Operator Theory : Lin’s Theorem Stephen Hardy May 2012

The goal of this presentation is to sketch the proof of Lin’s Theorem, due to Friis and Rordam. Much of this introductory information was gleaned from a survey article by Davidson and Szarek [DS01]. Theorem 0.1 (Lin’s Theorem). For any ε > 0 there is a δ > 0 such that for any n ∈ N and any pair a, b ∈ Mn (C) of self-adjoint matrices such that ||a||, ||b|| ≤ 1 and ||ab − ba|| < δ

“a and b almost commute”

Then there exists a pair of matrices a0 , b0 ∈ Mn (C) which are self-adjoint, a0 and b0 commute, and ||a − a0 || + ||b − b0 || < ε

“a and b are close to commuting elements”

Note that it is an important part of the theorm that δ is independent of the dimension n of the matrices. This problem dates back to the 1960s or earlier. The earliest the basic question was posed in print was in [Ros69] “If A and B ‘almost commute’ must there exist ‘nearby’ matrices A0 and B 0 which commute? More precisely, does there exist a function f : R+ → R+ with limt→0+ f (t) = 0 such that whenever δ > 0 and A, B ∈ B(H) where H is finite dimensional complex vector space, such that ||AB − BA|| < δ then there exists A0 , B 0 which commute with ||A − A0 || < f (δ)

||B − B 0 || < f (δ)

The answer to the basic question is almost trivial, Luxemburg and Taylor for instance showed the result held so long as one bounds the possible norms of A and B in [LT70] via non-standard analysis. It is also easy to show via sequencial compactness: Proof. For sake of contradiction, suppose not. Then there is a ε > 0 such that for all k, there is some xk ∈ Mn with ||xk || ≤ 1 and || [xk , x∗k ] || < 1/k but for every normal y ∈ Mn , ||xk − y|| ≥ ε. Since Mn is finite dimensional, the unit ball of Mn is compact. Thus by considering a subsequence of the xk we can assume the xk converge to some x ∈ Mn . Then ||x|| ≤ 1 and for all k
|| [x, x∗ ] || ≤ 2 ||x|| + ||xk || ||xk − x|| + || [xk , x∗k ] || Since the quantity on the right tends to zero as k grows, x is normal. This is a contradiction since the xk get arbitrarily close to x. However, this still leaves open what sort of dependence there are between the ε and δ and the dimension n. Halmos raised interest in the problem in [Hal77]. Positive results were published Pearcy and Shields [PS79] when A was self-adjoint, B was arbitrary and ||A||, ||B|| ≤ 1, then one can take δ = 2ε2 /(n − 1) but this result is still dimension-dependent. 1

Berg and Olsen showed the answer os false for arbitrary operators on a Hilbert space due to Fredholm index [BO81]. The weighted shift on `2 with basis {ek : k ≥ 1} given by Sn ek =

