Banach Journal of Mathematical Analysis ISSN: (electronic)

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1 Banach J Math Anal 7 (2013), no 1, Banach Journal of Mathematical Analysis ISSN: (electronic) wwwemisde/journals/bjma/ GEOMETRY OF THE LEFT ACTION OF THE -SCHATTEN GROUPS MARÍA EUGENIA DI IORIO Y LUCERO Communicated by M Frank Abstract Let H be an infinite dimensional Hilbert sace, B (H) the - Schatten class of H and U (H) be the Banach-Lie grou of unitary oerators which are -Schatten erturbations of the identity Let A be a bounded selfadjoint oerator in H We show that O A := {UA : U U (H) is a smooth submanifold of the affine sace A + B (H) if only if A has closed range Furthermore, it is a homogeneous reductive sace of U (H) We introduce two metrics: one via the ambient Finsler metric induced as a submanifold of A + B (H), the other, by means of the quotient Finsler metric rovided by the homogeneous sace structure We show that O A is a comlete metric sace with the rectifiable distance of these metrics 1 Introduction Let H be an infinite-dimensional Hilbert sace; denote by B (H) the algebra of bounded linear oerators acting in H, by U (H) the set of unitary oerators in H and by B (H) the -Schatten class (1 < + ), that is B (H) := {A B (H) : tr ( A ) < where tr is the usual trace in B (H) If A B (H), define A = tr ( A ) Date: Received: 23 March 2012; Acceted: 20 June Mathematics Subject Classification Primary 47B10; Secondary 46T05, 57N20, 58B20 Key words and hrases Analytic submanifold, Finsler metric, Riemannian metric, Schatten oerator 73

2 74 ME DI IORIO Y LUCERO Throughout this aer, denotes the usual oerator norm following grou of oerators U (H) := {U U (H) : U I B (H) Consider the where I B (H) denotes the identity oerator This grou is an examle of what in the literature [8] is called a classical Banach-Lie grou It has differentiable structure when it is endowed with the metric U 1 U 2 (note that U 1 U 2 B (H)) For instance, the Banach-Lie algebra of U (H) is the (real) Banach sace B (H) ah := {X B (H) : X = X Hereafter, the subscrit h (resectively ah) will be used to denote the sets of hermitic (resectively antihermitic) oerators Fix a selfadjoint oerator A B (H) If A / B (H), then A and B (H) are linearly indeendent Denote by A + B (H) := {A + X : X B (H) Note that every element S A + B (H) has a unique decomosition S = A + X, X B (H) We shall endow this affine sace with the metric induced by the -norm: if S 1 = A + X 1 y S 2 = A + X 2, then S 1 S 2 := X 1 X 2 The main goal of this aer is the study of the set O A := {UA : U U (H) It is shown that it is a differential submanifold if only if A has closed range Therefore it is regarded as a metric sace with two natural metrics, obtained as length metrics (ie, comuted as infimum of length of curves inside O A ) Since any oerator U U (H) can be written as U = I + X, with X B (H), it can be seen that O A is contained in A+B (H) Therefore, O A will be considered with the toology induced by the -metric of A + B (H) Unitary orbits of oerators have been studied before For instance, Andruchow and Stojanoff [2] studied the geometry of the orbit {ubu : u U where b is a fixed element of a comlex unital C -algebra and U is the unitary grou of this C -algebra In [7] can be found a thorough study of the geometric structure of the sace Q of idemotent elements of a C -algebra In this aer, Corach, Porta and Recht studied, among other things, the unitary orbits induced by the inner automorhisms of the grou of elements which are unitary for the non-degenerate conjugate-bilinear symmetric form determined by a selfadjoint element of Q, on the sace of selfadjoints elements of Larotonda in [12] studied the geometry of unitary orbits in a Riemannian, infinite dimensional manifold modeled on the full-matrix algebra of Hilbert-Schmidt oerators In this articular framework, restricting the action to classical grous, certain results can be found in [6] by Carey, [5] by Bóna and [4] by Beltiţă Let us describe the contents of the sections Section 2 contains some reliminary results In section 3 we establish that the necessary and sufficient condition for O A to be a submanifold of A + B (H) is that the selfadjoint oerator A has closed range In section 4, we will endow O A with a Finsler metric given by the Banach quotient norm of the Lie algebra of U (H) by the Lie algebra of the isotroy grou of the action and study the case = 2, in articular, we show that

