Jordan triple endomorphisms and isometries of unitary groups

JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF UNITARY GROUPS LAJOS MOLN´ AR Abstract. In this paper we present the general form of all continuous endomorphisms of the group Unof n×ncomplex unitary matrices with respect to the Jordan triple product. These are the continuous maps φ:Un→Unwhich satisfy φ(V W V ) = φ(V)φ(W)φ(V), V, W ∈Un. The result is applied to determine the structure of certain isometries of Un. These include the isometries relative to any metric given by a unitarily invariant norm on the space Mnof all n×ncomplex matrices and also the isometries relative to any member of a new class of metrics on Unrecently introduced by Chau, Li, Poon and Sze [6]. 1. Introduction and statement of the main results The famous Mazur-Ulam theorem states that every surjective isometry (i.e., surjective distance preserving mapping) between real normed spaces is automatically affine, it preserves the operation of convex combination. Motivated by this important result, in the paper [9] Hatori, Hirasawa, Miura and Moln´ar made attempts to generalize it for the noncommutative setting, especially for metric groups and for certain substructures of them. The authors managed to obtain results saying that under certain conditions, the surjective isometries of groups equipped with translation and inverse invariant metrics locally preserve an operation called inverted Jordan triple product. In some cases the local preservation of that operation can be shown to extend globally. These results demonstrate that in the considered cases the surjective isometries have a certain remarkable algebraic property, they are some sort of isomorphisms between the underlying groups. In [10] the results given in [9] were utilized to describe the structure of surjective isometries of the unitary group of an arbitrary complex Hilbert space relative to the metric induced by the usual operator norm. In [15] Moln´ar and ˇ Semrl determined the structure of surjective isometries of the unitary group of a complex infinite dimensional separable Hilbert space with respect to 2010 Mathematics Subject Classification. Primary: 15A60, 15A86. Secondary: 47B49. Key words and phrases. Unitary group, isometries, unitarily invariant norm, Jordan triple product. The author was supported by the ”Lend¨ulet” Program (LP2012-46/2012) of the Hungarian Academy of Sciences and by the Hungarian Scientific Research Fund (OTKA) Reg.No. K81166 NK81402. 1 2 LAJOS MOLN´ AR any unitarly invariant uniform norm on the full operator algebra over the underlying Hilbert space. By employing different analytical tools but using the same algebraic properties of surjective isometries between groups that were obtained in [9], Hatori and Moln´ar presented results in [11] on the structure of surjective isometries (relative to the usual norm) of unitary groups in C∗-algebras and in von Neumann algebras. An interesting recent result on the form of certain isometries of the special orthogonal group is to appear in [1]. In [15] the problem of describing the surjective isometries of the unitary group under unitarily invariant norms in the finite dimensional case was left as an open problem, see [15, 4. Remarks, examples, open problems]. One of the aims of this paper is to give a solution of that problem. On the other hand, below we determine the structure of all isometries of the unitary group Unrelative to a new class of metrics on Unthat has been introduced by Chau, Li, Poon and Sze very recently [6]. As can be suspected from the discussion above the isometries we are going to consider turn to be isomorphisms under a certain algebraic operation on Un. Indeed, this operation is the inverted Jordan