Extension Theorems of Hahn-Banach Type for Nonlinear Disjointly ...

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338 CONNES AND KKIlXEK then (S - &JnGN E 1 ~ a E iyAw, and if (H,s),,,k E B E .P, (B,,‘),,sk E 8’ E 99, then (B, n B,L’) E B n i?. p is given by P(J z- iftt #+L), (~~rr)rt&! EAES+ . (W, ~(1’) is a measure algebra. Every automorphism II: of (,Y, p) induces an automorphism c’lU of P bv ( cA,),EN E l,.~,JA, (AnLv t A- E .&P . (‘onsidcr now an ergodic automorphism T of (-I-, CL). L\‘e denote by .H
YIEASURE SPACE AUTOMORPIIISMS 339 We remark that for all a E gU , thefa EL,(S, CL) that is given b! fA m: weak*-lim xa, , 12 ‘W is T-invariant, and therefore a constant. It follows that and also that p*,(A) =: weak*-lim xa,, , (A,),& E a E ?g,, , n-iw P&J, = pi 44, MJEN EA-ES, 0 > L’ -I% 4X) = 1. (3) For R E .,Y[T], Rw leaves &Jo, invariant and we denote the restriction of P to S?‘,,, by R,,, . By (3), R, is a pw-preserving automorphism of gn, . (2.2) LEMMA. Let (An)nsN E a E dc9,, , and let B C AY. Then Proof. From (2), ,,(A) P(B) = fi; &% n B). ifez ~(-4n n B) -= ‘,i; j-x Xa,XE? 4-l = .c, CL&f) XB 4 ~~ ,A4 P(B). Q.I:.D. (2.3) I,EMMA. Let R E .N[T] be such that for some ff E .H(,., , a + 0, Then R,,, == I . R,i? = i?, B c a. (4) Proof. U:\;C’e assume that there is a c C 1 .--- d, f? +: 0 such that R,,,C C 1 - e, and we proceed to arrive at a contradiction. be such that Let (Av)nEv E a and let (C,),t.,N E 6 RC,< C S - C,; , REN. (5) By lemma (2.2) there exists a subsequence I PL(BJ ) MJn) ,4~1). 12 EN. (BnLtw represents an element B of 8, fi ~/- 0, and fi C A^. From (5) RB, C S ~ B,, , TlEiw. Hence R,.,B C 1 - B, contradicting (4) QED. (2.4) LEMMA. For R E .N[T], R2,,, I ;f and on.[~s if R E [T]. (2)
Page 1 and 2: Extension Theorems of Hahn-Banach T
Page 3 and 4: EXTENSIOK THEOREMS FOR FUNCTIOK.4LS
Page 5 and 6: EXTENSION THEOREMS FOR FL'NCTIONALS
Page 7 and 8: EXTENSION THEOREMS FOR FLJNCTIONALS
Page 9 and 10: EXTEIiSION THEOREMS FOR FUSCTIONALS
Page 11 and 12: EXTENSION THEOREMS FOR FUNCTIONALS
Page 13 and 14: Hence, by Proposition 4.6, EXTENSIO
Page 17 and 18: ESTENSION THEOREMS FOR FIVVCTIONALS
Page 19 and 20: EXTENSION THEORERIS FOR FUNCTIONALS
Page 21 and 22: EXTENSION TfIEOREMS FOR FUNCTIONALS
Page 23 and 24: EXTENSION THEOREMS FOR FUNCTIONAW 3
Page 27 and 28: EXTENSION THEOREMS FOR FUNC’l’I
Page 29 and 30: l\jow, let f be a function in F,/ ,
Page 33 and 34: KXTEXSION THEOREMS FOR FLWCTIOXA1.S
Page 35: MEASURE SPACE AI-TOiVIORPHIBRIS 337
Page 39 and 40: C‘hoosc a representative (F,,),,N
Page 41 and 42: MEASURE SPACY AUTOMORPIIIS~I~ 343 (
Page 43 and 44: MEASURE SPACE AYI’O~IORPHIS~ld 34
Page 45 and 46: MEASURE SPACE AUTOMORPHISMS 347 (3.
Page 47 and 48: such that and Pvoqf. We denote On C
Page 49 and 50: 4. ~WPLICATIONS \\‘c want to desc
Page 51 and 52: The Theta Transform and the Heisenb
Page 53 and 54: THE THETA TRANSFORRZ 355 The origin
Page 55 and 56: THE THETA TRANSFORM 357 It is conve
Page 57 and 58: THE THETA TRAXSFORM 359 zohew x1 T
Page 59 and 60: TIIE TfIEI‘A ‘lxANSFOK~1 361 Fo
Page 61 and 62: THE THETA TRANSFORhI 363 where J u,
Page 63 and 64: A DUNFORD-PETTIS THEOREM FOR L1/Hal
Page 65 and 66: A DlJNFORD-PETTIS THEOREM FOR L1/Hu
Page 67 and 68: the mapping A DVNFORD-PETTIS THEORE
Page 69 and 70: A DCNFORD-PETTIS THEOREM FOR LIIHu
Page 71 and 72: .t DCNFORD-PETTIS THEOREM FOR L’,
Page 73 and 74: A DUNFORD-PETTIS THEOREM FOR Ll/H=-
Page 75 and 76: Lebesgue’s theorem implies that A
Page 77 and 78: ,omivnr. OF FLW~~IONAI. ANALYSIS 24
Page 79 and 80: REPRESENTATIONS OF NILPOTENT LIE GR
Page 81 and 82: IIEPRESENTATIOKS OF NILPOTENT LIE G
Page 83 and 84: I
Page 85 and 86: REPRESENTATIONS OF NILPOTENT LIE GR
Page 87 and 88:
REPRESENTATIONS OF SII.PoTEST LIE C
Page 89 and 90:
REPRESENTATIONS OF NILPOTENT LIE GR
Page 91 and 92:
KWKESE~TATIOSS OF NILPOTENT LIE (:I
Page 93 and 94:
REPRESENTATIONS OF NILPOTENT LIE GR
Page 95 and 96:
JOURNAL OF FUNCTIONAL ANALYSIS 24,
Page 97 and 98:
TOPOLOGI('AI. ASPECTS 399 maps i,,
Page 99 and 100:
where TOPOLOGICAL ASPECTS 401 1,’
Page 101 and 102:
TOPOLOGICAL ASPECTS 403 PROPOSITION
Page 103 and 104:
TOPOLOGICAL ASPECTS 405 (iv) S is b
Page 105 and 106:
TOPOLOGICAL ASPECTS 407 ever!- o-co
Page 107 and 108:
TOPOLOGICAL ASPECTS 409 I,et a = (J
Page 109 and 110:
The polar decomposition of the map
Page 111 and 112:
TOPOLOGICAL ASPECTS 413 Proof. It i
Page 113 and 114:
TOPOLOGICAL ASPECTS 415 algebras a
Page 115 and 116:
TOPOLOGICAL ASPECTS 417 where M, <
Page 117 and 118:
‘I’OPOLOCICAL ASPECTS 419 RIore
Page 119 and 120:
By the same method: TOPOLOGICAL ASP
Page 121 and 122:
TOPOLOGICAL ASPECTS 423 Since f bel
Page 123 and 124:
TOPOLOGICAL ASPECTS 425 20. H. H. S
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