JORDAN MANIFOLDS

L.L. STACHÓ and W. WERNER

Abstract. We introduce Jordan manifolds as Banach manifolds whose tangent spaces are endowed with Jordan triple products depending smoothly on the underlying points. As chief examples we study in detail the natural complex geometry bounded symmetric domains with their Harish-Chandra realizations as unit balls of JB*-triples and we extend the results to Jordan manifolds where the chart transition maps are locally generalized Möbius transformations.

AMS Subject Classification: ????????????????????????

Key words: Jordan manifold, Jordan triple, JB*-triple, Bergman operator, Möbius transformation.

1. Introduction

Recently, in [9] the second author investigated the natural complex geometry of the unit ball of a complex Hilbert C*-module. Among others, he proved the existence of a unique symmetry invariant (Levi Civitá) connection in these structure along with its explicit form in terms of the non-commutative scalar product. Our first goal in Section 2 will be to show with a different Jordan theoretical approach that the results in [9] can be extended to the natural complex geometry of any bounded symmetric domain in a complex Banach space. Actually, such domains are biholomorphically equivalent to the unit balls of the so-called JB*-triples that is Banach spaces equipped with a three variable operation (x, y, z) ↦ {xyz} satisfying the generalized C*-condition ∥{xxx}∥ = ∥x∥^{3} along with the Jordan identity and the fact that its elementary inner derivations are positive Hermitian operators. Using the deep topological Jordan algebraic results in [3, Section 2] we achieve simple explicit formulas for the Levi Civitá connection in terms of the Bergman operator associated with the triple product {…}. Taking into account that the context in [3, Section 2] extends to general real (not only complex) topological Jordan*-triples (even Jordan pairs) our techniques can also be applied in a fairly more general setting. Namely we establish analogous results in Section 3 for real Banach manifolds which admit an atlas consisting of maps into some Jordan*-triple such that the transitions between charts around closely situated couple of points are Möbius transformations of the product triple product (Jordan-Möbius manifolds in out terminology). These results apply automatically to all complex symmetric Hermitian Banach manifolds treated in [2] but go far beyond the complex setting. In Section 4 we start a more general approach. We introduce the concept of Jordan manifold as a real or complex Banach manifold equipped with smoothly varying Jordan*-triple products on the tangent spaces. This generality includes even classical Riemann spaces (with triple product \{uvw\}_{p} := \frac{1}{2}⟨u|v⟩w + \frac{1}{2}⟨w|v⟩u in terms of the scalar product on the tangent space at the point p) in a natural manner. To approach the real background of the previous results, homogeneous and symmetric Jordan manifolds are more suited. These concepts are at hand because Jordan manifolds form a category with morphisms being smooth mappings whose derivatives are homomorphisms for the pointwise triple products. Namely a Jordan manifold is homogeneous if its automorphism group is transitive and symmetric if every point admits an automorphism whose derivative there is minus-identity. For the moment it seems to be an open question if the automorphisms of a symmetric Jordan manifold form a Banach-Lie group with an analogous construction as Upmeier’s topology [8] for symmetric Hermitian Banach manifolds. We close the paper with examples of various Jordan manifolds.

3. The open unit ball of a complex JB*-triple

Throughout this work let Z denote an arbitrarily fixed complex Banach space with norm ∥ · ∥. We shall write Ball(Z) for its open unit ball Ball(Z) := {z ∈ Z : ∥z∥ < 1}. By a Jordan triple product on Z we mean a continuous 3-variable operation (x, y, z) ↦ {xyz} being symmetric bilinear in the outer variables x, z and conjugate-linear in the inner variable y satisfying the Jordan identity

(J)  {ab{xyz}} = {{abx}yz} − {x{bay}z} + {xy{abz}}

for all a, b, x, y, z ∈ E. We say that the Jordan triple (Z, {…}) is a JB*-triple if the generalized C*-axiom

∥{zzz}∥ = ∥z∥^{3} , z ∈ Z

holds and all the operations

D(a) : z ↦ {aaz} , a ∈ Z

are Z-Hermitian with non-negative spectrum, that is

∥ exp (ζD(a))∥ ≤ 1 for every ζ ∈ C with Re ζ ≤ 0 .

2.1 Remark. (1) The Jordan identity is equivalent with the fact that all the operators iD(a) (a ∈ E) are derivations of the triple product, that is we have the identities

iD(a){xyz} = {[iD(a)x]yz} − {x[iD(a)y]z} + {xy[iD(a)z]}.

(2) C*-algebras are Jordan triples with the triple product

\{xyz\} := \frac{1}{2}xy^{∗}x + \frac{1}{2}zy^{∗}x ,

moreover complex C*-algebras are JB*-triples with their natural norm.

(3) Complex Hilbert C*-modules are also Jordan triples with

\{xyz\} := \frac{1}{2}⟨x|y⟩z + \frac{1}{2}⟨z|y⟩x.

(4) If the unit ball D := Ball(Z) is symmetric (that is for each point a ∈ D, there is a biholomorphic automorphism S_{a} : D ↔ D with Fréchet derivative S′_{a}(a) = −Id_{Z} at a) then E can be equipped with a unique JB*-triple product.

(5) Any bounded symmetric domain in a complex Banach space can be mapped biholomorphically onto some bounded balanced convex symmetric domain (a so-called Harish-Chandra realization), that is onto the unit ball of some equivalent JB*-norm.

In the sequel let (Z, {…}) denote any given JB*-triple and let

D(a, b) : z ↦ {a, b, z}, D(d) := D(a, a),
Q(a, b) : z ↦ {a, z, b}, Q(a) := Q(a, a),
B(a, b) := 1 − 2D(a, b) + Q(a)Q(b), B(a) := B(a, a)

be the usual skew-derivations, quadratic representations and Bergman operators, respectively. Recall that the transformation

T : c ↦ [ exp ([c − Q(z)c]\frac{∂}{∂z} )]0

that is T (x) = [the value z_{1} for the initial value problem \frac{d}{dt}z_{t} = c − Q(z_{t})c, z_{0} = 0] is a well-defined real bianalytic mapping

T : Z ⟷ Ball(Z) .

Given any point a ∈ Ball(Z), it is well-known [6, p. 27, 4] that the mapping

g_{a} := exp ([T^{−1}(a) − Q(z)T^{−1}(a)]∂/∂z)

is a holomorphic automorphism of Ball(Z) and we have

(2.2)  g_{a}(z) = a + B(a)^{1/2}[1 + D(z, a)]^{−1}z,  ∥z∥, ∥z∥ < 1.