triple product that we define by the formula V W −1V. Morphisms with respect to this product are very closely related to morphisms with respect to a much more common and important operation which is called the Jordan triple product. This is defined by the formula V WV . Transformations preserving this operation or alike operations are extensively investigated in ring theory and its applications. The second main aim of this paper is to obtain the full description of all continuous Jordan triple endomorphisms of the group Un. We begin with presenting the notation and definitions that we shall use throughout the paper. We denote by Mnthe space of all n×ncomplex matrices, by Hnthe space of all self-adjoint elements of Mnand by Unthe group of all unitary elements of Mn. A unitary matrix is called a symmetry if it is self-adjoint (it has eigenvalues ±1). It is well-known that every unitary matrix is the exponent of a skew-symmetric matrix, i.e., every U∈Uncan be written of the form U=eiH with some H∈Hn. In what follows k.k denotes the usual operator norm (or, in another words, spectral norm) on Mn(i.e., kAkis the square-root of the largest eigenvalue of the positive semi-definite matrix A∗A). If not specified otherwise, when we speak of metrical or topological properties related to Unwe always mean the metric induced by the norm k.k. In what follows Istands for the identity matrix, tr denotes the transpose of matrices, Tr is the usual trace functional, and refers to complex conjugate. Recall that a norm N(.) on Mnis called unitarily invariant if N(UAV ) = N(A) holds for all A∈Mn,U, V ∈Un. In what follows we assume that n≥2 (in the case n= 1 the results below follow from classical mathematical analysis). Our first main result which gives the complete description of Jordan triple endomorphisms of Unreads as follows. JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF UNITARY GROUPS 3 Theorem 1. Let φ:Un→Unbe a continuous map which is a Jordan triple endomorphism, i.e., assume that φsatisfies φ(V WV ) = φ(V)φ(W)φ(V), V, W ∈Un. Then there exist a unitary matrix U∈Un, an integer k, a number c∈ {−1,1}, and a set {P1, . . . , Pn}of mutually orthogonal rank-one projections in Mn, a set {k1, . . . , kn}of integers and a set {c1, . . . , cn} ⊂ {−1,1}such that φis of one of the following forms: (j1) φ(V) = c(det V)kUV U−1,V∈Un; (j2) φ(V) = c(det V)kUV −1U−1,V∈Un; (j3) φ(V) = c(det V)kUV trU−1,V∈Un; (j4) φ(V) = c(det V)kUV U−1,V∈Un; (j5) φ(V) = Pn j=1 cj(det V)kjPj,V∈Un. From the above result one can immediately deduce the structure of all continuous Jordan triple automorphisms of Un. Corollary 2. Let φ:Un→Unbe a continuous Jordan triple automorphism, i.e., a continuous bijective map which satisfies φ(V WV ) = φ(V)φ(W)φ(V), V, W ∈Un. Then there exist a unitary matrix U∈Unand a number c∈ {−1,1}such that φis of one of the following forms: (a1) φ(V) = cUV U−1, V ∈Un; (a2) φ(V) = cUV −1U−1, V ∈Un; (a3) φ(V) = cUV trU−1, V ∈Un; (a4) φ(V) = cUV U−1, V ∈Un. As our second main aim in this paper, in the next theorem we determine the structure of all isometries of the unitary group Unwith respect to any unitarily invariant norm given on Mn. Theorem 3. Let N(.)be a unitarily invariant norm on Mn. If φ:Un→Un is an isometry, i.e., φis a map which satisfies N(φ(V)−φ(W)) = N(V−W), V, W ∈Un, then there exists a pair U, U0∈Unof unitary matrices such that φis of one of the following forms: (i1) φ(V) = UV U0, V ∈Un; (i2) φ(V) = UV −1U0, V ∈Un; (i3) φ(V) = UV trU0, V ∈Un; (i4) φ(V) = UV U0, V ∈Un. In our fourth theorem we determine the isometries of Unwith respect to a recently defined class of interesting metrics on Un. Motivated by considerations in quantum information processing, in [5] Chau introduced a certain