min(k, n) ek+1 n

k≥1

Has self-commutator [Sn Sn∗ ] ≤ n1 Pn Where Pn is the projection onto span(e1 , . . . , en ) so || [Sn Sn∗ ] || ≤ 1/n → 0, but Sn is a compact perturbation of the unilateral shift S, so this operator is Fredholm of index −1 (and this properties are invarient under compact perturbations). It can be shown that if T is an operator with ||S1 − T || < 1 also has Fredholm index -1, so this is true for Sn . A normal operator N has Fredholm index dim(ker(N )) − dim(ker(N ∗ )) = 0, because ker(N ) = ker(N ∗ ) Thus Sn is at least distance 1 from a normal operator for all n (in fact this is sharp). The problem is known to be false for arbitary matrices: Choi presented a sequence of A, B ∈ Mn with ||A|| = 1 − 1/n, ||B| ≤ 1, ||AB − BA|| ≤ 2/n but ||A − R|| + ||B − S|| ≥ 1 − 1/n for all commuting R, S ∈ Mn [Cho88]. The problem is known to be false for unitary matrices: Voiculescu demonstrated gave a sequence of unitary matrices {un , vn } such that limn→∞ ||un vn − vn un || = 0 ) but un , vn is bounded away from commuting pairs of matrices [Voi83]. Exel and Loring showed that the crux of Voiculescu’s counterexample is K-theoretical, related to the second cohomology of the two-torus [EL89]. Some improvements to Pearcy and Shields’ result were published by Szarek [Sza90], but the δ was still dimension-dependent. Lin announced his paper in January 1995, although it was not published until 1996 [Lin96]. In the mean time, Friis and Rørdan drastically shortened the proof to 5 pages [FR96]. Lin’s Theorem is a corollary from the following theorem about approximately normal matrixes being close to normal matrices: Theorem 0.2. For every ε > 0, there is a δ > 0 such that for any n ∈ N, if x ∈ Mn (C) has ||x|| ≤ 1 and x is almost normal, i.e. ||xx∗ − x∗ x|| < δ Then there is a normal matrix x0 ∈ N which is close to x: ||x − x0 || < ε In other words, “almost normal implies close to normal”. Proof. To see how Lin’s theorem follows from this theorem, let ε > 0 be given, suppose a and b are self-adjoint matrices in MN (C) with , ||a||, ||b|| ≤ 1, with ||ab − ba|| < δ with δ as provided by the above theorem. Consider the element x = (a + ib)/2, then since 2||x|| = 2||a + ib|| ≤ ||a|| + ||b|| ≤ 2 Thus ||x|| ≤ 1. Then since a and b are self-adjoint 2x∗ = (a + ib)∗ = a∗ − ib∗ = a − ib Then we have 4||x∗ x − xx∗ || = ||(a − ib)(a + ib) − (a + ib)(a − ib)|| = || |a|2 + iab − iba + |b|2 − [ |a|2 − iab + iba + |b|2 ] || = ||2i(ab − ba)|| = 2||ab − ba|| < 2δ 2

Thus ||x∗ x − xx∗ || < δ/2 So by the above theorem, there is a normal matrix x0 ∈ Mn (C) with ||x − x0 || < ε. Then there is the unique decomposition of x0 = a0 +ib0 where a0 , b0 ∈ Mn (C) where a0 and b0 are self-adjoint. This comes from noting the functions z 7→ 0. Then there is a normal element y ∈ M/A with finite spectrum and ||x − y|| < ε. The idea: since sp(x) is a compact set, it is closed and bounded, so sp(x) ⊂ Γδ ∩ B(0, r) for some r > 0, so by applying the previous lemma finitely many times (with errors ε/4, ε/8, etc.) we can find a normal element x0 of M/A whose spectrum is a subset of sp(x) where the connected components Vi of sp(x0 )have diameter less than ε/2, and furthermore ||x − x0 || < ε/2. Then pick representatives λi for each connected component Vi , then there is a continuous retraction f : sp(x0 ) → {λ1 , . . . , λn } satisfying supz∈sp(x0 ) |f (z)−z| < ε/2. Then y = f (x0 ) is a normal element with spectrum sp(y) = sp(f (x0 )) = f (sp(x0 )) = {λ1 , . . . , λn } with ||y − x0 || = ||f − id ||∞ < ε/2, so ||y − x|| ≤ ||y − x0 || + ||x0 − x|| < ε as claimed. Putting the above three lemmas together, we have that Lemma 0.5. For any ε > 0, if x is a normal element of M/A, then there is a normal element y in M/A with finite spectrum and ||x − y|| < ε. This is refered to as the FN property, that normal elements can be well-approximated by normal elements with finite spectrum. Now we are ready to prove Lin’s theorem, or rather the theorem about approximately normal matrices being close to normal matrices which implies Lin’s theorem: Proof. Suppose Lin’s theorem is false, then there would be a sequence {nj } of natural numbers and a sequence {xj } of matrices xj ∈ Mnj (C) with ||xj || ≤ 1 such that ||x∗j xj − xj x∗j || → 0

as j → ∞

but there is a ε0 > 0 such that for all j ∈ N, xj is distance at least ε0 from any normal matrix in Mnj (C). Let M and A be the C∗ -algebras of bounded sequences and null sequences corresponding 4