2.3 Definition. In the sequel we shall call the mappings g_{a} ∘ L composed with linear unitary operators of Z the Möbius transformations associated with the triple product {…}. It is well-known [3] the group Aut Ball(Z) of all holomorphic automorphisms of the unit ball of a complex JB*-triple coincides with the set of all Möbius transformations of the underlying triple product.

2.4 Remark. In [3] one can find g_{a}(z) = a + B(a)^{1/2}[1 − D(x, a)]^{−1}z which is obviously incorrect in the sign of the term D(z, a) as on can see on the 1-dimensional example of the classical Möbius transformation g_{a}(z) = (z + a)/(1 + \overline{a}z) (with a, z ∈ Ball(C)).

Next we are going to consider Ball(Z) as a complex manifold equipped with the charts {g_{a}^{−1} : a ∈ Ball(Z)}. In this manner we get a natural generalization of the complex Poincaré model on the unit disc Ball(C) for the real 2-dimensional hyperbolic geometry.

Due to the possible lack of non-trivial smooth functions vanishing outside a ball, for a Banach manifold M, it is no longer convenient to apply the usual definition of a connection as a mapping ∇ : T M × T M → T M as being a derivation for the first and linear in the second variable, both with respect to multiplication with smooth functions.

2.5 Definition [5]. Let M be a manifold, modeled over the Banach space E, and denote the space of bounded bilinear mappings E × E → E by L^{2}(E, E). Then M is said to possess a connection if there is an atlas U for M so that for each (U, Φ) ∈ U (where U is some open subset of M and Φ is a homeomorphism of U onto some open subset of E) there is a smooth mapping Γ_{Φ} : Φ(U) → L^{2}(E, E), called the Christoffel symbol of the connection on U, which under a change of coordinates ψ : Φ(U) → E transforms according to

Γ_{ψ∘Φ}(ψ′u, ψ′v) = Ψ′′(u, v) + ψ′Γ_{Φ}(u, v)

for smooth vector fields u, v on Φ(U) ⊂ E. The covariant derivative of a vector field Y in the direction of the vector field X is, locally, defined to be the principal part of

∇_{X}Y = dX(Y) − Γ(X, Y),

that is, if X, Y are smooth vector fields on M with u := Φ^{#}X( : Φ(U) ∋ Φ(p) ↦ Φ′(p)X(p)) and v := Φ^{#}X then

Φ^{#}∇_{X}Y = ∇_{u}v + Γ_{Φ}(u, v) : Φ(U) ∋ q ↦ \frac{d}{dt}|_{t=0}v(q + tu(q)) + Γ_{Φ}(u(q), v(q)) .

If a Banach Lie group G acts smoothly on M then, for each g ∈ G a connection g^{∗}∇ is defined by letting

g^{∗}∇_{X}Y = ∇_{g^{∗}X}g^{∗}Y, g^{∗}X(gm) = d_{m}gX(m).

The Christoffel symbols then transform as in the definition above,

\begin{matrix}Γ_{Φ∘g}(g(m)) & (g′(m)X(m), g′(m)Y (m)) = \\ = g′′(m)(X(m), Y (m)) + g′(m)[Γ_{Φ}(m)(X(m), Y (m))], m ∈ M \end{matrix}

and we call ∇ invariant under the action of G whenever g^{∗}∇ = ∇ for all g ∈ G.

2.6 Theorem. On U := Ball(Z), there exists exactly one Möbius invariant connection whose Christoffel symbol at a is given by

Γ_{Id}(a)(x, y) = 2B(a)^{1/2}{B(a)^{−1/2}x, a, B(a)^{−1/2}y} = 2{x, g′_{a}(0)a, y}_{a},

Proof. Notice first the invariance of ∇ with respect to the symmetry S_{0}(z) := −z (z ∈ U) at the origin implies −Γ_{Id}(0)(x, y) = S′_{0}(0)Γ_{Id}(0)(x, y) = Γ_{Id}(0)(S′_{0}(0)x, S′_{0}(0)y) = Γ_{Id}(0)(x, y) that is

Γ_{Id}(0)(x, y) = 0.

Then, for any Möbius transformation g leaving the origin fixed, g′′ = 0 and so Γ_{g}(0) remains zero under the transformation g. Recall [6,4] that the Fréchet derivative of g_{a} can be expressed as

g′_{a}(z) = B(a)^{1/2}B(z, −a)^{−1}, ∥z∥ < 1.

Using that Q′(z)(h) : x ⟼ 2{z, x, h}, and writing the second derivative as a bilinear mapping, we similarly arrive at

\begin{aligned}g′′_{a}(z)(u, v) & = −B(a)^{1/2}B(z, −a)^{−1}∂_{1}B(u, a)B(z, −a)^{−1}(v) \\ & = 2B(a)^{1/2}B(z, −a)^{−1}[D(u, a) − Q(u, z)Q(−a)]B(z, −a)^{−1}(v), \end{aligned}

applying the chain rule. If we evaluate this expression at z = 0, then

Γ_{Id}(a)(x, y) = g′′_{a}(0)(g′_{−a}(0)x, g′_{−a}(0)y)
= 2B(a)^{1/2}D(g′_{−a}(0)x, a)(g′_{−a}(0)y),
= 2B(a)^{1/2}{B(a)^{−1/2}x, a, B(a)^{−1/2}y} .

From the above calculations it is immediate that a Christoffel symbol thus defined gives rise to a connection being invariant under all Möbius transformations. Qu.e.d.

Next we proceed to the geodesic equation of the connection ∇. Recall that a smooth curve γ : I → Ball(Z) on an open real interval I around 0 is a ∇-geodesic if its derivative satisfies the equation ∇_{γ̇(t)}γ̇(t) = 0. According to Theorem 2.6, this means

γ̈(t) + 2B(γ(t))^{1/2}D(B(γ(t))^{−1/2}γ̇(t), γ(t))B(γ(t))^{−1/2}γ̇(t) = 0.

