MEANS OF UNITARY OPERATORS, REVISITED

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1 MATH. SCAND. 100 (007), MEANS OF UNITARY OPERATORS, REVISITED UFFE HAAGERUP, RICHARD V. KADISON, ad GERT K. PEDERSEN (Dedicated to the memory of Gert K. Pederse by the first-amed author, his pupil, ad the secod-amed author, his metor, with admiratio, affectio, ad respect) Abstract It is proved that a operator with boud ot exceedig ( ) 1 i a C -algebra is the mea of uitay operators i that algebra. 1. Itroductio I [3], it is proved that if A < 1, the A = 1 (U 1 + +U ), where A lies i a C -algebra ad U 1,...,U are i the uitary group U( ) of. The Russo-Dye theorem [6], each A i ( ) 1, the closed uit ball ({A : A 1,A })i, is the orm limit of covex combiatios of uitary operators i, is a immediate cosequece of this much sharper result. The lauch platform for the ivestigatio i [3] was the observatio by L. T. Garder [1] that ( ) U( ) + ( ) o 1 U( ) + U( ), where ( 1 ) o ={A : A < 1, A } (the ope uit ball of ). To see this, ote that, with T i ( ) o 1 ad V i U( ), 1 (V + T) = 1 V(I + V T) ad V T = T < 1. Thus I + V T, ad hece 1 (V + T)are ivertible. So, 1 (V +T)= UH, with U i U( ) ad H 0i.Now, H = UH 1, whece H = 1 (U 1+U ), with U 1 ad U i U( ). Thus V +T = UU 1 +UU, with UU j i U( ), ad ( ) follows. Garder proceeds from this observatio to his short proof of the Russo-Dye theorem. At a lecture, about Garder s proof, to the Operator Algebra Semiar i Copehage o 7 October 1983, the secod-amed author oted that a differet departure from Garder s observatio allowed oe to coclude that each T i ( ) o 1 is a fiite, covex combiatio of elemets i U( ), from which the Russo-Dye theorem is immediate. A few days of discussio after that lecture, Received Jue 13, 006.

2 194 uffe haagerup, richard v. kadiso, ad gert k. pederse led to the followig argumet from ( ) to the result i [3] oted at the begiig of this article. With V i U( ) ad T i ( ) o 1, ( ) V + ( 1)T = V + T + ( )T = U 1 + V 1 + ( )T = U 1 + U + V + ( 3)T = =U 1 + +U + V + T = U 1 + +U + U 1 + U, with each U j ad V j i U( ). If 3 ad S (1 )( )0 1, the ( 1) 1 (S I) ( 1) 1 ( S +1) <1. Replacig T by ( 1) 1 (S I) ad V by I i ( ), we have S = U 1 + +U (U U( )). As oted i [3], is as good a estimate as possible of the least umber of elemets of U( ) eeded i a covex sum equal to T i whe T < 1, for with V a o-uitary isometry o a Hilbert space H, ad 1 1 < a < 1, a V has orm a ad is a mea of uitary operators o H but o fewer. There are a umber of other topics discussed, results proved, ad questios raised i [3]. Those questios are aswered i a hail of further results by M. Rørdam i his brilliat [7]. Oe questio, raised by C. Olse ad G. K. Pederse i [4], remaied uaswered: Is T i a mea of elemets of U( ) whe T =1?For a vo Neuma algebra, this questio is aswered i the affirmative i [4]; ideed, the uitary rak of each T i ( ) 1 is determied as well i terms of Olse s idex for T [5] ad the distace of T from the group of ivertible elemets i. For the geeral C -algebra, this questio was dautig to may of us. There were partial results; for example, the first-amed author aswered the questio affirmatively whe is commutative. (See Propositio 3.6 of [4].) The secod-amed author proved (upublished otes) that if { z : z 1 } is ot the spectrum of T (that is, if a sigle poit of this disk is missig from the spectrum of T ), The T i is the mea of elemets of U( ) whe T =1. The argumet was itricate. It could be made much simpler usig later results ad techiques of Rørdam [7]. The full cojecture, however, remaied elusive util the first-amed author proved it [] (at the ed of 1987). That proof was quite ivolved. Pederse, o receivig a copy of that proof, was able to simplify it cosiderably. The simplified proof was still so complex that Pederse remarked to the secodamed author, that despite havig simplified it, he still did ot uderstad it. Whe the Pederse versio reached the secod-amed author, it was simplified

3 meas of uitary operators, revisited 195 ad restructured further. It became uderstadable, well-motivated, but still ot simple. This last versio of the first-amed author s proof is the oe we preset i the ext sectio. It remais attached to the same structure as the origial argumet of the first-amed author.. The proof We begi with some otatio, i additio to the otatio established i the precedig sectio. Throughout, is a uital C -algebra, ( ) + 1 ={H : H 0,H ( ) 1 }, ad P = {UH : H ( ) + 1 }. We deote by sp(t ) the spectrum of T (i relative to ). We prove the mai theorem of this article i what follows. Theorem. If A ad A 1 ( = 3, 4,...), the A = 1 (U 1 + +U ) with U 1,...,U i U( ). With the aid of the lemma that follows: Lemma 1. If T ( ) 1 ad H is i ( ) + 1, the T + H = U + V + V for some U ad V i U( ), where sp(u V) { e iθ : π θ π}, we ca prove: Lemma. If T ( ) 1 ad S P, the T + S = U + R, where U U( ) ad R P. With the aid of Lemma, we ca prove our theorem. We prove the theorem from Lemma first. Proof of Theorem. Let B be A. The B ( ) 1. From Lemma, with S i P, A + S = ( )B + S = ( 3)B + B + S = ( 3)B + U 1 + S 1 = U 1 + ( 4)B + B + S 1 = U 1 + U + ( 4)B + S = =U 1 + +U + S, where each U j U( ) ad each S j P. Whe T P, T = UH, with U i U( ) ad H i ( ) + 1, whece T = UV + UV, where V = H + i(i H ) 1 U( ). Thus S = U 1 + U, with U 1 ad U i U( ), ad A + S = U 1 + U + +U.