to the sequence {nj } as defined above, and π : M  M/A be the canonical surjective projection. Then let x = {xj } ∈ M and y = π(x). Then y is a normal element of M/A, since x∗ x − xx∗ = {x∗j xj − xj x∗j } ∈ A

since ||x∗j xj − xj x∗j || → 0

as j → ∞

Then by our above lemmas, there is a normal elemenet y 0 ∈ M/A with finite spectrum and with ||y −y 0 || < ε0 /4. We claim there is a normal element x0 = {x0j } ∈ M with π(x0 ) = y 0 . The existance of an x0 with π(x0 ) = y 0 is trivial, of course, but the fact there is a normal preimage is key. Claim: there are polynomials p(z), q(z) ∈ C[z] with p(sp(y 0 )) ⊆ R and q(p(λ)) = λ for all λ ∈ sp(y 0 ). Idea: use something like Newton polynomials: if the elements of sp(y 0 ) = {λ1 , . . . , λn } where the λi are distinct, let p(z) =

n X i=1

z − zi + i Y Q (z − zj ) j6=i (zi − zj ) j6=i

Then p(zi ) = i for 1 ≤ i ≤ n and p(sp(zi )) ⊂ R. Similarly, let q(z) =

n X z − i + zi Y Q (z − j) j6=i (i − j) i=1

j6=i

Then q(i) = zi so q(p(λi )) = λi for 1 ≤ i ≤ n as required. Thus we know that q(p(y)) = y. Now it is clear that p and q are continuous everywhere, so in particular, since y 0 is normal and p is continuous on sp(y 0 ), p(y 0 ) is defined, and since sp(p(y 0 )) = p(sp(y 0 )) ⊂ R, p(y 0 ) is self-adjoint. Let z be such that π(z 0 ) = p(y 0 ). Then consider (z ∗ − z)/2, this elemenent is clearly self-adjoint. Since π is a ∗ -homomorphism, and p(y 0 ) is self-adjoint π(z ∗ ) = π(z)∗ = p(y 0 )∗ = p(y 0 ) Thus

 π

z∗ − z 2

 =

π(z ∗ ) − π(z) π(z)∗ − π(z) (p(y 0 ))∗ − p(y 0 ) = = = p(y 0 ) 2 2 2

Note that q is a continuous function on sp(z ∗ + z/2),   ∗ z +z 0 x =q 2 is defined, and furthermore since q is a polynomial it commutes with the ∗ -homomorphism π, and thus   ∗    ∗  z +z z +z 0 π(x ) = π q =q π = q(p(y 0 )) = y 0 2 2 Similarly,   ∗   ∗   ∗    ∗ z +z ∗ z +z z +z z +z ∗ (x ) x − x (x ) = q q −q q 2 2 2 2  ∗ ∗   ∗   ∗   ∗   z +z z +z z +z z +z ∗ = q¯ q −q q¯ 2 2 2 2  ∗   ∗   ∗   ∗  z +z z +z z +z z +z = q¯ q −q q¯ 2 2 2 2 =0 0 ∗ 0

0

0 ∗

5

Now by the definitition of the norm on the quotient topology, we have ||π(x) − π(x0 )||M/A = inf ||x − x0 − a|| = ||y − y 0 || < ε0 /4 a∈A

Thus there is an a ∈ A such that ||x − x0 − a|| < ||y − y 0 || + ε0 /4 < ε0 /2 Then since a = {aj } has ||aj || → ∞, there is some j ∈ N such that ||aj || < ε0 /2, so ||xj − x0j − aj || < ε0 /2 so ||xj − x0j || ≤ ||xj − x0j − aj || + ||aj || < ε0 So xj is within ε0 of the normal element x0j , a contradiction of our choice of the xj . This result is related to the question of whether normal elements in ultraproducts have representing sequences of normal elements. Friis and Rordam also give some generalizations of this result. More recent work by Hastings has studied how δ should be related to ε and how to construct nearby commuting elements [Has09]. Lin’s problem for the Hilbert-Schmidt norms has been studied by Filonov and Kachkovskiy [FK10]. Loring has studied semiprojectivity which is closely related to these types of lifting problems [Lor98].

