled subset of К such that r{J) = J We aim to show that the induction hypothesis applies to J, and then deduce the theorem for ЩК) from that for H{J)

Let the nonzero vector ^еШР represent xp, so that

Let AT be a positive integer By Dinchlet's pigeon-hole principle [7, Chap 11], there exist an n-tuple ß = {ßi, ,iS„) of integers and an integer к such that IgfcgN", while