3 GEOMETRY OF THE LEFT ACTION OF THE -SCHATTEN GROUPS 75 for this, is a Riemannian metric In section 5, we endow O A with the Finsler metric rovided by the ambient norm of B (H) and show that it is comlete metric sace In section 6, we will consider de Finsler quotient metric introduced in section 4 and characterize the rectifiable distance induced by this metric We show this rectifiable distance coincides with the metric for the quotient toology of U (H)/G A, where G A is the isotroy grou We also show that O A is a comlete metric sace with the rectifiable distance given by the quotient metric 2 Preliminaries Let B ( ) H, H be the algebra of bounded linear oerators from H to H From now on, N(T ) and R(T ) will denote the nullsace and range of T, resectively, for every oerator T B ( ) H, H Let T B ( ) H, H be an oerator with closed range T will denote the Moore Penrose seudoinverse of T, ie, the unique bounded linear oerator from H to H such that: (1) T T = ( T T ), (2) T T = ( T T ), (3) T T T = T, (4) T T T = T This is an oerator defined by T (T x) = x if x N(T ) and T R(T ) = 0 Note that T T coincides with the orthogonal rojection onto N(T ) and T T with the orthogonal rojection onto R(T ) For further information about Moore Penrose seudoinverse, the reader is referred to Penrose s aer [14] or the book by Groetsch [10] If H 0 is a closed linear subsace of H, then H = H 0 H0 If X B (H), then X can be written as a 2 2 matrix with oerators entries, ( ) X 11 X X = 12 X 21 X 22 where X 11 B (H 0 ), X 12 B ( ) H0, H 0, X 21 B ( ) H 0, H0, X 22 B ( ) H0 Given any x, y H, the oerator x y on H will be defined by (x y)(z) = z, y x for all z H If x, y H and T B (H), then the following equalities are readily verified: (x y) = y x T (x y) = T (x) y (x y)t = x T (y) Given A B (H) and X B (H), denote by R A the ma given by: R A (X) = XA Since B (H) is two-sided ideal, R A : B (H) B (H)

4 76 ME DI IORIO Y LUCERO 3 Differentiable structure of O A In this section we study under what conditions the set O A = {UA : U U (H) A + B (H) is a differentiable (real analytic) submanifold of the affine sace A+B (H), where A is any fixed oerator in B (H) and A / B (H) The Banach-Lie grou U (H) acts on O A on the left, by means of U V A UV A for any U U (H) and V A O A This action induces the ma π A U (H) OA U UA This ma, regarded as a ma on A + B (H), is real analytic Its differential at the identity is the linear ma δ A B (H) ah B (H) X XA Here we have identified the Banach-Lie algebra of U (H) with the sace B (H) ah of antihermitian elements in B (H) We start with a lemma Lemma 31 For any A B (H) selfadjoint, the following assertions are equivalent: (1) R(A) is closed in H (2) R(R A ) is closed in B (H) (3) R(δ A ) is closed in B (H) Proof 1 2: Let (X n ) B (H) such that R A (X n ) Y in B (H) Since R(A) is closed, A is well defined and since X n A Y, Y N(A) = 0 Then if X := Y A, we have that R A (X) = Y A A = Y 2 1: Let (x n ) H such that Ax n y Let X n = y x n B (H) It is easy to see that R A (X n ) = X n A = y Ax n y y in B (H) Now, using the fact that R(R A ) is closed in B (H) and that A is selfadjoint, there exists X B (H) such that y y = AX Evaluating the last equality in y, we conclude that y R(A) 2 3: Let (X n ) B (H) ah such that δ A (X n ) = X n A Y in B (H) Since R(R A ) is closed in B (H), there is a Z B (H) such that Y = ZA (Note that Z is not necessarily anti-hermitic) If Y is reresented as a 2 2 oerator matrix relative to H = H 0 H0, then ( ) Z Y = 11 A 0 0 Z 21 A 0 0 where H 0 = (N(A)) = R(A) and A 0 = A H0 : H 0 H 0 Note that R(A) is closed because R(R A ) is closed in B (H) and it was roved that assertion 2 imlies assertion 1 Also note that Z 11 is anti-hermitic, since R(A) is closed and Z 11 is the limit of anti-hermitics oerators

