The field of C∗-algebras over the interval [0,2] for which the fibers are

arXiv:funct-an/9212004v1 19 Dec 1992THE SOFT TORUS IIA Variational Analysis ofCommutator NormsRuy Exel†Abstract.The field of C∗-algebras over the interval [0,2] for which the fibers arethe Soft Tori is shown to be continuous. This result is applied to show that any pairof non-commuting unitary operators can be perturbed (in a weak sense) in such a wayto decrease the commutator norm.

Perturbations in norm are also considered and acharacterization is given for pairs of unitary operators which are local minimum pointsfor the commutator norm in the finite dimensional case.1. Introduction.As in [3], for every ε in the real interval [0,2] we let Aε be the universal unital C∗-algebra generated by unitary elements uε and vε subject to the relation||uεvε −vεuε|| ≤ε.Clearly, if ε1 ≤ε2 there is a unique homomorphism Aε2 −→Aε1 sending uε2 and vε2respectively to uε1 and vε1.

In case ε2 = 2 and ε1 = ε we shall denote this map by φε, i.e.φε : A2 −→AεOne of the main results of the present work (Theorem 3.4) is the fact that there existsa continuous field of C∗-algebras over the interval [0,2] such that Aε is the fiber over ε andmoreover such thatε ∈[0, 2] 7→φε(a) ∈Aεis a continuous section for every a in A2. We refer the reader to [2] for a treatment of thetheory of continuous fields of C∗-algebras.The central point in proving our main result is to show that ||φε(a)|| is a continuousfunction of ε for all a in A2.

If we letJε = Ker(φε)then we have that ||φε(a)|| = dist(a, Jε) and, since the Jε’s clearly form a decreasing chainof ideals, ||φε(a)|| is seen to be an increasing function of ε.Let us denote by J+εthe closure of the union of all Jε′ for ε′ > ε and by J−εtheintersection of all Jε′ for ε′ < ε. That isJ+ε =[ε′>εJε′† Partially supported by FAPESP, Brazil.

On leave from the University of S˜ao Paulo.1991 MR Subject Classification: 46L05, 49R20.1

andJ−ε =\ε′<εJε′.1.1 Proposition.Let ε be in [0,2) (resp. (0,2]).

A sufficient condition forε′ −→||φε′(a)||to be right (resp. left) continuous at ε for all a in A2 is that Jε = J+ε (resp.

Jε = J−ε ).Proof. Note that for a in A2 we havedist(a, J+ε ) = infε′>ε dist(a, Jε′)and thatdist(a, J−ε ) = supε′<εdist(a, Jε′).The first identity follows from trivial Banach space facts.

On the other hand, thesecond one cannot be generalized to Banach spaces so let’s prove it.Letφ : A2 −→Yε′<εAε′be given by φ(a) = (φε′(a))ε′<ε and observe that Ker(φ) = J−ε sodist(a, J−ε ) = ||φ(a)|| = supε′<ε||φε′(a)|| = supε′<εdist(a, Jε′).The conclusion now follows without much trouble.⊓⊔En passant, let’s give an example to support our statement that the above fact doesnot generalize to Banach spaces.Let E = C[0, 1] and for f in E put||f|| = |f(0)| + supt∈[0,1]|f(t)|.LetEn = {f ∈E : f([1/n, 1]) = 0}and let g be the constant function g = 1. Thendist(g, En) = 1for all n whiledist(g,\n≥1En) = ||g|| = 2.2

1.2 Proposition.For all ε ∈[0, 2) one has J+ε = Jε.Proof. If we denote by u+ε and v+ε the images of u2 and v2 in A2/J+ε then it is clear that||u+ε v+ε −v+ε u+ε || ≤εhence by the universal property of Aε there is a homomorphismAε = A2/Jε −→A2/J+εsending uε and vε to u+ε and v+ε .

Therefore Jε ⊆J+ε . But since the reverse inclusion istrivial, our proof is complete.⊓⊔The main technical result of this work, which shall be proven in the next two sections,is the following.1.3 Theorem.For all ε in (0, 2] one has J−ε = Jε.Therefore we have1.4 Corollary.For all a in A2 the functionε ∈[0, 2] −→||φε(a)||is continuous.2.