References [BD87]

I. David Berg and Kenneth R. Davidson. Almost commuting matrices and the BrownDouglas-Fillmore theorem. Bull. Amer. Math. Soc. (N.S.), 16(1):97–100, 1987.

[BD91]

I. David Berg and Kenneth R. Davidson. Almost commuting matrices and a quantitative version of the Brown-Douglas-Fillmore theorem. Acta Math., 166(1-2):121–161, 1991.

[BO81]

I. D. Berg and Catherine L. Olsen. A note on almost-commuting operators. Proc. Roy. Irish Acad. Sect. A, 81(1):43–47, 1981.

[Cho88] Man Duen Choi. Almost commuting matrices need not be nearly commuting. Proc. Amer. Math. Soc., 102(3):529–533, 1988. [DS01]

Kenneth R. Davidson and Stanislaw J. Szarek. Local operator theory, random matrices and Banach spaces. In Handbook of the geometry of Banach spaces, Vol. I, pages 317–366. North-Holland, Amsterdam, 2001.

[EL89]

Ruy Exel and Terry Loring. Almost commuting unitary matrices. Proc. Amer. Math. Soc., 106(4):913–915, 1989.

[FK10]

N. Filonov and I. Kachkovskiy. A Hilbert-Schmidt analog of Huaxin Lin’s Theorem. ArXiv e-prints, August 2010.

[FR96]

Peter Friis and Mikael Rørdam. Almost commuting self-adjoint matrices—a short proof of Huaxin Lin’s theorem. J. Reine Angew. Math., 479:121–131, 1996.

[Hal77]

P. R. Halmos. Some unsolved problems of unknown depth about operators on Hilbert space. Proc. Roy. Soc. Edinburgh Sect. A, 76(1):67–76, 1976/77. 6

[Has09] M. B. Hastings. Making almost commuting matrices commute. Comm. Math. Phys., 291(2):321–345, 2009. [HL97]

Don Hadwin and Terry A. Loring. Normal operators in C ∗ -algebras without nice approximants. Proc. Amer. Math. Soc., 125(1):159–161, 1997.

[HL08]

D. Hadwin and W. Li. A Note on Approximate Liftings. ArXiv e-prints, April 2008.

[KS14]

I. Kachkovskiy and Y. Safarov. On the distance to normal elements in $Cˆ*$-algebras of real rank zero. ArXiv e-prints, March 2014.

[Lin96]

Huaxin Lin. Approximation by normal elements with finite spectra in C ∗ -algebras of real rank zero. Pacific J. Math., 173(2):443–489, 1996.

[Lor97]

Terry A. Loring. Lifting solutions to perturbing problems in C ∗ -algebras, volume 8 of Fields Institute Monographs. American Mathematical Society, Providence, RI, 1997.

[Lor98]

Terry A. Loring. When matrices commute. Math. Scand., 82(2):305–319, 1998.

[LT70]

W. A. J. Luxemburg and R. F. Taylor. Almost commuting matrices are near commuting matrices. Nederl. Akad. Wetensch. Proc. Ser. A 73=Indag. Math., 32:96–98, 1970.

[PS79]

Carl Pearcy and Allen Shields. Almost commuting matrices. J. Funct. Anal., 33(3):332– 338, 1979.

[Ros69] Peter Rosenthal. Research Problems: Are Almost Commuting Matrices Near Commuting Matrices? Amer. Math. Monthly, 76(8):925–926, 1969. [Sza90]

Stanislaw J. Szarek. On almost commuting Hermitian operators. In Proceedings of the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987), volume 20, pages 581–589, 1990.

[Voi83]

Dan Voiculescu. Asymptotically commuting finite rank unitary operators without commuting approximants. Acta Sci. Math. (Szeged), 45(1-4):429–431, 1983.

[ZHC14] Y. Zhang, D. Hadwin, and Y. Chen. Lifting Normal Elements in Nonseparable Calkin Algebras. ArXiv e-prints, March 2014.

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