5 GEOMETRY OF THE LEFT ACTION OF THE -SCHATTEN GROUPS 77 Then, if we consider ( Z 11 (Z X = 21 ) ) Z 21 0 we have that X B (H) ah and Y = XA = δ A (X), then R(δ A ) is closed in B (H) 3 2: Let us consider again the decomosition H = H 0 H0, where H 0 = (N(A)) = R(A) and A 0 = A H0 : H 0 H 0 Since R(δ A ) is closed in B (H), the set {X 0 A 0 : X 0 B (H 0 ) h is closed in B (H 0 ) Now, let B (H 0 ) h δ 0 {X0 A 0 : X 0 B (H 0 ) h X 0 X 0 A 0 Since R(δ 0 ) = {X 0 A 0 : X 0 B (H 0 ) h is closed in B (H 0 ) and A 0 is injective, by the inverse maing theorem, there is a constant C such that X 0 B(H 0 ) C X 0A 0 B(H 0 ) (31) for all X 0 B (H 0 ) h Let Y 0 B (H 0 ) Relacing X 0 by Y 0 B (H 0 ) h in (31), and using the fact that Y 0 B (H 0 ) = Y 0 B (H 0 ) and Y 0 A 0 B (H 0 ) = Y 0A 0 B (H 0 ), we see that Y 0 B(H 0 ) C Y 0A 0 B(H 0 ) Then R(R A0 ) is closed in B (H 0 ) It follows, using the fact that assertion 1 is equivalent to assertion 2, that R(R A ) is closed in B (H) For the next roosition, the following notation will be used Let x = U 0 A, π x will denote U (H) π x OA U Ux Note that π x is real analytic and it s differential at the identity is the linear ma δ x B (H) ah B (H) X Xx Proosition 32 Let A be a selfadjoint oerator with closed range Then π x has continuous local cross sections with uniform radius That is, for any x = U 0 A O A there is a continuous ma { σ x : V x = UA O A : UA U 0 A < 1 U 2 A (H) such that π x σ x = id Vx In articular, π x is a locally trivial fibre bundle

6 78 ME DI IORIO Y LUCERO Proof Let s denote P A = AA Since A is selfadjoint and R(A) is closed, P A is the orthogonal rojection onto R(A) Let φ : O A {UP A : U U (H) be the ma given by φ(ua) = UP A Note that φ is well defined and that, since U n P A UP A (U n U) A A, φ (U n A) φ (UA) 0 when U n A UA 0 Also note that given R > 0, φ (B OA ( A, R A Indeed, )) lies in V PA := { UP A : U U (H), UP A P A < R φ(ua) P A = (U I)AA UA A A < R Now, let U V PA and call Q 0 and Q the rojections Q 0 = P A y Q = UP A U Since Q Q 0 = UP A U UP A + UP A P A = UP A (P A U P A ) + UP A P A UP A P A U P A + UP A P A < 2R, if R 1, there is Z U 2 (H) such that ZQ 0 Z = Q, namely, ZP A Z = UP A U (See, for instance, [11]) Let W = (U Z) P A + Z Then W is unitary and W I B (H), namely, W U (H) Also note that W P A = UP A Let σ PA (UP A ) = W U (H) for any UP A V PA Then σ PA is well defined on V PA, is continuous and σ PA (UP A )P A = UP A Let σ = σ PA φ Then σ is a continuous { ma (since it is the comosition of continuous mas) defined in the set UA O A : UA A < and σ (UA) U (H) More over, π A (σ (UA)) = σ PA (UP A ) A = σ PA (UP A ) P A A = UP A A = UA R A Thus σ is a continuous local cross section for π A on a neighbourhood of A Finally, note that since σ can be translated with the action to any x = U 0 A O A, one obtains cross { sections defined on translated balls with the same radius Namely, calling V x := UA O A : UA U 0 A < one can define σ x in any UA V x by σ x (UA) := U 0 σ ( U 1 0 UA ) U 1 0 R A To establish the equivalence between the existence of the submanifold structure for O A A + B (H) and the closed R(A) condition, the following general result on homogeneous saces is useful It is contained in the aendix of the aer [15] by I Raeburn, and it is a consequence of the imlicit function theorem in Banach saces