4 196 uffe haagerup, richard v. kadiso, ad gert k. pederse As 0 P ad S is a arbitrary elemet of P, we may use 0 for S. The A = 1 (U 1 + +U ). Proof of Lemma. Sice S P, S = VH for some V i U( ) ad H i ( ) + 1. From Lemma 1, T + S = V(V T + H) = V(W + V 0 + V0 ) for some W ad V 0 i U( ), where sp(w V 0 ) C 0 ad C 0 = { e iθ : π θ π }. The fuctio f o C 0, defied by f(e iθ ) = e 1 iθ, is cotiuous. Thus f(w V 0 ) is a elemet U 0 i, U0 = W V 0, ad sp(u 0 ) lies i the right half-plae. Thus U 0 + U0 = K, where K ( )+ 1 ad T + S = V(W + V 0 + V0 ) = VW(I + W V 0 + W V0 ) = VW(I + U0 + W V0 ) = VWU 0(U0 + U 0 + U0 W V0 ) = VWU 0 (K + U0 W V0 ) = VV 0 + VWU 0K = U + R, where U = VV0 U( ) ad VWU 0K = R P. Proof of Lemma 1. If we have foud U ad V, the T U = V +V H, which is self-adjoit. Thus 1 i (U U ) must be B, where T = A + ib with A ad B self-adjoit. Defie U to be B + ib, where the otatio D will be used to deote (I D ) 1, whe I D I. The T +H U = A B +H = V + V. Defie V to be C + ic, where C = 1 (A B + H). For this, we must show that I C I. Sice A = 1 (T + T ) ad B = 1 i (T T ), we have that A + B = 1 (T T + T T) I, sice T 1 (so that TT I ad T T I). Thus A B ad A B. I particular, A B, whece A B + H H, ad C H I. At the same time, C 1 (A B ) I, sice A B (for A T 1 ad B 1). To establish the spectrum coditio o U V, we assume that cos θ i si θ (= λ)isisp(u V), where 0 <θ< 1 π. The U V λi ad, hece, V λu are ot ivertible i. Some maximal left or right ideal i cotais V λu, so that 0 = ρ(v λu) for some (pure) state ρ of. Now, V λu = C + cos θb si θb + i(c + cos θb + si θb ). Sice ρ is a state, ρ(c + cos θb si θb) = 0 = ρ(c + cos θb + si θb ).

5 Thus ad meas of uitary operators, revisited 197 ρ(cos θc cos θ si θb + cos θb ) = 0 ρ(si θc + si θ cos θb + si θb ) = 0, 0 = ρ(cos θc+ B + si θc ) = ρ(cos θ(c+ B ) + (1 cos θ)b + si θc ). Note that C + B = 1 (A + B + H) 0, sice A A B, from our earlier observatios. By assumptio 0 <θ< 1 π, so that cos θ,1 cos θ, ad si θ are positive umbers. As C + B, B, ad C are positive operators ad ρ is a state, we have that ad 0 = ρ(c ) = 1 ρ(c ). Hece But the ρ(c + B ) = ρ(b ) = ρ(c ) = 0, 0 = ρ(cb ) = ρ(b C) = ρ(b ) = ρ((c + B ) ). 0 = ρ((c + B ) ) = ρ(c + CB + B C + B ) = ρ(c ) = 1, a cotradictio. Thus λ, of the form described, is ot i sp(u V). REFERENCES 1. Garder, L. T., A elemetary proof of the Russo-Dye theorem, Proc. Amer. Math. Soc. 90 (1984), Haagerup, U., O covex combiatios of uitary operators i C -algebras,imappigs of Operator Algebras, Prog. Math (1990), Kadiso, R., ad Pederse, G. K., Meas ad covex combiatios of uitary operators, Math. Scad. 57 (1985), Olse, C., ad Pederse, G. K., Covex combiatios of uitary operators, J. Fuct. Aal. 66 (1986), Olse, C., Idex theory i vo Neuma algebras, Mem. Amer. Math. Soc. 47 (1984). 6. Russo, B., ad Dye, H., A ote o uitary operators i C -algebras, Duke Math. J. 33 (1966), Rørdam, M., Advaces i the theory of uitary rak ad regular approximatio, A. of Math. 18 (1988), MATHEMATICS DEPARTMENT UNIVERSITY OF PENNSYLVANIA PHILADELPHIA, PA USA kadiso@math.upe.edu INSTITUT FOR MATEMATIK OG DATALOGI SYDDANSK UNIVERSITET CAMPUSVEJ 55 DK-530 ODENSE M DENMARK haagerup@imada.sdu.dk