First we look for particular solutions with the property γ(0) = 0. Let 0 ≠ v ∈ Z be any fixed vector and let F_{v} denote the real JB*-subtriple generated by v. That is F_{v} = \overline{Span_{IR}}\{D(v)^{n}v : n = 0, 1, … \}. Since (Z, \{…\}) is JB*-triple, the spectrum of the operator D(v)|F_{v} is non-negative, and, by writing Ω_{v} := [SpD(v)^{1/2}|F_{v}| \ {0}, the commutative Gelfand-Naimark Theorem of JB*-triples asserts that there is a real JB*-isomorphism H_{v} : Re C_{0}(Ω_{v}) → F_{v} such that H_{v}(Id_{Ω_{v}}) = v and H_{v}(φψχ) = {H_{v}(φ), H_{v}(ψ), H_{v}(χ)} for all functions φ, ψ, χ ∈ Re C_{0}(Ω_{v}). In particular, given any solution φ : I → ReC_{0}(Ω_{v}), of the equation

\frac{d^{2}}{dt^{2}} φ(t, ω) + 2[1 − φ(t, ω)^{2}]^{−1}φ(t, ω)[\frac{d}{dt}φ(t, ω)]^{2}, ω ∈ Ω_{v},

the curve γ(t) := H_{v}φ(t), t ∈ I is a ∇-geodesic. Notice that, for any real constant α, the function x(t) = artanh(αt) := (e^{αt} − e^{−αt})(e^{αt} + e^{−αt})^{−1} is the solution of the initial value problem ẍ + (1 − x^{2})^{−1}xẋ^{2}, x(0) = 0, ẋ(0) = α. Indeed, we have ẋ = 1 − x^{2} and hence ẍ = −2xẋ = −2x(1−x^{2}) and ẍ+(1−x^{2})^{−1}xẋ^{2} = −2xẋ = −2x(1−x^{2})+2(1−x^{2})^{−1}x(1− x^{2})^{2} = 0. Therefore we get the following.

2.7 Theorem. The ∇-geodesic curves passing through the origin have the form

γ_{0,v}(t) = H_{v} artanh(t Id_{Ω_{v}}), t ∈ R, v ∈ Z

in terms of the Gelfand-Naimark representations H_{v}.

2.8 Corollary. The (unique) maximal ∇-geodesic γ_{a,w} with the properties γ_{a,w}(0) = a and γ̇(0) = w has the form

R ∋ t ↦ g_{a}(H_{g′_{a}(0)^{−1}w} artanh(t Id_{Ω_{g′_{a}(0)^{−1}w}})).

Proof. It is a well-known consequence of the Möbius invariance of the connection ∇ that h ∘ γ is ∇-geodesic whenever γ is a ∇-geodesic and h ∈ Aut Ball(Z). The curve in the statement of the Corollary is thus a ∇-geodesic. It is straightforward to see that its starting point is a and its starting speed vector is w. The classical Piccard-Lindelöf Theorem ensures the uniqueness of the solution of the geodesic equation with given starting point and starting speed vector. Qu.e.d.

The expression for the geodesic in the above can be conveniently rewritten in terms of the power series for artanh. In fact, we have

γ_{0,v}(t) = ∑_{n=0}^{∞} \frac{t^{2n+1}}{2n + 1}D(v, v)^{2n}v = artanh tv.

In the same vein,

γ_{a,w}(t) = a + B(a)^{1/2}[1 + D(artanh tB(a)^{1/2}v, a)]^{−1} artanh tB(a)^{1/2}v.

3. Local Möbius transformations in real Jordan-Banach triples and Jordan-Möbius manifolds

Throughout this section E denotes a real Jordan-Banach triple with the norm ∥.∥ and triple product {…}, respectively. Thus we only assume that E is a real Banach space and {…} is a continuous real trilinear mapping E^{3} → E satisfying the Jordan identity (J). We can take over the notations D(a, b), Q(a, b), B(a, b) in the real setting without formal changes. In particular, all the operators D(a, b) − D(b, a) : z ↦ {abz} − {baz} belong to Der(E, {…}) the set of the derivations of the triple product. Though not explicitly stated, a straightforward inspection of [3, Corollary 2.20] establishes the existence of a constant ε > 0 such that the transformations

H_{v} := exp V_{v} with the vector fields V_{v} := [v − {zvz}]∂/∂z

are well-defined on the ball εBall(E) whenever ∥v∥ < ε, moreover they have the fractional linear Möbius form

(3.1)

H_{v}(z) = g_{H_{v}(0)}(z) where
g_{a}(z) := a + λ(a)[1 + D(z, a)]^{−1}z where λ(a) := H′_{v}(0) ∈ L(E) .

Notice that in the case of JB*-triples we can write λ(a) = B(a)^{1/2} in termd of the Bergman operator. Besides the vector fields V_{v} of polynomial degree 2, let us introduce also the linear vector fields

L_{ℓ} := [ℓz]∂/∂z, ℓ ∈ Der(E, {…}).

For their Poisson commutators [f(z)∂/∂z, g(z)∂/∂z] := (f′(z)g(z) − g′(z)f(z))∂/∂z, it is straightforward to check that we have

[L_{ℓ}, V_{v}] = V_{ℓv}, [L_{ℓ}, L_{m}] = L_{[ℓ,m]}, [V_{u}, V_{v}] = L_{D(v,u)−D(u,v)}.

Therefore the real linear space

{V_{v} + L_{ℓ} : v ∈ Z, ℓ ∈ Der(E, {…})}

equipped with the norm ∥V_{v} + L_{ℓ}∥ := sup{∥v − {zvz} + ℓz∥ : ∥z∥ ≤ 1} is a Banach-Lie algebra with the Poisson commutator. Thus, according to the Campbell-Hausdorff formula, for some sufficiently small constant δ > 0 we have

[ exp(V_{u} + L_{ℓ}) exp(V_{v} + L_{m})]z = [ exp(V_{w(u,v,ℓ,m)} + L_{Δ(u,v,ℓ,m)}]z

with two suitable real-analytic mappings

w : [δ Ball(E)]^{2} × [δ Ball(Der(E, {…}))]^{2} ⟶ E,
Δ : [δ Ball(E)]^{2} × [δ Ball(Der(E, {…}))]^{2} ⟶ Der(E, {…})

whenever ∥z∥, ∥u∥, ∥v∥, ∥ℓ∥, ∥m∥ < δ.

3.2 Definition. By a Möbius transformation of (E, {…}) we mean a bianalytic mapping Φ : U → E defined in a neighborhood U of the origin in E such that for some ℓ ∈ Der(E, {…}) we have Φ(z) = g_{a}(exp ℓz), z ∈ U with the customary notation in g_{a}(z) := a + λ(a)[1 − D(z, a)]^{−1} established in (3.1).