7 GEOMETRY OF THE LEFT ACTION OF THE -SCHATTEN GROUPS 79 Lemma 33 Let G be a Banach-Lie grou acting smoothly on a Banach sace X For a fixed x 0 X, denote by π x0 : G X the smooth ma π x0 (g) := g x 0 Suose that (1) π x0 is an oen maing, when regarded as a ma from G onto the orbit {g x 0 : g G of x 0 (with the relative toology of X) (2) The differential d (π x0 ) 1 : (T G) 1 X slits: it s kernel and range are closed comlemented subsaces Then the orbit {g x 0 : g G is a smooth submanifold of X, and the ma π x0 : G {g x 0 : g G is a smooth submersion Theorem 34 The orbit O A is a real analytic submanifold of A + B (H) if and only if R(A) is closed in H In this case, the ma π x : U (H) O A is a real analytic submersion and O A is a homogeneous sace of U (H) Proof Suose first that R(A) is closed in H and use lemma 33 In this case, G = U (H) and X = A + B (H) Let x = U 0 A O A Since R(A) is closed in H, π x is oen because, by roosition 32, π x has continuous local cross sections In order to see that δ x has closed comlemented range and kernel, let s consider the ma E x given by: E x (Z) = 1 2 xx Zx 1 2 (x ) Z xx + ( 1 xx ) Zx (x ) Z ( 1 xx ) for any Z B (H) Note that since R(A) is closed, x is well defined and also note that E x (Z) is anti-hermitic, then E x : B (H) B (H) ah Note that δ x E x δ x = δ x This imlies that δ x E x and E x δ x are both idemotent oerators in the Banach Sace B (H), therefore their ranges and kernels are comlemented Since the range of δ x E x equals the range of δ x and the kernel of E x δ x equals the kernel of δ x, δ x slits Then, by lemma 33 and using the fact that, in this context, smooth means real analytic (the grou and the action are real analytic), O A is a real analytic submanifold of A + B (H) and π x is a real analytic submersion Conversely, if O A is a submanifold of A + B (H), then the tangent sace T A (O A ) = {XA : X = X B (H) is closed in B (H) Then, by lemma 31, the range of A is closed in H 4 Finsler and Riemannian metrics in O A Since O A is a homogeneous sace, it is natural to endow each the tangent sace with the quotient metric We will show that, in the case = 2, this metric is a Riemannian metric Fix x = U 0 A O A Since π x is a submersion, the tangent sace of O A at x is: T x (O A ) = R(δ x ) = { Zx : Z B (H) ah As noted before, it is a closed linear subsace of B 2 (H) ah