family of metrics on Un. In the paper [6] Chau, Li, Poon and Sze have extended this class significantly and presented a number of its interesting 4 LAJOS MOLN´ AR properties. It is a remarkable fact that starting from a very much different origin in [2], Antezana, Larotonda and Varela have been led practically to the same class of distances on Un. As for the definition of the metrics in question, we first remark the following. To any V∈Unthere corresponds a unique self-adjoint matrix H∈Hn with spectrum in ]−π, π] such that V= exp(iH). Indeed, Hcan be obtained in the following way. Applying an appropriate unitary similarity transformation, Vis transformed into a diagonal matrix. The diagonal elements of this matrix are complex numbers of modulus 1. For each such diagonal element take the corresponding unique angle that belongs to ] −π, π]. From the so obtained angles form the corresponding diagonal matrix and finally transform it with the inverse of the previously mentioned unitary similarity transformation. What we get is just the self-adjoint matrix Hthat we have been looking for. For temporary use, in this paper we call this self-adjoint matrix Hthe angular matrix of V. Now, given a unitarily invariant norm N(.) on Mn, for any pair V, W ∈Unof unitary matrices pick the angular matrix Hof V W−1and define dN(V, W) = N(H). It has been proven in [6] that dNis a metric on Unand several interesting properties of dNhave been derived. In our last result we determine the structure of the corresponding isometries of Un. Theorem 4. Let N(.)be a unitarily invariant norm on Mn. The structure of the isometries of Unwith respect to the metric dNdefined above is exactly the same as in Theorem 3. Remark 5.At this point let us remark the following. The results in the previous statements can all be reversed, meaning that all transformations of any of the forms which appear in the conclusions are in fact isometries, continuous Jordan triple automorphisms, and continuous Jordan triple endomorphisms, respectively. To verity these one needs to apply only simple observations. 2. Proofs In this section, after verifying some auxiliary results, we present the proofs of our main theorems. Our first lemma that follows states that the continuous Jordan triple endomorphisms of Unare all Lipschitz functions. The result could also be derived following the argument given in [13] (p. 177, Satz 1) relating to group endomorphisms of linear groups. For the sake of completeness below we present a more direct and simple proof in the case of Jordan triple endomorphisms of Un. Lemma 6. Let φ:Un→Unbe a continuous Jordan triple endomorphism. Assume φ(I) = I. Then φis a Lipschitz function. JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF UNITARY GROUPS 5 Proof. We begin with the following observation. For an arbitrary V∈Un let Hbe the angular matrix of Vand denote the operator norm kHkof H by m(V). Apparently, we have the inequalities kV−Ik ≤ m(V)≤2kV−Ik. Moreover, if m(V)< π and kis a positive integer such that k m(V)< π, then we have m(Vk) = k m(V). Turning to the proof of the lemma, we first assert that there exists a positive real number Lsuch that kφ(U)−Ik ≤ LkU−Ikholds for all U∈Un. Assume on the contrary that we have a sequence (Uk) in Unand a sequence (ck) of positive integers such that ck→ ∞ and (1) kφ(Uk)−Ik> ckkUk−Ik holds for every k∈N. Since Unis a compact metric space, (Uk) has a convergent subsequence. Without serious loss of generality we may and do assume that already the original sequence (Uk) is convergent. If its limit were different from I, by (1) we would have kφ(Uk)−Ik→∞which contradicts kφ(Uk)−Ik ≤ 2. Therefore, Uk→Iand φ(Uk)→Ias k→ ∞. We can also assume that kφ(Uk)−Ik=k, k<1/2 holds for all k∈N. Choose positive integers lksuch that 1/(lk+ 1) ≤k<1/lk. Clearly, lk≥2. Since k> ckkUk−Ik, we have kUk−Ik< k/ckand this implies that m(Uk)≤2kUk−Ik<(2k)/ck. On the other hand, we have 2klk ck <2 ck < π. Therefore, we infer m(Ulk k) = lkm(Uk)<2/ckwhich implies kUlk k−Ik ≤ m(Ulk k)<2/ck. Consequently, Ulk k→Iand since Uk→Ialso holds, we have Ulk+1 k→Ias k→ ∞. We continue with the inequalities m(φ(Uk)) ≤2kφ(Uk)−Ik= 2k and 2k(lk+ 1) <2(lk+ 1)/lk< π, where in the last inequality we have used lk≥2. These imply that m(φ(Uk)lk+1) = (lk+ 1)m(φ(Uk)). 