The next two auxiliary results 3.3-4 establish in particular that composition preserve Möbius tranformations.

3.3 Proposition. There exists δ′ > 0 such that

∥v∥, ∥ℓ∥ < δ′ and [exp(V_{v} + L_{ℓ})]0 = 0 ⟹ v = 0.

Proof. We can choose δ′ to be so small that the mapping u ↦ [exp(V_{u})]0 be injective on δ′ Ball(E) and the terms

[exp(tL_{m})][exp(V_{u} + L_{m})][exp(−tL_{m})]z, |t|, ∥z∥, ∥u∥, ∥m∥ < δ′

be all well-defined. Consider any couple v, ℓ with ∥v∥, ∥ℓ∥ < δ′ and [exp(V_{v} + L_{ℓ})]0 = 0. Then, for all −δ′ < t < δ′,

0 = [exp(tL_{ℓ})][exp(V_{v} + L_{ℓ})][exp(−tL_{ℓ})]0 =
= [ exp ([ exp(tL_{ℓ}^{#})](V_{v} + L_{ℓ}))]0 =
= [ exp (V_{exp tℓ)v} + L_{ℓ})]0.

Notice that the points

x_{s,t} := [ exp (s[V_{(exp tℓ)v} + L_{ℓ}])]0, |s| < 1 + ε′, |t| < δ′

are well-defined for some ε′ > 0 and they satisfy the differential equation

\frac{∂}{∂s}x_{s,t} = (exp tℓ)v − \{x_{s,t}[(exp tℓ)v]x_{s,t}\} + ℓx_{s,t}.

Moreover, since [ exp (V_{exp tℓ)v} + L_{ℓ})]0 = 0 for |t| < δ′, it follows

\begin{aligned}0 & = \frac{∂^{2}}{∂s∂t}|_{s=1, t=0}x_{s,t} = \frac{∂^{2}}{∂t∂s}|_{s=1, t=0}x_{s,t} = \\ & = ℓv − 2\{(\frac{∂}{∂t}|_{t=0, s=1}x_{s,t})vx_{s,t}\} + ℓ\frac{∂}{∂t}|_{t=0, s=1}x_{s,t}. \end{aligned}

However, here we have \frac{∂}{∂t} |_{t=0, s=1}x_{s,t} = 0 which implies that necessarily ℓv = 0. It follows

[L_{ℓ}, V_{v}] = V_{ℓv} = 0 and therefore

0 = [exp(V_{v} + L_{ℓ})]0 = [exp V_{v}][exp L_{ℓ}]0 = [exp V_{v}]0.

The choice of δ′ with ∥v∥ < δ′ ensures that v = 0. Qu.e.d.

3.4 Corollary. There exists 0 < δ′′(< δ′) such that given any v ∈ Z and ℓ ∈ Der(E, {…}) with ∥v∥, ∥ℓ∥ < δ′′, for some m ∈ Der(E, {…}) we have

[ exp(V_{v} + L_{ℓ})]z = g_{a(v,ℓ)}([exp m]z), ∥z∥ < δ′′

where a(v, ℓ) := [ exp(V_{v} + L_{ℓ})]0. In particular, for sufficiently small vectors p, q ∈ E,

g_{p} ∘ g_{q} = g_{g_{p}(q)} ∘ [exp m_{p,q}]

with a real-analytic mapping (p, q) ↦ m_{p,q} ∈ DerZ, {…}).

3.5 Definition. By a Jordan-Möbius manifold modeled with (E, {…}) we mean a a real-analytic manifold M with an inverse atlas X = {X_{p} : p ∈ M} such that for each point p ∈ M we have

(3.6) X_{p} : U_{p} → M, X_{p}(0) = p, X_{p}^{−1} ∘ X_{q} Möbius transformation for q ∈ X_{p}(U_{p})

where U_{p} is an open connected neighborhood of the origin in E the

A Jordan-Möbius manifold M with an inverse atlas X = {X_{p} : p ∈ M} satisfying (3.6) is said to be a uniform Jordan-Möbius manifold if there exists a common constant ε > 0 such that

dom(X_{p} ∘ g) ⊃ ε Ball(E) if g ∈ {Möbius transformations} and ∥g(0)∥ < ε.

In particular the region ⋃_{L∈Aut(E,{...})} εLBall(E) is contained in dom(X_{p}) for all p ∈ M.

3.7 Example. If E := Z is a JB*-triple then its unit ball M := Ball(Z) is a uniform Jordan manifold with the charts X_{p} := g_{p}|M, p ∈ M. In this case we may choose ε = 1.

3.8 Example. Let (E, {…}) be any Jordan-Banach triple. Then we can find a constant ϱ > 0 such that the sections of the mapping [ϱBall(E)]^{2} ∋ (p, z) ↦ g_{p}(z) are real-bianalytic for any fixed p and z, respectively. Then the ball M := ϱBall(E) with the topology from E and with the charts X_{p}(z) := g_{p}(z) defined for z ∈ U_{p} := {u ∈ E : ∥u∥, ∥g_{p}(u)∥ < ϱ} is a Jordan manifold which is not uniform in general.

3.8a Example. (Special case of 3.8 with Lorenz space). E := Mat(1, 2, IR), S := ({1 0 \atop 0 −1}); indefinite scalar product ⟨x|y⟩_{S} := xSy^{∗} ∈ IR (x, y ∈ E), tripe product {xyz}_{S} := \frac{1}{2}⟨x|y⟩_{S}z + \frac{1}{2}⟨z|y⟩_{S}x = \frac{1}{2}xy^{∗}Sz + \frac{1}{2}zy^{∗}Sx on E. Then the local Möbius transformations have the form

M_{a}(x){= (1 − aSa^{∗})^{−1/2}(x + a)(1 + Sa^{∗}x)^{−1}(1 − Sa^{∗}a)^{1/2} = \atop = (1 − aSa^{∗})^{−1/2}(1 + xSa^{∗})^{−1}(x + a)(1 − Sa^{∗}a)^{1/2} .}

Proof. By definition, M_{a}(x) = a + B(a)^{1/2}[1 + D(x, a)]^{−1}x. Here we have

B(a)z = z − 2D(a)z + Q(z)^{2} = z − aSa^{∗}z − zSa^{∗}a + aSa^{∗}zSa^{∗}a =
= (1 − aSa^{∗})z(1 − Sa^{∗}a),
D(x, a)^{n}x = (xSa^{∗})^{n}x = x(Sa^{∗}x)^{n} (n = 0, 1, …).

