MEANS OF UNITARY OPERATORS, REVISITED
UFFE HAAGERUP, RICHARD V. KADISON, and GERT K. PEDERSEN (Dedicated to the memory of Gert K. Pedersen by the first-named author, his pupil,
and the second-named author, his mentor, with admiration, affection, and respect)
Abstract
It is proved that an operator with bound not exceeding(n−2)n−1in aC∗-algebra is the mean of nunitay operators in that algebra.
1. Introduction
In [3], it is proved that ifA< 1− 2n, thenA= n1(U1+ · · · +Un), where Alies in aC∗-algebraᑛandU1, . . . , Unare in the unitary groupU(ᑛ)ofᑛ. The Russo-Dye theorem [6], “eachAin(ᑛ)1, the closed unit ball ({A:A ≤ 1, A∈ᑛ}) inᑛ, is the norm limit of convex combinations of unitary operators inᑛ,” is an immediate consequence of this much sharper result. The launch platform for the investigation in [3] was the observation by L. T. Gardner [1]
that
(∗) U(ᑛ)+(ᑛ)o1⊆U(ᑛ)+U(ᑛ),
where(ᑛ1)o= {A: A< 1, A∈ᑛ}(theopenunit ball ofᑛ). To see this, note that, withT in(ᑛ)o1and V in U(ᑛ), 12(V +T )= 12V (I +V∗T )and V∗T = T< 1. ThusI+V∗T, and hence 12(V +T )are invertible. So,
1
2(V+T )=UH, withUinU(ᑛ)andH ≥0 inᑛ. Now,H = UH ≤1, whenceH = 12(U1+U2), withU1andU2inU(ᑛ). ThusV+T =UU1+UU2, withUUj inU(ᑛ), and (∗) follows. Gardner proceeds from this observation to his short proof of the Russo-Dye theorem.
At a lecture, about Gardner’s proof, to the Operator Algebra Seminar in Copenhagen on 7 October 1983, the second-named author noted that a different departure from Gardner’s observation allowed one to conclude that eachT in (ᑛ)o1 is a finite, convex combination of elements in U(ᑛ), from which the Russo-Dye theorem is immediate. A few days of discussion after that lecture,
Received June 13, 2006.
led to the following argument from (∗) to the result in [3] noted at the beginning of this article. WithV inU(ᑛ)andT in(ᑛ)o1,
(∗∗)
V +(n−1)T =V +T +(n−2)T
=U1+V1+(n−2)T
=U1+U2+V2+(n−3)T
= · · · =U1+ · · · +Un−2+Vn−2+T
=U1+ · · · +Un−2+Un−1+Un,
with eachUj andVj in U(ᑛ). Ifn ≥ 3 andS ∈ (1− 2n)(ᑛ)01, then(n− 1)−1(nS−I) ≤(n−1)−1(nS+1) <1. ReplacingTby(n−1)−1(nS−I) andV byIin (∗∗), we have
nS =U1+ · · · +Un (Un∈U(ᑛ)).
As noted in [3],nis as good an estimate as possible of the least number of elements ofU(ᑛ)needed in a convex sum equal toT inᑛwhenT<1−2n, for with V a non-unitary isometry on a Hilbert space H, and 1− n−21 <
an < 1− 2n,anV has norman and is a mean ofnunitary operators onH but no fewer. There are a number of other topics discussed, results proved, and questions raised in [3]. Those questions are answered in a hail of further results by M. Rørdam in his brilliant [7]. One question, raised by C. Olsen and G. K. Pedersen in [4], remained unanswered: IsT inᑛa mean ofnelements ofU(ᑛ)whenT =1− 2n? Forᑛ a von Neumann algebra, this question is answered in the affirmative in [4]; indeed, the “unitary rank” of eachT in (ᑛ)1is determined as well in terms of Olsen’s index forT [5] and the distance ofT from the group of invertible elements inᑛ. For the generalC∗-algebra ᑛ, this question was daunting to many of us. There were partial results; for example, the first-named author answered the question affirmatively whenᑛ is commutative. (See Proposition 3.6 of [4].) The second-named author proved (unpublished notes) that if
z:|z| ≤1−2n
is not the spectrum ofT (that is, if a single point of this disk is missing from the spectrum ofT), ThenT inᑛis the mean ofnelements ofU(ᑛ)whenT =1− 2n. The argument was intricate.