The Case ε < 2.Throughout this section we shall fix a real number ε with 0 < ε < 2.2.1 Lemma.Suppose u0, u1, ..., un is a finite sequence of unitary elements in a C∗-algebra B such that ||uk−1 −uk|| ≤ε for all k. Then, for every δ > 0, there are unitariesv0, v1, ..., vn in B such that(i) ||vk −uk|| ≤δfork = 0, ..., n(ii) ||vk−1 −vk|| < εfork = 1, ..., n.Proof. Since ||uk−1 −uk|| ≤ε < 2, it follows that ||uku−1k−1 −1|| < 2 so that −1 is not inthe spectrum of uku−1k−1.

Therefore we may definehk = log(uku−1k−1)where log is the principal branch of the logarithm. Each hk is then a skew adjoint elementin B and we haveuk = ehkuk−1fork = 1, 2, ..., n.3

We shall choose our sequence v0, ..., vn of the formv0 = u0andvk = etkhkuk−1fork ≥1where each tk will be a suitably chosen positive real number approaching 1 from below.A real function which will be useful in our estimates isd(x) = |1 −eix| = 2 sinx2forx ∈R.For instance, observe that if h is skew adjoint and ||h|| ≤π one has||1 −eh|| = d(||h||)by the spectral theorem. Moreover, for all k = 1, ..., nd(||hk||) = ||1 −ehk|| = ||uk−1 −uk|| ≤εwhich implies (since d is increasing in [0, π]) that||hk|| ≤θwhere θ = d−1(ε).

Note that θ < π because ε < 2.In search of the correct choice of the tk’s observe that||v0 −v1|| = ||u0 −et1h1u0|| = d(t1||h1||) ≤d(t1θ)and that for k ≥1 we have||vk −vk+1|| ≤||vk −uk|| + ||uk −vk+1|| =||etkhkuk−1 −ehkuk−1|| + ||uk −etk+1hk+1uk|| =d((1 −tk)||hk||) + d(tk+1||hk+1||) ≤d((1 −tk)θ) + d(tk+1θ).Note that d′(x) = cos(x/2) so for x ∈[0, θ] we have m ≤d′(x) ≤1 where m = cos(θ/2)is strictly positive since θ < π. By the mean value theorem we then havem|t −s| ≤|d(t) −d(s)| ≤|t −s|for all t and s in [0, θ].If this last fact is used in our previous computations, we obtain||v0 −v1|| ≤d(t1θ) = ε −(d(θ) −d(t1θ)) ≤4

ε −m(θ −t1θ) = ε −mθ(1 −t1)while for k ≥1||vk −vk+1|| ≤d((1 −tk)θ) + d(tk+1θ) ≤(1 −tk)θ + ε −(d(θ) −d(tk+1θ)) ≤(1 −tk)θ + ε −m(θ −tk+1θ) =ε + (1 −tk −m(1 −tk+1))θ.Therefore the condition that ||vk−1 −vk|| < ε will be fulfilled as long as1 −t1 > 0and1 −tk+1 > 1 −tkm.If we thus put tk = 1 −2kσ/mk for σ < (m/2)n we have that each tk is in (0,1) and||vk−1 −vk|| < ε.Clearly, as σ tends to zero, each vk approaches the corresponding uk so a suitablechoice for σ yields||vk −uk|| ≤δfor all k = 0, 1, ..., n.⊓⊔Recall from [3] that Aε is isomorphic to the crossed productAε ≃Bε ×τ Zwhere Bε is the universal C∗-algebra generated by a sequence {u(ε)n: n ∈Z} of unitariessatisfying the relations||u(ε)n −u(ε)n+1|| ≤εfor all n.Moreover τ is the automorphism of Bε given byτ(u(ε)n ) = u(ε)n+1forn ∈Z.2.2 Proposition.There exists a sequence (ψn)n∈N of endomorphisms of Bε, convergingpointwise to the identity map, such thatsupk∈Z||ψn(u(ε)k ) −ψn(u(ε)k+1)|| < ε.Proof. By the previous Lemma, let for every nv(n)−n, v(n)−n+1, ..., v(n)0 , v(n)1 , ..., v(n)n5