8 80 ME DI IORIO Y LUCERO Denote by G x the isotroy grou and by G x the Banach-Lie algebra of G x Let X T x (O A ), the Finsler quotient metric is usually defined (cf [1]) as: { X x = inf Y : Y B (H) ah, δ x (Y ) = X = inf { Z + W : Z G x, (41) where W B (H) ah, δ x (W ) = X Note that since π x is a surjective ma, such W exists Also note that this metric is invariant under the left action of the grou Namely, for each fixed U U (H), x O A and X T x (O A ), we have that X x = UX Ux Let us now study the case = 2 B 2 (H) is a two-sided ideal of B (H) and a Hilbert sace with the inner roduct Z, W 2 = tr (W Z) Note that the norm induced by this inner roduct coincides with the 2 reviously defined In the revious section, it was noted that E x δ x is an idemotent real linear oerator on B (H) ah (in articular, on B 2 (H) ah ) Let s call this oerator Q x Exlicitly, Q x : B 2 (H) ah B 2 (H) ah Q x (Z) = Zxx + xx Z xx Zxx Then, Q x is a real linear idemotent oerator of B 2 (H) ah with N(Q x ) = N(δ x ) and R (Q x ) = { Z B 2 (H) ah : (1 xx )Z(1 xx ) = 0 Moreover, since Q x is symmetric for the inner roduct, 2, Q x is a tr-orthogonal rojection Note that if x O A, X T x (O A ) and Z B (H) ah is a lifting for X (ie, δ x (Z) = X), then X x = Q x (Z) 2 It is easy to see that { ( ) I0 0 ( ) G A = U = : I 0 U 0 is the identity of B (H 0 ) and U 22 U H 0 22 and since it is an algebraic subgrou, it is a Banach-Lie subgrou of U (H) (see [3]) It s Lie algebra is given by {( ) 0 0 ( G A = : x 0 x 22 B H 0 22 )ah Thus G x is exonential for any given x O A Consider the decomosition of the Banach-Lie algebra B 2 (H) ah of U 2 (H): exlicitly, B 2 (H) ah = G x R(Q x ) B 2 (H) ah = {Z B 2 (H) ah : ZP = 0 {Z B 2 (H) ah : (1 P )Z(1 P ) = 0, where P = xx

9 GEOMETRY OF THE LEFT ACTION OF THE -SCHATTEN GROUPS 81 Note that R(Q x ) is invariant under the inner action of the isotroy grou Then, this decomosition is what in differential geometry is called a reductive structure of the homogeneous sace [16] Let s denote by η x the inverse of the linear isomorhism δ x := δ x R(Qx) : R(Q x ) T x O A Note that δ x η x = I TxO A and that η x δ x = Q x There is a natural Riemann-Hilbert metric induced by the isomorhisms η x, x O A Given any X, Y T x O A define the inner roduct Note that X, Y x = tr (η x (Y ) η x (X)) = tr (η x (Y )η x (X)) (42) X, X x = η x (X) 2 2 Then if z is a lifting for X, we have that X, X x = η x (X) 2 2 = η x(δ x (z)) 2 2 = Q x(z) 2 2 = X 2 x so the metric (42) coincides with the quotient metric (41), and the quotient metric is Riemannian when = 2 Also note that if X, Y are tangent vector fields of O A, then the ma that sends any x O A to X x, Y x x is smooth, thus, this inner roduct defines a Riemann- Hilbert metric in O A Now, let us recall two natural linear connections for this tye of homogeneous reductive saces defined in [13], the reductive connection and the classifying connection First the reductive connection r If X, Y are a tangent vector fields of O A then the field r X Y in x O A is given by η x ( r X Y(x)) = η x (X x )(η x (Y x )) + [η x (Y x ), η x (X x )], where X(Y ) denotes the derivative of Y in the direction of X and [, ] the commutator of oerators in B (H) Note that the reductive connection is comatible with the metric defined since the quotient metric coincides with the metric (42) The clasifying connection c is defined by: η x ( c X Y(x)) = (η x δ x )(η x (X x )(η x (Y x ))) = Q x (η x (X x )(η x (Y x ))), where X, Y are a tangent vector fields of O A Remark 41 The Levi Civita connection of the metric (42) is = 1 2 ( r + c ) The geodesics of these connection were comuted in [13]: γ(t) = e tηx(x) x t R, is the geodesic with γ(0) = x and γ (0) = X