6 LAJOS MOLN´ AR Hence we compute 1 = (1/k)kφ(Uk)−Ik ≤ (lk+ 1)kφ(Uk)−Ik ≤(lk+ 1)m(φ(Uk)) = m(φ(Uk)lk+1)≤2kφ(Ulk+1 k)−Ik. Consequently, φ(Ulk+1 k)6→ Iand this contradicts Ulk+1 k→I. Therefore, we have a positive real number Lsuch that kφ(U)−Ik ≤ LkU−Ikholds for every U∈Un. To complete the proof pick arbitrary unitaries W, W 0∈Un. We can choose V∈Unsuch that V2=W0and then find U∈Unsuch that V UV =W. We infer kφ(W)−φ(W0)k=kφ(V UV )−φ(V2)k=kφ(V)φ(U)φ(V)−φ(V)Iφ(V)k =kφ(U)−Ik ≤ LkU−Ik=LkV UV −V2k=LkW−W0k. This proves that φis a Lipschitz function.  In the next auxiliary result we show that every continuous Jordan triple endomorphism of Unwhich is unital (i.e., maps Ito I) gives rise to a linear transformation on Hn. The use of one-parameter groups in the proof that originates from [12] has already been exploited in the papers [11] and [1]. Lemma 7. Let φ:Un→Unbe a continuous Jordan triple endomorphism with φ(I) = I. Then there exists a linear transformation f:Hn→Hnsuch that φ(eitA) = eitf(A), t ∈R, A ∈Hn. Moreover, fsatisfies f(V AV ) = φ(V)f(A)φ(V) for every A∈Hnand symmetry V∈Un. Proof. Since φis a unital Jordan triple endomorphism, it is easy to check that φ(Vk) = φ(V)kholds for every V∈Unand positive integer k. We show that φpreserves the inverse operation. To prove this, let W∈Unbe such that W2=V. We compute φ(W)φ(V−1)φ(W) = φ(WV −1W) = φ(I) = I which implies that φ(V−1) = φ(W)−2=φ(W2)−1=φ(V)−1. Therefore, we obtain that φ(Vk) = φ(V)kholds for every integer kand V∈Un. In the rest of the paper we shall use several times that, in particular, φmaps symmetries to symmetries. In the next step, following an argument similar to the proof of Theorem 7 in [11] we show that φmaps one-parameter unitary groups to one-parameter unitary groups. Pick an arbitrary self-adjoint matrix T∈Hnand define ST:R→Unby ST(t) = φ(eitT ), t ∈R. JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF UNITARY GROUPS 7 We assert that STis a continuous one-parameter unitary group in Mn. Since φis continuous, we only need to prove that ST(t+t0) = ST(t)ST(t0) holds for every pair t, t0of real numbers. First select rational numbers rand r0 such that r=k mand r0=k0 m0with integers k, k0, m, m0. We compute ST(r+r0) = φ(eikm0+k0m mm0T) = φ(ei1 mm0T)km0+k0m =φ(ei1 mm0T)km0φ(ei1 mm0T)k0m=ST(r)ST(r0). Since φis continuous, we deduce that ST(t+t0) = ST(t)ST(t0) holds for every pair t, t0of real numbers. By Stone’s theorem we obtain that there exists a unique self-adjoint matrix f(T)∈Hn(the generator of the one-parameter unitary group ST) such that φ(eitT ) = ST(t) = eitf(T), t ∈R. We next prove that f:Hn→Hnis in fact a linear transformation. Pick A, B, C ∈Hn. We compute ei(t/2)AeitBei(t/2)A−eitC it =(ei(t/2)A−I)eitBei(t/2)A+ (eitB −I)ei(t/2)A+ (ei(t/2)A−I)−(eitC −I) it →A/2 + B+A/2−C=A+B−C as t→0. It follows that lim t→0 ei(t/2)AeitBei(t/2)A−eitC it = 0 ⇐⇒ C=A+B. If C=A+B, then using the Lipschitz property of φproven in Lemma 6 we have ei(t/2)f(A)eitf(B)ei(t/2)f(A)−eitf(C) it =φ(ei(t/2)A)φ(eitB)φ(ei(t/2)A))−φ(eitC) it =φ(ei(t/2)AeitBei(t/2)A))−φ(eitC) it →0 as t→0. On the other hand, just as above we deduce ei(t/2)f(A)eitf(B)ei(t/2)f(A)−eitf(C) it →f(A) + f(B)−f(C). This gives us that