be unitaries in Bε such that||v(n)k−u(ε)k || ≤1nfork = −n, ..., nand||v(n)k−1 −v(n)k || < εfork = −n + 1, ..., n.Define ψn : Bε −→Bε byψn(u(ε)k ) =v(n)−nifk < −nv(n)kif−n ≤k ≤nv(n)nifk > n.It is then clear thatlimn→∞ψn(u(ε)k ) = u(ε)kwhich implies that ψn converges pointwise to the identity.The condition thatsupk∈Z||ψn(u(ε)k ) −ψn(u(ε)k+1)|| < εis also clearly satisfied.⊓⊔2.3 Definition.Let Kε be the ideal in B2 given by the kernel of the canonical mapφε : B2 −→Bε.2.4 Theorem.One hasTε′<ε Kε′ = Kε.Proof. Let x be inTε′<ε Kε′ and put y = ψε(x).

We havey = limn→∞ψn(y) = limn→∞ψn(φε(x)).Now letε′n = supk||ψnφε(u(2)k ) −ψnφε(u(2)k+1)|| = supk||ψn(u(ε)k ) −ψn(u(ε)k+1)||which by (2.2) is strictly less than ε. So ψnφε factors through Bε′n and since x is in Kε′nwe have ψnφε(x) = 0.

So y = 0 which implies that x is in Kε. The converse inclusion istrivial.⊓⊔2.5 Lemma.Let φ : A −→B be a C∗-algebra homomorphism and suppose φ isequivariant with respect to automorphisms α and β of A and B respectively.6

Let K = Ker(φ) and J = Ker(φ) where φ is the canonical extension of φ to thecorresponding crossed products by Z. ThenJ = {x ∈A ×α Z : EA(xu−n) ∈Kforn ∈Z}whereEA : A ×α Z −→Ais the associated conditional expectation [5] and u is the unitary implementing α.Proof. It is clear thatφ ◦EA = EB ◦φwhere EB is the conditional expectation forB ×β Z.Let v be the implementing unitary for B ×β Z.

Given x ∈A ×α Z we have that x isin J if and only if φ(x) = 0 which is equivalent to the fact that EB(φ(x)v−n) = 0 for alln, or that φ(EA(xu−n) = 0. But this is to say that EA(xu−n) is in K.⊓⊔2.6 Theorem.For every ε ∈(0, 2) one has J−ε = Jε.Proof.

Let E : A2 −→B2 be the conditional expectation induced by the isomorphismA2 ≃B2 ×τ Z.Given x in J−ε we have that E(xu−n) is in Kε′ for all n and ε′ < ε soE(xu−n) ∈\ε′<εKε′ = Kεfor all n, which shows that x ∈Jε. The converse inclusion is trivial.⊓⊔3.

The case ε = 2.The purpose of this section is to prove that J−2 = J2 or, since J2 = (0), that\ε<2Jε = (0).Note that the techniques employed in the previous section do not work here becauseone couldn’t take logarithms in the proof of (2.1) if ε = 2. A different approach is thusnecessary.3.1 Lemma.Let w1 and w2 be n × n unitary matrices.

Then ||w1 −w2|| = 2 if and onlyif det(w1 + w2) = 0.Proof. We have that ||w1 −w2|| = 2 if and only if ||w1w−12−1|| = 2 which is equivalentto −1 being in the spectrum of w1w−12which is to say that det(w1w−12+ 1) = 0 or thatdet(w1 + w2) = 0.⊓⊔7

3.2 Proposition.Given unitary n × n matrices u and v such that ||uv −vu|| = 2 thereis, for every δ > 0, a unitary u′ with||u′ −u|| < δand||u′v −vu′|| < 2.Proof. Write u = eh for some skew adjoint h and let u(t) = ue−th for all real t. Putf(t) = det (u(t)v + vu(t))and observe thatf(1) = det(2v) ̸= 0whilef(0) = det(uv + vu) = 0by (3.1).