Thus B(a)^{1/2}z = (1−aSa^{∗})^{1/2}z(1−Sa^{∗}a)^{1/2} and [1+D(x, a)]^{−1}x = ∑_{n=0}^{∞}(−1)^{n}D(x, a)^{n}x =

(1 − xSa^{∗})^{−1}x = x(1 − Sa^{∗}x)^{−1}. Since also (1 − aSa^{∗})^{1/2}a = a(1 − Sa^{∗}a)^{1/2} as one can see from the Newtonian expansion, it follows

M_{a}(x) = a + (1 − aSa^{∗})^{1/2}x(1 + Sa^{∗}x)^{−1}(1 − Sa^{∗}a)^{1/2} =

= (1 − aSa^{∗})^{−1/2}a(1 − Sa^{∗}a)^{1/2} + (1 − aSa^{∗})^{1/2}x(1 + Sa^{∗}x)^{−1}(1 − Sa^{∗}a)^{1/2} =

= (1 − aSa^{∗})^{−1/2}[a + (1 − aSa^{∗})x(1 + Sa^{∗}x)^{−1}](1 − Sa^{∗}a)^{1/2} =
= (1 − aSa^{∗})^{−1/2}[a(1 + Sa^{∗}x) + (1 − aSa^{∗})x](1 + Sa^{∗}x)^{−1}(1 − Sa^{∗}a)^{1/2} =
= (1 − aSa^{∗})^{−1/2}(x + a)(1 + Sa^{∗}x)^{−1}(1 − Sa^{∗}a)^{1/2}.

The proof for the alternative expression of M_{a}(x) is analogous.

3.9 Example [1,7]. Let (Z, {…}) be any complex JB*-triple such that the family Tri(Z, {…}) := {e ∈ Z : {eee} = e ≠ 0} of its non-trivial tripotents is not void. Two tripotents e, f ∈ Tri(Z, {…}) are said to be equivalent (e ∼ f in notation) if D(e, e) = D(f, f). Actually ∼ is an equivalence relation on Tri(Z, {…}). Let M be a connected component of Tri(Z, {…}) (with respect to the topology inherited from Z). Then the Peirce spaces Z_{1/2}(e) := {u ∈ Z : D(e, e)u = u/2} (e ∈ M) are all isomorphic and the set IM := {e : e ∈ M} (e := {f ∈ M : f ∼ e}) of its equivalence classes can be regarded as a complex hermitian symmetric manifold modeled on Z_{1/2}(e_{0}) with any e_{0} ∈ M and being such that the automorphisms exp t[D(e, u) − D(u, e)] (t ∈ IR, u ∈ Z_{1/2}(e)) of Z acting on IM form a continuous one-parameter subgroup of the Banach-Lie group Aut(IM) of all biholomorphic automorphisms of IM (with Upmeier’s topology [8]). Given any e ∈ M, there is a neighborhood U of its equivalence class e in IM along with a holomorphic chart map Φ : U → Z_{1/2}(e) such that each map Φ^{#} exp[D(e, u) − D(u, e)] (u ∈ Z_{1/2}(e)) is a Möbius transformation with the triple product

\{u_{1}u_{2}u_{3}\} := \frac{∂^{3}}{∂ζ_{1}∂\overline{ζ_{2}}∂ζ_{3}} |_{ζ_{1}=ζ_{2}=ζ_{3}=0} Φ^{#}([P_{ζ_{1}u_{1}}, [P_{ζ_{2}u_{2}}, P_{ζ_{3}u_{3}}]]_{e})

in terms of the vector fields P_{u}(f) := [t ↦ exp t[D(e, u) − D(u, e)]f] ∈ T_{e}IM.

3.10 Lemma. Assume U, V are domains in a Banach space W and let T : U ↔ V be a smooth diffeomorphism between them. Then given any couple X, Y : U → W of smooth vector fields on U, for their transforms X̃ := T^{#}X respectively Ỹ := T^{#}Y on V we have

Ỹ′X̃(v) = T′′(T^{−1}(v))X(T^{−1}(v))Y (T^{−1}(v)) + T′(T^{−1}(v))Y′(T^{−1}(v))X(T^{−1}(v)), v ∈V.

Proof. By definition, Ỹ′X̃(v) = \frac{d}{dt} |_{t=0}Ỹ(v + tX̃(v)). Therefore

Ỹ′X̃(v) = \frac{d}{dt}|_{t=0}T′(T^{−1}(v + tX̃(v)))Y (T^{−1}(v + X̃(v))) =

= [\frac{d}{dt}|_{t=0}T′(T^{−1}(v + tX̃(v)))]Y (T^{−1}(v))+

+ T′(T^{−1}(v))\frac{d}{dt}|_{t=0}Y (v + tX̃(v)) =

= T′′(T^{−1}(v))[\frac{d}{dt}|_{t=0}T^{−1}(v + tX̃(v))]Y (T^{−1}(v))+

+ T′(T^{−1}(v))Y′(T^{−1}(v))\frac{d}{dt}|_{t=0}T^{−1}(v + tX̃(v)) =

= T′′(T^{−1}(v))[T′(T^{−1}(v))^{−1}X̃(v)]Y (T^{−1}(v))+

+ T′(T^{−1}(v))Y′(T^{−1}(v))[T′(T^{−1}(v))]^{−1}X̃(T^{−1}(v)) =

= T′′(T^{−1}(v))X(T^{−1}(v))Y (T^{−1}(v))+

+ T′(T^{−1}(v))Y′(T^{−1}(v))X(T^{−1}(v)). Qu,e.d.

In the sequel let (M, {X_{p} : p ∈ M}) be a Jordan-Möbius manifold modeled on (E, {…}) with inverse charts X_{p} : E ⊃ U_{p} → M satisfying X_{p}(0) = p, p ∈ M and with Möbius transition maps X_{p}^{−1} ∘ X_{q} for couples of point lying sufficiently close together.