It could be made much simpler using later results and techniques of Rørdam [7]. The full conjecture, however, remained elusive until the first-named author proved it [2] (at the end of 1987). That proof was quite involved. Pedersen, on receiving a copy of that proof, was able to simplify it considerably. The
“simplified” proof was still so complex that Pedersen remarked to the second- named author, that despite having “simplified it,” he still did not understand it.
When the Pedersen version reached the second-named author, it was simplified
and restructured further. It became “understandable,” well-motivated, but still not “simple.” This last version of the first-named author’s proof is the one we present in the next section. It remains attached to the same structure as the original argument of the first-named author.
2. The proof
We begin with some notation, in addition to the notation established in the preceding section. Throughout,ᑛis a unitalC∗-algebra,(ᑛ)+1 = {H : H ≥ 0, H ∈ (ᑛ)1}, and P = {UH : H ∈ (ᑛ)+1}. We denote by ‘sp(T )’ the spectrum ofT (inᑛrelative toᑛ). We prove the main theorem of this article in what follows.
Theorem. IfA∈ᑛandA ≤1− 2n (n= 3,4, . . .), thenA= 1n(U1+
· · · +Un)withU1, . . . , UninU(ᑛ). With the aid of the lemma that follows:
Lemma1. IfT ∈(ᑛ)1andH is in(ᑛ)+1, then T +2H =U +V +V∗ for someU andV inU(ᑛ), wheresp(U∗V )⊆
eiθ :−π2 ≤θ ≤π
, we can prove:
Lemma2. IfT ∈(ᑛ)1andS∈P, then T +2S=U +2R, whereU ∈U(ᑛ)andR ∈P.
With the aid of Lemma 2, we can prove our theorem. We prove the theorem from Lemma 2 first.
Proof of Theorem. Let B be n−n2A. ThenB ∈ (ᑛ)1. From Lemma 2, withSinP,
nA+2S=(n−2)B+2S=(n−3)B+B+2S=(n−3)B+U1+2S1
=U1+(n−4)B+B+2S1=U1+U2+(n−4)B+2S2
= · · · =U1+ · · · +Un−2+2Sn−2, where eachUj ∈U(ᑛ)and eachSj ∈P.
WhenT ∈ P,T = UH, withU inU(ᑛ)andH in(ᑛ)+1, whence 2T = UV+UV∗, whereV =H+i(I−H2)12 ∈U(ᑛ). Thus 2Sn−2=Un−1+Un, withUn−1andUninU(ᑛ), and
nA+2S=U1+U2+ · · · +Un.
As 0 ∈ P andS is an arbitrary element ofP, we may use 0 forS. Then A= 1n(U1+ · · · +Un).
Proof of Lemma2. SinceS ∈P,S=V H for someV inU(ᑛ)andH in(ᑛ)+1. From Lemma 1,
T +2S=V (V∗T +2H)=V (W+V0+V0∗) for someW andV0inU(ᑛ), where sp(W∗V0)⊆ C0andC0 =
eiθ : −π2 ≤ θ ≤π
. The functionf onC0, defined byf (eiθ)=e12iθ, is continuous. Thus f (W∗V0)is an elementU0in ᑛ,U02 = W∗V0, and sp(U0) lies in the right half-plane. ThusU0+U0∗=2K, whereK∈(ᑛ)+1 and
T +2S=V (W +V0+V0∗)=V W(I +W∗V0+W∗V0∗)
=V W(I +U02+W∗V0∗)=V WU0(U0∗+U0+U0∗W∗V0∗)
=V WU0(2K+U0∗W∗V0∗)=V V0∗+2V WU0K
=U +2R,
whereU =V V0∗∈U(ᑛ)andV WU0K=R∈P.