f(A) + f(B)−f(A+B) = 0, i.e., fis additive. The homogeneity of fis trivial to see. Indeed, we have eitλf(A)=φ(eitλA) = eitf(λA) for every t, λ ∈Rwhich implies λf(A) = f(λA). Consequently, fis a linear transformation on Hn. 8 LAJOS MOLN´ AR To obtain the last statement of the result we compute eitφ(V)f(A)φ(V)=φ(V)eitf(A)φ(V) =φ(V)φ(eitA)φ(V) = φ(V eitAV) = eitf(V AV ). Since this holds for every t∈Rwe easily get the desired equality f(V AV ) = φ(V)f(A)φ(V) for every A∈Hnand symmetry V∈Un. In what follows Tdenotes the circle group which is just the unitary group in the one-dimensional case. The next auxiliary result describes the structure of continuous Jordan triple functionals on Un. It can be viewed also as a characterization of the determinant function on the unitary group. Lemma 8. Let ϕ:Un→Tbe a continuous Jordan triple functional, i.e., assume that ϕis continuous and satisfies ϕ(V WV ) = ϕ(V)ϕ(W)ϕ(V), V, W ∈Un. Then there is an integer kand a number c∈ {−1,1}such that ϕ(V) = c(det V)k, V ∈Un. Proof. Clearly, ϕ(I)3=ϕ(I) implying that ϕ(I) = ±1. There is no loss of generality in assuming that ϕ(I) = 1. Since, by the transformation λ7→ diag(λ, 1,...,1), the group Tembeds trivially into Un, the functional ϕ: Un→Tgives rise to a continuous unital Jordan triple endomorphism of Un. Applying Lemma 7 to this transformation one can easily check that there is a linear functional l:Hn→Rsuch that ϕ(eitA) = eitl(A),t∈R, A ∈Hn. By the second statement in Lemma 7 we further have that l(V AV ) = l(A) holds for all A∈Hnand symmetry V∈Un. Proceeding further, since lis a linear functional on the real Hilbert space Hn, by Riesz representation theorem there is an element H∈Hnsuch that l(A) = Tr(AH), A∈Hn. Then Tr(AH) = l(A) = l(V AV ) = Tr(V AV H) = Tr(AV HV ) holds for every A∈Hnwhich implies that H=V HV for every symmetry V∈Hn. Multiplying by V, this gives us that Hcommutes with all symmetries in Un. Since any symmetry Vis of the form V= 2P−Iwith some projection, it follows that Hcommutes with every projection and we obtain that His necessarily a scalar multiple of the identity. Let h∈Rbe such that H=hI. We have ϕ(eitA) = eith Tr(A), t ∈R, A ∈Hn. Pick a rank-one projection P∈Mn. Since exp iπP is a symmetry, it follows that its image under ϕis a number that has square equal to 1. Therefore, exp(iπh Tr(P)) = exp(iπh) equals ±1 which yields that his an integer. Denote it by k. We compute ϕ(eitA) = eitk Tr(A)= (eTr(itA))k= (det(eitA))k which shows that ϕ(V) = (det V)kholds for every V∈Un. JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF UNITARY GROUPS 9 In the case where ϕ(I) = −1 we apply the above argument to the Jordan triple functional −ϕ. After these preliminaries we are now in a position to prove our first main theorem. The basic idea of the proof is the use of a structural result concerning commutativity preserving linear transformations of Hn. That result holds only when n≥3. The two-dimensional case requires some special considerations. Proof of Theorem 1. Let φ:Un→Unbe a continuous Jordan triple endomorphism. Clearly, we have φ(I) = φ(I)3which implies that I=φ(I)2, i.e., φ(I) is a symmetry. We have φ(V) = φ(IV I) = φ(I)φ(V)φ(I) for every V∈Un. Multiplying by φ(I) from either side we obtain that φ(I) commutes with every element φ(V) of the range of φ. Defining ψ(V) = φ(I)φ(V), the transformation ψ:Un→Unis a continuous map which is easily seen to be a Jordan triple endomorphism of Unwhich sends Ito I. In what follows we assume that already our original map φfixes the identity I. By Lemma 7 we have a linear transformation f:Hn→Hnsuch that (2) φ(eitA) = eitf(A), t ∈R, A ∈Hn. We prove that fpreserves commutativity meaning that if A, B ∈Hnare such that AB =BA, then f(A)f(B) = f(B)f(A) holds, too. Pick