10 82 ME DI IORIO Y LUCERO Remark 42 If γ(t), t [0, 1] is a smooth curve in O A such that γ(0) = x then the horizontal lifting is a curve Γ in U (H) that is the unique solution of the linear differential equation in B 2 (H): { Γ = ηγ ( γ)γ, (43) Γ(0) = 1 The existence and uniqueness is guaranteed because γ is smooth and then the ma that sends t [0, 1] to η γ(t) ( γ(t)) B 2 (H) ah is smooth Similarly, as it was roved in [16] in the context of classical homogeneous reductive saces, it can be roved that if γ(t), t [0, 1] is a smooth curve in O A, then the unique solution Γ of (43) verifies: (1) Γ(t) U 2 (H) t [0, 1] (2) Γ lifts γ: π γ (Γ) = γ (3) Γ is horizontal: Γ Γ R(Qx ) 5 Comleteness of the metric sace O A with the ambient Finsler metric In this section we will introduce the ambient Finsler metric induced as a submanifold of A+B (H) and rove that O A is comlete with the rectifiable distance given by this metric Let x O A and Zx T x (O A ) = { Zx : Z B (H) ah The Finsler ambient metric is defined by F amb (Zx) := Zx If γ(t) O A, t [0, 1] is a iecewise smooth curve, we measure its length L amb (γ) = 1 0 F amb ( γ(t))dt = Then the rectifiable distance is given by: 1 0 γ(t) dt d amb (b 0, b 1 ) = inf {L amb (γ) : γ is iecewise smooth curve and γ(0) = b 0, γ(1) = b 1 Since O A is a homogeneous sace of U (H) (theorem 34), by [15], the next remark follows Remark 51 Let x = U 0 A O A There exist r > 0 (r does not deend on x), a neighborhood U I of the identity in U (H) and a smooth ma σ x such that { σ x : V x = UA O A : UA x < r U I U (H) and π x σ b Vx = id Vx The next lemma will be used later on Lemma 52 Let (v n ) n be a sequence in O A such that v n v m 0 Then there is v O A such that v n v 0

11 GEOMETRY OF THE LEFT ACTION OF THE -SCHATTEN GROUPS 83 Proof Consider r > 0 given in the remark 51 and n 0 = n(r) N such that v n v n0 < r for all n n 0 { If we call x = v n0 and V x = UA O A : UA x < r, we have that v n V x for all n n 0 and that (σ x (v n )) n n0 U (H) is a Cauchy sequence because σ x is locally Lischitz Then, since U (H) is comlete, there is u U (H) such that σ x (v n ) u 0 Thus, v n ux 0 We end this section showing that (O A, d amb ) is a comlete metric sace Proosition 53 O A d amb is a comlete metric sace with the rectifiable distance Proof Let v n a Cauchy sequence O A for d amb Since v n v m d amb (v n, v m ), (v n ) is a Cauchy sequence also for Then, by lemma 52, there is x O A such that v n x 0 By roosition 32, π x has continuous local cross sections Then, there is n 0 N such that v n is in the domain of σ x for all n n 0 and also σ x (v n ) I 0 Note that there is Z n B (H) ah such that σ x (v n ) = e Zn and Z n 0 This haens because U (H) is a Banach-Lie grou and then the exonential ma is a local diffeomorhism Let γ n (t) = e tzn x O A Note that γ n (0) = x and γ n (1) = v n for all n N, then d amb (v n, x) L amb (γ n ) Z n x 6 A characterization of the rectifiable distance Comleteness of O A with the quotient Finsler metric In this section we rove the comleteness of O A as a metric sace with the rectifiable distance given by the Finsler quotient metric introduced in section 4 In order to rove this, the rectifiable distance induced by this metric will be characterized as a quotient distance of grous, furthermore, we show that it coincides with the metric for the quotient toology of U (H)/G A Let Γ(t) with t [0, 1] be a iecewise C 1 curve in U (H) The length of Γ can be measured as 1 L (Γ) = Γ(t) dt 0 Note that since, for any U U (H), the tangent sace of U (H) can be identified with UB (H) ah, as well as with B (H) ah U, this length functional is well defined Then, the rectifiable distance in U (H) is given by: d (U 0, U 1 ) = inf {L (Γ) : Γ U (H), Γ(0) = U 0, Γ(1) = U 1