Proof of Lemma1. If we have foundUandV, thenT−U =V+V∗−2H, which is self-adjoint. Thus21i(U−U∗)must beB, whereT =A+iBwithA andBself-adjoint. DefineUto beB+iB, where the notationDwill be used to denote(I−D2)12, when−I ≤D≤I. ThenT+2H−U =A−B+2H = V +V∗. DefineV to beC+iC, whereC = 12(A−B+2H ). For this, we must show that−I ≤C ≤I. SinceA = 21(T +T∗)andB = 21i(T −T∗), we have that
A2+B2= 1
2(T T∗+T∗T )≤I,
sinceT ≤1 (so thatT T∗≤IandT∗T ≤I). ThusA2≤B2and|A| ≤B. In particular,A≤B, whenceA−B+2H ≤2H, andC ≤H ≤I. At the same time,C ≥ 12(A−B)≥ −I, sinceA−B ≤2 (forA ≤ T ≤1 andB ≤1).
To establish the spectrum condition onU∗V, we assume that−cosθ − isinθ(=λ) is in sp(U∗V ), where 0< θ < 12π. ThenU∗V −λI and, hence, V−λUare not invertible inᑛ. Some maximal left or right ideal inᑛcontains V −λU, so that 0=ρ(V −λU)for some (pure) stateρofᑛ. Now,
V −λU =C+cosθB−sinθB+i(C+cosθB+sinθB).
Sinceρis a state,
ρ(C+cosθB−sinθB)=0=ρ(C+cosθB+sinθB).
Thus ρ(cosθC−cosθsinθB+cos2θB)=0 ρ(sinθC+sinθcosθB+sin2θB)=0, and
0=ρ(cosθC+B+sinθC)=ρ(cosθ(C+B)+(1−cosθ)B+sinθC).
Note thatC+B = 12(A+B+2H )≥0, since−A≤ |A| ≤B, from our earlier observations. By assumption 0< θ < 12π, so that cosθ, 1−cosθ, and sinθ are positive numbers. AsC+B,B, andCare positive operators andρ is a state, we have that
ρ(C+B)=ρ(B)=ρ(C)=0, and 0=ρ(C2)=1−ρ(C2). Hence
0=ρ(CB)=ρ(BC)=ρ(B2)=ρ((C+B)2).
But then
0=ρ((C+B)2)=ρ(C2+CB+BC+B2)=ρ(C2)=1, a contradiction. Thusλ, of the form described, is not in sp(U∗V ).
REFERENCES
1. Gardner, L. T.,An elementary proof of the Russo-Dye theorem, Proc. Amer. Math. Soc. 90 (1984), 181.
2. Haagerup, U.,On convex combinations of unitary operators inC∗-algebras, inMappings of Operator Algebras, Prog. Math (1990), 1–13.
3. Kadison, R., and Pedersen, G. K.,Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249–266.
4. Olsen, C., and Pedersen, G. K.,Convex combinations of unitary operators, J. Funct. Anal. 66 (1986), 365–380.
5. Olsen, C.,Index theory in von Neumann algebras, Mem. Amer. Math. Soc. 47 (1984).
6. Russo, B., and Dye, H.,A note on unitary operators inC∗-algebras, Duke Math. J. 33 (1966), 413–416.
7. Rørdam, M.,Advances in the theory of unitary rank and regular approximation, Ann. of Math.
128 (1988), 153–172.
MATHEMATICS DEPARTMENT UNIVERSITY OF PENNSYLVANIA PHILADELPHIA, PA 19104-6395 USA
E-mail:kadison@math.upenn.edu
INSTITUT FOR MATEMATIK OG DATALOGI SYDDANSK UNIVERSITET
CAMPUSVEJ 55 DK-5230 ODENSE M DENMARK
E-mail:haagerup@imada.sdu.dk