3.11 Proposition, Let p ∈ M be an arbitrarily given point, let X, Y be two smooth vector fields on M and define R := ∇_{X}Y. Then, for any vector w ∈ U_{p}, the image R := [X_{p}^{−1}]^{#}R of the vector field R by means of the local coordinate X_{p} can be expressed in terms of the image vector fields X := [X_{p}^{−1}]^{#}X and Y := [X_{p}^{−1}]^{#}Y as

R(w) = H′_{w}(w)^{−1}H′′_{w}(w)X(w)Y (w) + Y′(w)X(w) where H_{w} := X_{X_{p}(w)}^{−1} ∘ X_{p}.

Proof. Let w ∈ U_{p} be arbitrarily fixed and write q := X_{p}(w). Then, with the mapping T := H_{w} we have

T (w) = X_{q}^{−1} ∘ X_{p}(w) = X_{q}^{−1}(q) = 0.

Also in terms of T we can write

[X_{q}^{−1}]^{#}X = [X_{q}^{−1}]^{#}X_{p}^{#}[X_{p}^{−1}]^{#}X =
= [X_{q}^{−1} ∘ X_{p}]^{#}[X_{p}^{−1}]^{#}X =
= T^{#}X

and similarly [X_{q}^{−1}]^{#}Y = T^{#}Y and [X_{q}^{−1}]^{#}R = T^{#}R. Hence

[X_{q}^{−1}]^{#}R(0) = [T^{#}([X_{p}^{−1}]^{#}R)](0) =
= T′(T^{−1}(0))[[X_{p}^{−1}]^{#}R](T^{−1}(0)) =
= T′(w)[[[X_{p}^{−1}]^{#}R](w)] =
= T′(w)R(w).

By axiomatic assumption we have [X_{q}^{−1}]^{#}R(0) = [[[X_{q}^{−1}]^{#}Y]′(0)][[[X_{q}^{−1}]^{#}X](0)]. That is

R(w) = [X_{p}^{−1}]^{#}R(w) =
= T′(w)^{−1}[[[X_{q}^{−1}]^{#}Y]′(0)][[[X_{q}^{−1}]^{#}X](0)] =
= T′(w)^{−1}[(T^{#}Y)′(0)][T^{#}X(0)] =
= T′(w)^{−1}T′′(w)X(w)Y (w) + Y′(w)X(w)

in view of Lemma 3.7 and the fact that w = T^{−1}(0). Qu.e.d.

Our technical results 3.3-4, 3.10-11, establish immediately that the calculations for the proof of Theorem 2.6 can be carried out locally even in the setting of general real Jordan-Möbius manifolds. As a first straightforward consequence we get the following.

3.10 Theorem. Let M be a Jordan-Möbius manifold modeled on (E, {…}) with a system of inverse charts {X_{p} : p ∈ M} having the properties (3.6). Then there exists a (necessarily unique) connection ∇ on M such that its Christoffel symbol Γ with the charts Φ_{p} := X_{p}^{−1} satisfies

Γ_{Φ_{p}}(0) = 0 , p ∈ M.

Namely, if p ∈ M is any point and the constant δ > 0 is so chosen that δ Ball(E) ⊂ dom(X_{p}) then for any couple of vectors x, y ∈ E and any point a ∈ δ Ball(E) we have

\begin{aligned}Γ_{Φ_{p}}(a)(x, y) & = g′′_{a}(0)(g′_{−a}(0)x, g′_{−a}(0)y) = \\ & = 2λ(a)\{[λ(a)^{−1}x]a[λ(a)^{−1}y]\} . \end{aligned}

In particular Γ_{Φ_{p}}(a)(x, y) = 2B(a)^{1/2}{[B(a)^{−1/2}x]a[B(a)^{−1/2}y]} in the case of (E, {…}) being a complex JB*-triple.

In the above theorem we could not include a statement about symmetry invariant connections because, unlike the unit ball of a complex JB*-triple, Jordan-Möbius manifolds need not be necessarily symmetric. We close this section by showing that the assumption of uniformness implies this property.

3.11 Theorem. Connected uniform Jordan-Möbius manifolds are symmetric and they admit a unique symmetry invariant connection.

Proof. SKETCHED

Let M be a connected uniform Jordan-Möbius manifold modeled with (E, {…}) along with a constant ε > 0 and a system X = {X_{p} : p ∈ M} as in Definition 3.5. By Corollary 3.3, there exists δ ∈ (0, ε) such that g_{a}(δ Ball(E)) ⊂ εBall(E) for any vector a ∈ δ Ball(E). Observe that, given any couple of points p, q ∈ M, we can find a finite sequence v_{1}, …, v_{N} ∈ δ Ball(E) such that the recursively defined sequence

p_{0} := p, p_{n+1} := X_{p_{n}}(v_{n})

ends in q = p_{N}. Let us fix any point p ∈ M. We can see by induction that there exists a sequence q_{0} = p, q_{1}, …, q_{N} ∈ M of points along with (linear) automorphisms L_{0} = Id_{E}, L_{1}, …, L_{N} ∈ Aut(E, {…}) such that for the modified charts

Y_{n} := X_{q_{n}} ∘ L_{n}, n = 1, … n = 0, …, N

we have

q_{n+1} = Y_{n}(−v_{n}), Y_{n+1}^{−1} ∘ Y_{n} = X_{q_{n}}^{−1} ∘ X_{q_{n+1}}.

In view of 3,3-4, if we have another sequence ṽ_{1}, …, ṽ_{N} and consider the corresponding points p̃_{1}, …, p̃_{N} respectively q̃_{1}, …, q̃_{N} with the above construction then the coincidence p_{N} = p̃_{N} of the endpoints implies the coincidence q_{N} = q̃_{N} as well. It is not hard to check that the thus well-defined transformation S_{p} : p_{N} ↦ q_{N} is a symmetry through the point p such that the maps X_{r}^{−1} ∘ S ∘ X_{p} (r ∈ M) are Möbius transformations. Finally we notice that S_{p} ∘ S_{q} is a Möbius transformation if its domain contains the origin of E. Hence the statement of the theorem is immediate. Qu.e.d.

4. Jordan manifolds

4.1 Definition. A manifold M modeled on a (real) Jordan-Banach triple (E, {…}) is a Jordan manifold if its tangent spaces T_{p}M (p ∈ M) are endowed with a triple product {…} being isomorphic to {…} on E and the mapping (p, u, v, w) ↦ {uvw}_{p} is continuously differentiable. In the sequel we shall write (M, \overline{A}, P) for the triple of the carrier space, atlas and system of triple products. A morphism F : M → M̃ between two Jordan manifolds is a smooth mapping such that its derivatives give rise to triple product homomorphisms on the tangent spaces.