Therefore f is not a constant function and since it is analytic, its zeros areisolated. So there are arbitrarily small values of t for which f(t) ̸= 0, which is to say, by(3.1) again, that||u(t)v −vu(t)|| < 2.Taking t sufficiently small will also ensure that ||u(t) −u|| < δ.⊓⊔3.3 Theorem.One has J−2 = J2.Proof.

Assume by way of contradiction that a ∈J−2 is non-zero.Note that A2 is isomorphic to the full C∗-algebra of the free group on two generators sothat by [1] A2 has a separating family of finite dimensional representations. Let thereforeπ : A2 −→Mn(C)be a representation such that π(a) ̸= 0.Let u = π(u2) and v = π(v2) and writeu = limi→∞u′iwhere ||u′iv −vu′i|| < 2 by (3.2).

For each i let πi be the representation of A2 such thatπi(u2) = u′iandπi(v2) = v.Then, if εi = ||u′iv −vu′i||, we have that πi vanishes on Jεi and thus also on J−2 soπi(a) = 0.On the other hand it is clear that πi converges pointwise to π so π(a) = 0 which is acontradiction.⊓⊔8

3.4 Theorem.There exists a continuous field of C∗-algebras over the interval [0,2] suchthat Aε is the fiber over ε and such thatε ∈[0, 2] 7→φε(a) ∈Aεis a continuous section for every a in A2.Proof. Let S be the set of sectionsε ∈[0, 2] 7→φε(a) ∈Aεfor a in A2.

According to [2] (Propositions 10.2.3 and 10.3.2) all one needs to check is thatS is a *-subalgebra of the algebra of all sections, that the set of all s(ε) as s runs throughS is dense in Aε and that ||s(ε)|| is continuous as a function of ε for all s in S.The first two properties are trivial while the last one follows from (1.1), (1.2), (2.6)and (3.3).⊓⊔4. Local Minima for Commutator Norms.As clearly indicated by the results obtained above, the phenomena under considerationis related to the following question4.1 Question.Given unitary operators u and v which do not commute, when is itpossible to perturb u and v in order to obtain a new pair u′ and v′ such that||u′v′ −v′u′|| < ||uv −vu||?In other words (4.1) calls for a characterization of pairs of unitary operators which arenot local minimum points for the commutator norm.

Proposition (3.2) is clearly a partialanswer to (4.1) and it says that when ||uv −vu|| = 2 in finite dimensions then (u, v) isnever such a local minimum point.In formulating the question above we chose not to specify the precise meaning of “toperturb” in order to allow for different points of view.The following is a complete answer to (4.1) under quite a loose type of perturbationwhich we might call *-strong dilated perturbation.4.2 Theorem.Let u and v be non commuting unitary operators on a Hilbert space H.Then there are nets (ui)i and (vi)i of unitary operators on H∞(the direct sum of infinitelymany copies of H) satisfying||uivi −viui|| < ||uv −vu||and such that the compressionsprojH ◦ui|HandprojH ◦vi|H9

of ui and vi to H converge *-strongly to u and v respectively.Proof. After Theorems (2.6) and (3.3) this basically becomes a consequence of [4] andsome well known results on representation theory.

We therefore restrict ourselves to asketch of the proof.Let ε = ||uv −vu|| and consider the set N of states on A2 which vanish on some Jε′for ε′ < ε. Since Tε′<ε Jε′ = Jε it can be proved that N is weakly dense in the set of statesof A2 that vanish on Jε.Given u and v let π be the representation of A2 on H such that π(u2) = u andπ(v2) = v. Assume without loss of generality that π is cyclic with cyclic vector ξ and putf : a ∈A2 −→⟨π(a)ξ, ξ⟩.Since π factors through Aε we have that f vanish on Jε so there exists a net (fi)i∈I in Nconverging weakly to f. Let, for every i, πi be the GNS representation of A2 correspondingto fi.