12 84 ME DI IORIO Y LUCERO The quotient metric (41), reviously defined in O A, induces another length functional: L(γ) = 1 0 γ γ dt, where γ(t) with t [0, 1] is a continuous and iecewise smooth curve in O A Analogously, the rectifiable distance in O A is given by: d(b 0, b 1 ) = inf {L(γ) : γ O A, γ(0) = b 0, γ(1) = b 1 where the curves γ considered are continuous and iecewise smooth The following result (whose roof is adated from [1]) shows that the rectifiable distance in O A can be aroximated by lifting curves to the grou U (H) Lemma 61 Let x 0 and x 1 O A Then d(x 0, x 1 ) = inf {L (Γ) : Γ U (H), π x0 (Γ(0)) = x 0, π x0 (Γ(1)) = x 1, where the curves Γ considered are continuous and iecewise smooth Proof Let Γ(t) be a iecewise smooth curve in U (H) such that π x0 (Γ(0)) = x 0 and π x0 (Γ(1)) = x 1 Note that the existence of curves that verify π x0 (Γ(t)) = γ(t) for t [0, 1] is assured because π x0 : U (H) O A, π x0 (U) = Ux 0 is a real analytic submersion (lemma 33) Using the definition of the quotient metric (41), it can be seen that the differential ma of π x0 at the identity (denoted by δ x0 ) is contractive More over, is contractive at any U U (H) Then d(x 0, x 1 ) L(π x0 (Γ)) L (Γ) Now let us show that L(γ) can be aroximated by lengths of curves in U (H) joining the fibers of x 0 y x 1 Fix ɛ > 0 and consider 0 = t 0 < t 1 < < t n = 1 an uniform artition of [0, 1] satisfying: (1) γ(s) γ(s) < ɛ/4, if s, s [t i 1, t i ] n 1 (2) L (γ) γ (t i ) γ(ti ) t i < ɛ/2 i=0 For i = 0,, n 1, let Z i B (H) ah such that δ γ(ti ) (Z i ) = γ (t i ) and Z i γ (t i ) γ(ti ) + ɛ/2 Consider the following curve: e tz 0 t [0, t 1 ), e (t t 1)Z 1 e t 1Z 0 t [t 1, t 2 ), Γ(t) = e (t t 2)Z 2 e (t 2 t 1 )Z 1 e t 1Z 0 t [t 2, t 3 ), e (t t n 1)Z n 1 e (t 2 t 1 )Z 1 e t 1Z 0 t [t n 1, 1]