Given two Jordan manifolds (M, \overline{A}, P) and (N, \overline{B}, Q), a continuously differentiable map S : M → N is a Jordan morphism if ∀ p ∈ M ∀ u, v, w ∈ T_{p}(M) S′(p){uvw}_{p} = {[S′(p)u][S′(p)v][S′(p)w]}_{S(p)}. Isomorphism and automorphisms can be defined as usually. Aut(M) := {S : M ↔ M with S, S^{−1} Jordan morphisms}.

(D3) Given (M, \overline{A}, P) and (N, \overline{B}, Q) with U open ⊂ M and V open ⊂ N, a continuously differentiable map S : U ↔ V is a local Jordan automorphism if S is a Jordan isomorphism between U and V as Jordan submanifolds. Aut_{loc}(M) := {local Jordan automorphism in M}. Notice that S_{1} ∘ S_{2} ∈ Aut_{loc}(M) whenever S_{1}, S_{2} ∈ Aut_{loc}(M) with ran(S_{1}) ∩ dom(S_{2}) ≠ ∅.

Convention. Henceforth (M, \overline{A}, P) stands for a Jordan manifold.

Example. (E0) Riemann spaces can be regarded as Jordan manifolds: If ⟨.|.⟩_{p} is the inner product on T_{p}M then we take \{uvw\}_{p} := \frac{1}{2}⟨u|v⟩_{p}w + \frac{1}{2}⟨w|v⟩_{p}u.

Example. The triple products in P need not be isomorphic to each other:

(E1)  M := C with \{uvw\}_{p} := Re(p)u\overline{v}w.

Conjecture. (HARD)

If we assume all the {…}_{p} to be JB^{∗}-triple products then they are isomorphic.

Lemma. If for every couple of points p, q ∈ M there are neighborhoods U, V ⊂ M along with a diffeomorphism S : U ↔ V such that q = S(p) and S′(p) is an isomorphism between \{…\}_{p} and \{…\}_{q} then there is an atlas A ⊂ \overline{A} with the properties (N4).

Proof. Fix any o ∈ M and a local coordinate X ∈ \overline{A} with X(o) = 0. For any p ∈ M choose a local diffeomorhism S_{p} with S_{p}(o) = p and the properties in (D2). We redefine the triple product {…} on Z as the image of {…}_{o} by X′(0) and we set X_{p} := X ∘ S_{p}^{−1} (p ∈ M).

Definition. (D4) A Jordan manifold M is homogeneous if

Aut(M) is transitive on it: ∀ p, q ∈ M ∃ S ∈ Aut(M) S(p) = q.

(D5) A Jordan manifold is locally homogeneous if Aut_{loc}(M) is transitive on it.

Examples. (E2) [Corach-Porta-Recht].

M := A^{+} the positive invertible elements of a unital C^{∗}-algebra A. E := A with \{xyz\} := \frac{1}{2}xy^{∗}z + \frac{1}{2}zy^{∗}y as usually. U_{a} := M and X_{a}(b) := b − a for all a, b ∈ M. {uvw}_{a} = {uvw} (a ∈ M; u, v, w ∈ T_{a}M ≡ E). Then, for any invertible g ∈ A we have L_{g} := [a ↦ (g^{∗})^{−1}ag^{−1}] ∈ Aut(M). In particular M is homogeneous because {L_{g}(1) : e ∈ M} = {g^{−2} : e ∈ M} = M.

(E2′) [Corach-Porta-Recht_{2}]. A C^{∗}-algebra with unit 1, E := \{a ∈ A : a = a^{∗}\}, \{xyz\} := \frac{1}{2}xyz + \frac{1}{2}zyx M := {a ∈ A : a > 0} = {g^{∗}g : g invertible} L_{g} : M ↔ M, x ↦ g^{∗}xg linear S : p ↦ p^{−1} symmetry on M S_{g^{∗}g} := L_{g} ∘ S ∘ L_{g^{−1}} : p ↦ g^{∗}gp^{−1}g^{∗}g We can define

\{uvu\}_{g^{∗}g} : {= L_{g}\{(L_{g}^{−1}u)(L_{g}^{−1}u)(L_{g}^{−1}u)\} = \atop = u(g^{∗}g)^{−1}v(g^{∗}g)^{−1}u.}

Lemma. The transformations L_{g}, S_{p} are P-automorphisms.

Define X_{v} := \frac{d}{dτ} |_{τ=0+}S_{exp(τv/2)} ∘ S_{1}. Notice that exp(X_{v})p = exp(v/2)p exp(v/2).

Proposition. In the chart Y : [ exp(v − \{xvx\})\frac{∂}{∂x}]0 ↦ [exp X_{v}]1 the transformations exp(X_{w}) (w ∈ E) are not in general {…}-Möbius type. Proof. We would have Y : tanh v ↦ exp v that is Y (x) = exp (areath x) and Y^{−1}(y) = tanh log y. This implies

Y^{−1} ∘ [exp X_{w}] ∘ Y : x ↦^{Y}
exp(areath x) ↦^{exp X_{w}}
exp(w/2)[ exp(areath x)] exp(w/2) ↦^{Y^{−1}}

tanh [ log ( exp(w/2)[ exp(areath x)] exp(w/2))].

Such mappings are of Möbius type only in the commutative case.

(E3) [Chu-Isidro-Kaup-Stachó]. A case with equivalence classes of tripotents. H complex Hilbert space with inner product ⟨.|.⟩. S :=[unit sphere]. Triple product \{xyz\} := \frac{1}{2}⟨x|y⟩z + \frac{1}{2}⟨z|y⟩x (x, y, z ∈ H). E := e_{0}^{⊥} = {z ∈ H : ⟨z|e_{0}⟩ = 0 with a fixed e_{0} ∈ S. M := {TTe : e ∈ H, ⟨e|e⟩ = 1} with TT := {ζ ∈ C : |ζ| = 1}. For e ∈ S, let us write e^{∼} := TTe. T_{e^{∼}}M ≡ e^{⊥} = {z ∈ H : ⟨z|e⟩ = 0}. {uvw}_{e^{∼}} := −{uvw} ∈ T_{e^{∼}}M for u, v, w ∈ T_{e^{∼}}M. Given e ∈ S, let R_{e} be a rotation of H with R_{e}(e) = e_{0}. Φ_{e^{∼}}^{−1} : e ⊥ z ↦ [CR_{e}(e + z)] ∩ S (e ∈ S). Then X := {Φ_{e^{∼}} : e^{∼} ∈ M} = {Φ_{e^{∼}} : e ∈ S} is an atlas on M with holomorphic coordinate transitions. G^{∗} ⊂ Aut_{loc}(M) ∩ {X -holomorphic maps} is transitive on M. Aut(M) ∩ {X -holomorphic maps} = {Identity}.