Since each fi vanish on some Jε′i with ε′i < ε the same is true for πi hence||πi(u2)πi(v2) −πi(v2)πi(u2)|| ≤ε′i < ε.Using the methods of [4] one may assume that the space Hi where πi acts is a subspaceof H∞and that the conclusion holds withui = πi(u2) + 1 −piandvi = πi(u2) + 1 −piwhere pi is the projection onto Hi.⊓⊔We therefore see that there are no local minimum points for the commutator normother than the commuting pairs, as long as we consider *-strong dilated perturbations.The situation is quite different if norm perturbations are considered as we shall see inthe next Section.5. Norm Perturbations in Finite Dimensions.Let us now study question (4.1) for pairs of unitary operators on a finite dimensionalHilbert space.

From now on we shall only consider norm perturbations.For n ≥3 denote by Ωn and Sn the n × n Voiculescu’s unitary matrices (see [6])Ωn =ωω2ω3...ωnandSn =011010...1010

where ω = e2πi/n.5.1 Theorem.For n ≥3 there exists a neighborhood V of the pair (Ωn, Sn) in U(n) ×U(n) so that||uv −vu|| ≥||ΩnSn −SnΩn||for all pairs (u, v) in V .Proof. Note that ΩnSnΩ−1n S−1n= ωIn so that if (u, v) is close enough to (Ωn, Sn) thenthe spectrum of uvu−1v−1 is in a small neighborhood of ω in the complex plane.On the other hand, notice thatdet(uvu−1v−1) = 1so that if the spectrum of uvu−1v−1 is the set {eiθ1, ..., eiθn} with −π < θi < π one hasthat Pnk=1 θk is in 2πZ.

So, by continuity,nXk=1θk = 2π.Therefore, for some k0 we must have θk0 ≥2π/n and it follows that||uv −vu|| ≥|eiθk0 −1| ≥|ω −1| = ||ΩnSn −SnΩn||.⊓⊔In other words, the pair (Ωn, Sn) is a local minimum for the commutator norm. Thisshows that the situation is quite different from what we saw when we considered *-strongdilated perturbations in Section (4).

This result should also be compared with [6].Clearly the method used in Theorem (5.1) above applies to show that the conclusionis also true for any pair of unitary matrices whose multiplicative commutator is a scalarmultiple of the identity, but not equal to −I (see 3.2).In fact, among irreducible pairs there are no other examples as we shall prove shortly.We say that a pair of unitary operators is irreducible when there is no proper invariantsubspace for both elements of the pair.Denote by γ the mapγ : (u, v) ∈U(n) × U(n) 7→uvu−1v−1 ∈SU(n).5.2 Lemma.A point (u, v) ∈U(n) × U(n) is regular for γ (in the sense that γ is asubmersion at (u, v)) if and only if (u, v) is an irreducible pair.Proof. If h, k are in the Lie algebra u(n) of U(n), a simple computation shows thatdγ(u,v)(uh, vk) = uv(v−1hv −h + k −u−1ku)u−1v−1.11

So γ is a submersion at (u, v) if and only if the mapL : (h, k) ∈u(n) × u(n) −→v−1hv −h + k −u−1ku ∈su(n)is onto the Lie algebra su(n) of SU(n).Under the inner product on su(n) defined by⟨x, y⟩= Trace(xy∗)the orthogonal space to the image of L can easily be seen to be the set{x ∈su(n) : xu = ux and xv = vx}.Now, by Schur’s lemma, irreducibility of (u, v) can be characterized by the fact thatonly scalars commute with both u and v. Since su(n) contains no scalar matrices the resultfollows.⊓⊔5.3 Theorem.If (u, v) is an irreducible pair in U(n) × U(n) and at the same time alocal minimum for the commutator norm then uvu−1v−1 is a scalar.Proof. By the open mapping theorem the image under γ of a neighborhood of (u, v) is aneighborhood of γ(u, v).