commuting matrices A, B ∈Hn. Then for every t, s ∈Rwe have eitAei2sBeitA =eisBei2tAeisB implying φ(eitA)φ(ei2sB)φ(eitA) = φ(eisB)φ(ei2tA)φ(eisB) and hence eitf(A)ei2sf(B)eitf(A)=eisf(B)ei2tf(A)eisf(B). Fixing the real variable sand putting the complex variable zin the place of it we have that the equality ezf(A)ei2sf(B)ezf(A)=eisf(B)e2zf(A)eisf(B) between matrix valued holomorphic (entire) functions of the variable zholds along the y-axis. By the uniqueness theorem of holomorphic functions we infer that the above equality holds necessarily on the whole complex plane. Next, fixing zand inserting the complex variable win the place of is, the same reasoning leads to that the equality ezf(A)e2wf(B)ezf(A)=ewf(B)e2zf(A)ewf(B) holds for all values of the variables z, w ∈C. In particular, for arbitrary real numbers t, s setting z=t/2, w =s/2 we have (3) petf(A)esf(B)petf(A)=pesf(B)etf(A)pesf(B). 16 LAJOS MOLN´ AR small. Therefore, Unis compact also relative to the metric dNand hence any isometry φ:Un→Unwith respect to the metric dNis necessarily surjective. Let us check that the conditions in Proposition 9 are satisfied. We first show that dNis translation and inverse invariant. Indeed, dN(UW, V W) = dN(U, V ), U, V, W ∈Un holds trivially and from the equality V−1U=V−1(UV −1)V we deduce that dN(V−1, U−1) = dN(U, V ). Therefore, dNis inverse invariant and, since it is right translation invariant, we obtain that it is left translation invariant, too. We have noted in the proof of the previous theorem that N(.) is a symmetric norm equivalent to k.k. Hence we have a positive scalar csuch that ck.k ≤ N(.)≤N(I)k.k. Set α=πc/4. Pick V, W ∈Unwith dN(V, W)< α. Let X∈Unbe such that dN(X, V ) = dN(X, WV −1W) = dN(V, W). Then we have dN(X, W)≤dN(X, V ) + dN(V, W)=2dN(V, W)<2α=πc/2. It follows that the angular matrix Hof WX−1=eiH satisfies N(H)< πc/2, which implies that kHk< π/2. From this we infer that the angular matrix of (WX−1W)X−1= (WX−1)2is just 2H. This yields dN(WX−1W, X)=2dN(W, X). These show that the conditions in Proposition 9 are fulfilled (with constant K= 2) and we conclude that φ(V W−1V) = φ(V)φ(W)−1φ(V) holds for any V, W ∈Unwith dN(V, W)< α. Next, choose a positive number βsuch that β < α/(2N(I)). Assume kV−Wk< β. Then kV W−1− Ik< β and it easily follows that the angular matrix Hof V W−1satisfies kHk<2β. Hence dN(V, W) = N(H)≤N(I)kHk< N(I)2β < α holds which further implies the equality (7) φ(V W−1V) = φ(V)φ(W)−1φ(V). Consequently, for any pair V, W ∈Unwith kV−Wk< β (i.e., for elements close enough relative to the usual metric) we have the above equality. The argument given in the first part of the proof of Theorem 8 in [10] is about showing that the above property which tells us that (7) holds locally in Unin fact implies that it holds also globally. We can employ that argument here too and obtain that φsatisfies (7) for all pairs V, W ∈Un. Since φis an isometry with respect to the metric dNwhich induces the same topology as k.k, it follows that φis continuous in the operator norm. Applying Proposition 10 we obtain the desired conclusion.  JORDAN TRIPLE ENDOMORPHISMS AND ISOMETRIES OF UNITARY GROUPS 17 Remark 11.We finish with the following open problem and remarks. In Theorem 1 we have described the Jordan triple endomorphisms of Un. A natural problem arises which asks for the structure of all continuous Jordan triple endomorphisms from Uninto another unitary group Um. Of course, if m≤n, our result can be applied (Umembeds into Un), but what happens if m>n? Recall that in our proof above we have heavily used the structure of commutativity preserving linear maps of Hnwhich statement is no longer valid between different spaces. 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