13 GEOMETRY OF THE LEFT ACTION OF THE -SCHATTEN GROUPS 85 Γ is continuous and iecewise smooth curve in U (H) satisfying Γ(0) = 1 and n 1 L (Γ) = Z i t i i=0 ( n 1 ) γ (t i ) γ(ti ) + nɛ/2 t i L(γ) + ɛ i=0 and π(γ(1)) lies close to x 1 In order to rove the last statement, ( first) aly the mean value theorem in Banach saces [9] to the ma α(t) = π x0 e tz 0 γ(t) Then, for some s1 [0, t 1 ], we see that πx0 ( e t 1 Z 0 ) γ (t1 ) = α (t 1 ) α(0) α (s 1 ) t 1 = e s 1 Z 0 δ x0 (Z 0 ) γ (s 1 ) t 1 = e s 1 Z0 γ (0) γ (s 1 ) t 1 ( e s 1 Z0 γ (0) γ (0) ) + γ (0) γ (s 1 ) t 1 < ( e s 1 Z0 γ (0) γ (0) + ɛ/4) t 1 Note that the first summand is bounded by e s 1 Z0 γ (0) γ (0) = ( e s 1Z 0 I ) γ (0) γ (0) e s 1 Z 0 I M t 1, where M = max γ(t) Thus, t [0,1] π x0 (Γ(t 1 )) γ (t 1 ) (M t 1 + ɛ/4) t 1 Now, note that π x0 (Γ(t 2 )) γ(t 2 ) is less or equal than e (t 2 t 1 )Z 1 e t 1Z 0 x 0 e (t 2 t 1 )Z 1 γ (t 1 ) + e (t 2 t 1 )Z 1 γ (t 1 ) γ (t 2 ) and that, e (t 2 t 1 )Z 1 e t 1Z 0 x 0 e (t 2 t 1 )Z 1 γ (t 1 ) = e (t 2 t 1 )Z 1 ( e t 1 Z 0 x 0 γ(t 1 ) ) e (t 2 t 1 )Z 1 e t 1 Z 0 x 0 γ(t 1 ) (M t 1 + ɛ/4) t 1 Also, roceeding analogously as before, the second summand can be bounded by: e (t 2 t 1 )Z 1 γ (t 1 ) γ (t 2 ) (M t 2 + ɛ/4) t 2 Thus, Inductively, we obtain ( πx0 e (t 2 t 1 )Z 1 e t 1Z 0 ) γ (t2 ) 2 ( M n n + ɛ 4 π x0 (Γ (1)) γ (1) M n + ɛ 4 < ɛ/2 In the next theorem, the rectifiable distance will be characterized as the quotient distance of grous, identifying O A = U (H)/G A This fact will also imly the comleteness of O A with the rectifiable distance In the roof of this theorem, the next lemma (cf [17] age 109) will be used )

14 86 ME DI IORIO Y LUCERO Lemma 62 Let H be a metrizable toological grou, and G be a closed subgrou If d is a comlete distance function on H inducing the toology of H, and if d is invariant under the right translation by G, ie, d (xg, yg) = d (x, y) for any x, y H and g G, then the left coset sace H/G = {xg : x H is a comlete metric sace under the metric d given by d (xg, yg) = inf {d (xg 1, yg 2 ) : g 1, g 2 G Moreover, the distance d is a metric for the quotient toology In our context we shall take H = U (H), G = G A and d = d Theorem 63 Let A be a selfadjoint oerator with closed range Let U 0 and U 1 U (H) and d (U 0 A, U 1 A) = inf {d (U 0, U 1 U) : U G A Then, d = d Where d is the rectifiable distance in O A, reviously defined In articular, (O A, d) is a comlete metric sace and d metricates the quotient toology Proof It is known that (U (H), d ) is a comlete metric sace and that G A is d -closed in U (H), then the quotient distance d is well defined and it can be calculated: d (U 0 A, U 1 A) = inf {d (U 0, U 1 U) : U G A In order to rove the equality between distances first fix ɛ > 0 By lemma 61, there exists a curve Γ in U (H) satisfying: (1) Γ(0) = U 0 ; Γ(1) = U 1 U, U G A (2) L (Γ) < d(u 0 A, U 1 A) + ɛ Then d (U 0 A, U 1 A) d (U 0, U 1 U) L (Γ) < d(u 0 A, U 1 A) + ɛ Since ɛ is arbitrary, we have roved the first inequality To show the reversed inequality, note that given ɛ > 0, there exists U G A such that d (U 0, U 1 U) < d (U 0 A, U 1 A) + ɛ Then there exists a curve Γ in U (H) such that Γ(0) = U 0, Γ(1) = U 1 U and Thus, we get L (Γ) < d (U 0, U 1 U) + ɛ d(u 0 A, U 1 A) L (Γ) < d (U 0, U 1 U) + ɛ < d (U 0 A, U 1 A) + 2ɛ, hence the equality holds The comleteness of (O A, d) and the fact that d metricates the quotient toology follow from lemma 62 Acknowledgement The author would like to thank Professor E Andruchow for his guidance and the referees for their thoughtful suggestions

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