But since||uv −vu|| = ||γ(u, v) −1||it follows that γ(u, v) is a local minimum for the mapw ∈SU(n) −→||w −1||and this implies, as a moments thought will reveal, that γ(u, v) is a scalar.⊓⊔This completes the classification of local minima for irreducible pairs. So let us nowconsider a reducible pair (u, v) of unitary n × n matrices.

As usual writeu = ⊕ujandv = ⊕vjwhere each pair (uj, vj) is irreducible and observe that||uv −vu|| = maxj||ujvj −vjuj||.5.4 Theorem.Let u = ⊕uj and v = ⊕vj be as above and suppose that the pair (u, v)is a local minimum for the commutator norm. Then for some value of j for which||ujvj −vjuj|| = ||uv −uv||one has that ujvju−1j v−1jis a scalar.12

Proof.If this is not so then for all such j the pair (uj, vj) admits by (5.3) a smallperturbation decreasing the commutator norm.Together, these perturbations yield acontradiction to the hypothesis.⊓⊔A natural question which one could ask is, of course, whether the converse to Theorem(5.4) is also true. A good test case is given by the pair(Ωn ⊕Im, Sn ⊕Im)that is, the direct sum of Voiculescu’s unitaries with the m × m identity matrix.This pair clearly satisfies the conclusion of (5.4) and so it is natural to ask whetheror not it is a local minimum for the commutator norm.Despite strong favorable evidence given by some partial positive results and a largeamount of computer simulation supporting this thesis, we were unable to establish a prooffor this fact.

In fact we do not even know whether the above pair is a local minimum forn = 3 and m = 1. Nevertheless, we conjecture that5.5 Conjecture.The converse of (5.4) is also true.6.

An Example.Considering the apparent discrepancy between (4.2) and (5.1) it is perhaps interestingto see a concrete example of nets (ui)i and (vi)i, whose existence is guaranteed by (4.2),in case the given unitaries are taken to be Voiculescu’s unitaries, i.e. u = Ωn and v = Sn.For that purpose it is enough to find, for all δ > 0, unitary operators u′ and v′ on aseparable, infinite dimensional Hilbert space H, and an orthonormal set {ξk : k ∈Z/nZ}of vectors in H such that(i) ||u′v′ −v′u′|| < ||ΩnSn −SnΩn||(ii) ||u′(ξk) −ωkξk|| < δand(iii) ||v′(ξk) −ξk+1|| < δfor all k in Z/nZ where ω = e2πi/n.Let H = L2(S1) and let u′ be the unitary operator on H defined byu′(ξ)|z = zξ(z)forξ ∈H, z ∈S1and, for θ < 2π/n, let v′ be defined byv′(ξ)|z = ξ(e−iθz)forξ ∈H, z ∈S1.Let ξ0 be a unit vector in H represented by a function f on S1 supported in a neighbor-hood V of z = 1 which is small enough so that V is disjoint from eikθV for k = 1, 2, ..., n−1.For all such k let ξk = v′k(ξ0).

The reader may now check that (i), (ii) and (iii) hold aslong as θ is close to 2π/n and the diameter of V is small enough.13

References[1] M. D. Choi, The full C∗-algebra of the free group on two generators, Pacific J. Math.87(1980), 41-48. [2] J. Dixmier, C∗-Algebras, North Holland, 1982.

[3] R. Exel, The Soft Torus and applications to almost commuting matrices, Pacific J.Math, to appear. [4] J. M. G. Fell, C∗-algebras with smooth dual, Illinois J.

Math. 4(1960), 221-230.

[5] M. A. Rieffel, Induced representations of C∗-algebras, Advances in Math. 13(1974),176-257.

[6] D. Voiculescu, Asymptotically commuting finite rank unitary operators withoutcommuting approximants, Acta Sci. Math.

(Szeged) 45(1983), 429-431.Current address:Department of Mathematics and StatisticsUniversity of New MexicoAlbuquerque, NM 87131, USAe-mail: exel@math.unm.eduPermanent address:Departamento de Matem´atica,Universidade de S˜ao Paulo,Caixa Postal 20570,01498 S˜ao Paulo SP, Brasil14

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