Case (E2) Z := {z ∈ A : z^{∗} = z} self-adjoint elements. \{xyz\} := \frac{1}{2}xyz + \frac{1}{2}zyx (x, y, z ∈ Z). G := {g ∈ Z : g invertible}, M = A^{+} = {g^{∗}g : g ∈ G}. σ : a ↦ a^{−1} symmetry on M with fixed point 1. L_{g}(a) := g^{∗}ag (a ∈ M, g ∈ G); L_{g}^{−1} = L_{g^{−1}}, L_{g} : M ↔ M, 1 ↦ g^{∗}g. Definition. σ_{a} := L_{a^{1/2}} ∘ σ ∘ L_{a^{−1/2}} symmetry at e ∈ M. Remark. a = g^{∗}g ⇒ L_{g} ∘ σ ∘ L_{a^{−1}} = σ_{a} (g ∈ G). Definition. Infinitesimal translation on M with vector h ∈ Z ≡ T_{1}M at 1:

X_{h}(a) := \frac{d}{dτ} |_{τ=0}σ_{1+(τ/2)h} ∘ σ_{1}(a) (a ∈ M).

Lemma. We have

1) X_{h}(a) = \frac{1}{2}(ha + ah) = \{h1a\}, 2) X_{h}(a) = \frac{d}{dτ} |_{τ=0}σ_{exp(τh/2)} ∘ σ(a).

Proof. 1) σ_{1+(τh/2} ∘ σ_{1}(a) = = σ_{1+τh/2}(a^{−1}) = (1 + τh/2)^{1/2}[(1 + τh/2)^{−1/2}a^{−1}(1 + τh/2)^{−1/2}]^{−1}(1 + τh/2)^{1/2} = = (1 + τh/2)a(1 + τh/2).

2) exp(τh/2) = ∑_{n=0}^{∞} \frac{1}{2^{n}n!}τ^{n}h^{n} = 1 + τh/2 + o(τ).

Lemma. exp X_{h}(a) = [exp(h/2)]a[exp(h/2)].

Proof. The left and right multiplications Mult_{ℓ}^{h} : A ∋ x ↦ hx and Mult_{r}^{h} : A ∋ x ↦ hx commute. Observe that H_{h} = \frac{1}{2}[Mult_{ℓ}^{h} + Mult_{r}^{h}]. Therefore exp X_{2h}(a) = = exp[Mult_{ℓ}^{h} + Mult_{r}^{h}]a = exp[Mult_{ℓ}^{h}] exp[Mult_{r}^{h}]a = [Mult_{ℓ}^{exp h}][Mult_{ℓ}^{exp h}]a.

Remark. Since all elements from M = A^{+} admit a spectral resolution over some compact interval 0 < [α, β] < ∞, each element a ∈ M = A^{+} can be written as a = exp h with a unique self-adjoint h ∈ Z such that (log α) · 1 ≤ h ≤ (log β) · 1.

Definition. We write

Exp_{Z}(h) := exp(h) (h ∈ Z), Log_{M}a := Exp_{Z}^{−1}(a) (a ∈ M).

Let h ∈ Z and set a := exp(h) = Exp_{Z}(h) = exp(X_{h})1. We call the mapping

M_{a} := exp(X_{h}) = exp X_{Log_{M} a} : M ∋ x ↦ axa

the pseudo-Möbius transformation of the point a. Natural atlas on M:

Φ_{a} := M_{a} ∘ Log_{M} ∘ M_{a}^{−1} (a ∈ M).

Remark. M_{a}^{−1} = M_{a^{−1}}.

Our next aim is to calculate the image Φ_{1}^{#}X_{h} : Z → Z of the vector field in the atlas page of Φ_{1}. By definition,

[Φ_{1}^{#}X_{h}](z) := [Φ′_{1}(Φ_{1}^{−1}(z))]X_{h}(Φ_{1}^{−1}(z)) =

= \frac{d}{dτ} |_{τ=0}[Φ_{1}^{#}[σ_{exp(τh/2)} ∘ σ]](z) =

= \frac{d}{dτ} |_{τ=0}[Φ_{1}^{#}σ_{exp(τh/2)}] ∘ [Φ_{1}^{#}σ](z).

Notice that Φ_{1}^{#}σ(z) = −z. Therefore

[Φ_{1}^{#}X_{h}](z) = \frac{d}{dτ} |_{τ=0}[Φ_{1}^{#}σ_{exp(τh/2)}](−z) =

= \frac{d}{dτ} |_{τ=0}Log_{M}( exp (\frac{1}{2}τh) exp (z) exp (\frac{1}{2}τh)).

Let us fix z, h ∈ Z and define

w(τ) := Log_{M}( exp (\frac{1}{2}τh) exp (z) exp (\frac{1}{2}τh)).

Since w(0) = z trivially, we can write w(τ) = z + τb + o(τ) with a unique vector b ∈ Z. In terms of b, we have

\frac{d}{dτ} |_{τ=0} exp (w(τ)) = \frac{d}{dτ} |_{τ=0} exp (\frac{1}{2}τh) exp (z) exp (\frac{1}{2}τh).

On the right hand side we simply have

\frac{d}{dτ} |_{τ=0} exp (\frac{1}{2}τh) exp (z) exp (\frac{1}{2}τh) = \frac{1}{2}h exp(z) + \frac{1}{2} exp(z)h .

On the left hand side the derivation can be carried out with the power series of the exponential map termwise. That is

\frac{d}{dτ} |_{τ=0} exp (w(τ)) = ∑_{n=1}^{∞} \frac{1}{n!} \frac{d}{dτ} |_{τ=0}w(τ)^{n} = ∑_{n=1}^{∞} \frac{1}{n!} ∑_{k=1}^{n} z^{k−1}bz^{n−k} =

= ∑_{k,ℓ=0}^{∞} \frac{1}{(k + ℓ + 1)!}[Mult_{ℓ}^{z}]^{k}[Mult_{r}^{z}]^{ℓ}b .

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