On the Characters and the Plancherel Formula of Nilpotent Groups ...

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340 GARNETT AND GLICKSBERG since [I], [2] the Q”, 1 < j < n, span (Re R(K,))l. Since s”s”p = P/.L for m < n, and smqj = 0 = smvjn for j > m, we have with the same cj; thus p uniquely determines a sequence {cj} satisfying (2.6) for all m. We have by (2.4), and thus Cj”=, Cjrlj is an element of A(K)‘- (since each qj is). But now Sn /J - f Cjr)j = ( 1 j=l j=l Sn/Jd - C Iz Cjvj" = 0 for all 71, while p - cj”=r cj?7j is carried by uj”=, aHj, SO we conclude that the measure itself vanishes, and ,U = Cy=, Ciqj E A(K By [3, 2.5 infra], R(K) = A(K). When E\aH,, lies in a peak interpolation set for R(K) (and {ri} is any homology basis of smooth curves in K”) we can obtain R(K) = A(K) once we know that for each p in (Re li(K))l, {& cjqj} has a bounded subsequence, where {cj> satisfies (2.6): for if v is a w* cluster point of such a subsequence (hence in A(K then Pv = Cy=, Cj7)j" = S"p since S" is w* continuous, so p - v, which necessarily is carried by u,” aHj , vanishes as before. For example, it suffices to know that for infinitely many n at most a quarter of the mass of zy-, cjqj is carried by uz+, aHj since then so Thus it is easy to conclude that if we form K by removing from the closed unit disc a sequence of disjoint discs clustering on the unit circle then R(K) = A(K) p rovided the radii tend to zero rapidly enough.
ALGEBRAS WITH THE SAME MULTIPLICATIVE MEASURES 341 1. AHERN, P. R. AND SARASON, D., On some hypodirichlet algebras of analytic func- tions. Am. 1. Math. (to be published). 2. GLICKSBERG, I., Dominant representing measures and rational approximation Trans. Am. Math. Sot. (to be published). 3. GLICKSBERG, I., The abstract F. and M. Riesz theorem. /. Functiod Anal. 1 (1967), 109-122. 4. HOFFMAN, K. AND ROW, H., On the extension of positive weak* continuous functionals. Duke Moth. J (to be published). 5. LUMER, G., Herglotz transformation and HP theory. Bull. Am. Math. Sot. 71 (1965), 725-730. 6. ZYCMUND, A., “Trigonometrical Series I” (2nd ed.) Cambridge University Press, London.
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JOURNAL OF FUNCTIONAL ANALYSIS 1, 2
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PLANCHEREL FORMULA OF NILPOTENT GRO
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and PLANCHEREL FORMULA OF NILPOTENT
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PLANCHERRL FORMULA OF NILPOTRNT GRO
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PLANCHERRL FORMULA OF NILPOTENT GRO
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JOURNAL OF FUNCTIONAL ANALYSIS 1, 2
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VECTEURS SePARATEURs DES ALGhBRES D
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VECTEURS SfiPARATEURS DES ALGkBRES
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VECTEURS SltPARATEURS DES ALGhBRBS
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Page 37 and 38: SIZES OF COMPACT SUBSETS OF HILBERT
Page 57 and 58: Thus where SIZES OF COMPACT SUBSETS
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Page 107 and 108: Condition 10.2. SCATTERING THEORY W
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