The manifold of minimal tripotents in classical Cartan factors.

José M. Isidro^{1}

ABSTRACT. We prove that the manifold of minimal tripotents in classical Cartan factors is a fibre space with a symmetric Kähler manifold as base space and U(1) as typical fibre. We study the symmetries, the geodesics and the sectional curvature of the base manifold. This extends to the infinite dimensional case known results in the C^{n} setting.

0. Introduction

In [7], U. Hirzebruch proved that the manifold of minimal projections in a finite-dimensional simple formally real Jordan algebra is a compact Riemann symmetric space of rank 1, and that any such space arises in this way. Later on, in [13] Nomura established similar results for the manifold of minimal projections in a topologically simple real Jordan-Hilbert algebra.

In this paper we are concern with the Differential Geometry of certain manifolds in L(H, K), the space of all bounded linear operators z: H → K, where H and K are complex Hilbert spaces. More precisely, we shall consider certain subspaces Z ⊂ L(H, K) known as classical Cartan factors and the manifold consisting of the operators z ∈ Z, called the tripotents of Z, that satisfy the algebraic equation zz^{∗}z = z and a certain minimality condition. The classical Cartan factors are J^{∗}-algebras of operators (see [2], [4]), that is, norm closed complex subspaces Z ⊂ L(H, K) that are closed under the operation of taking the triple product zz^{∗}z for z ∈ Z. These spaces play a important role in the problem of the analytical classification of bounded symmetric domains in complex Banach spaces (see [3],[9],[8]). And the minimal tripotents are precisely the minimal partial isometries operators e: H → K that belong to Z. As one could expect, within this framework the algebraic structure of J^{∗}-algebra acts as a substitute for the Jordan algebra structure in the above papers of Hirzebruch and Nomura. However, the J^{∗}-algebra structure also provides us naturally with a local scalar product which, in turn, allows us to represent Min(Z), the manifold of minimal tripotents, as a fibre space with a suitable symmetric Kähler manifold as base space and U(1) as typical fibre. This extends to the infinite dimensional case well known results in the C^{n} case.

The paper is organized in this way: In the first section we collect the material about J^{∗}-algebras (and more generally JB^{∗}-triples) that is needed later. In section 2 we discuss the manifold Tri(Z) of tripotents in arbitrary JB^{∗}-triples. With the help of purely algebraic tools such as the Peirce projectors of a JB^{∗}-triple, we define an invariant torsionfree affine connection on Tri(Z) and discuss its geodesics. Different elements in Tri(Z) may have the same Peirce projectors, therefore it is reasonable to consider them as equivalent. The quotient set turns out to be a holomorphic manifold naturally endowed with an invariant torsionfree affine connection that is studied in section 3. In section 4 we discuss the manifold Min(Z) of minimal tripotents in a JB^{∗}- triple. The minimality condition allows us to consider the Levi form, which is a scalar product in the tangent space to Min(Z). If the tangent space is complete with respect to this scalar product, then Min(Z) becomes a holomorphic Riemann manifold, which is in fact a symmetric Kähler manifold. In section 5 we prove that for the special Cartan factors, the completeness assumption of the Levi norm is satisfied.

1. JB^{∗}-triples and tripotents.

For a complex Banach space Z, denote by L(Z) the Banach algebra of all bounded linear operators on Z. A complex Banach space Z with a continuous mapping (a, b, c) ↦ {abc} from

1 Supported by Comisión Hispano- Húngara de Cooperación Científica y Tecnológica

Z × Z × Z to Z is called a JB*-triple if the following conditions are satisfied for all a, b, c, d ∈ Z, where the operator a b ∈ L(Z) is defined by z ↦ {abz} and [ , ] is the commutator product:

1 {abc} is symmetric complex linear in a, c and conjugate linear in b.

2 [a b, c d] = {abc} d − c {dab}.

3 a a is hermitian and has spectrum ≥ 0.

4 ∥{aaa}∥ = ∥a∥^{3}.

If a complex vector space Z admits a JB*-triple structure, then the norm and the triple product determine each other. A derivation of a JB*-triple Z is an element δ ∈ L(Z) such that δ{zzz} = {(δz)zz}+{z(δz)z}+{zz(δz)} and an automorphism is a bijection φ ∈ L(Z) such that φ{zzz} = {(φz)(φz)(φz)} for z ∈ Z. The latter occurs if and only if φ is a surjective linear isometry of Z. The group Aut(Z) of automorphisms of Z is a real Banach-Lie group whose Banach-Lie algebra is the set of derivations of Z (see [5]). The connected component of the identity in Aut(Z) is denoted by Aut^{∘}(Z). Two elements x, y ∈ Z are orthogonal if x y = 0. An element e ∈ Z is called a tripotent if {eee} = e. The set Tri(Z) of tripotents is endowed with the induced topology of Z. If e ∈ Tri(Z), then e e ∈ L(Z) has the eigenvalues 0, \frac{1}{2}, 1 and we have the topological direct sum decomposition

Z = Z_{1}(e) ⊕ Z_{1/2}(e) ⊕ Z_{0}(e)

called the Peirce decomposition of Z. Here Z_{k}(e) is the k- eigenspace and the Peirce projections are

P_{1}(e) = Q^{2}(e),  P_{1/2}(e) = 2(e e − Q^{2}(e)),  P_{0}(e) = Id − 2e e + Q^{2}(e),

where Q(e)z = {eze} for z ∈ Z. We will use the Peirce rules {Z_{i}(e) Z_{j}(e) Z_{k}(e)} ⊂ Z_{i−j+k}(e) where Z_{l}(e) = {0} for l ≠ 0, 1/2, 1. We note that Z_{1}(e) is a complex unital JB*-algebra in the product a ∘ b: = {aeb} and involution a^{#}: = {eae}. Let

A(e): = {z ∈ Z_{1}(e) : z^{#} = z}.

Then we have Z_{1}(e) = A(e) ⊕ iA(e). JB*-triples include C*-algebras and JB*-algebras. A C*- algebra is a JB*-triple with respect to the triple product 2{abc}: = (ab^{∗}c + cb^{∗}a). Every JB*- algebra with Jordan product (a, b) ↦ a ∘ b and involution a ↦ a^{∗} is a JB*-triple with triple product {abc} = (a ∘ b^{∗}) ∘ c − (c ∘ a) ∘ b^{∗} + (b^{∗} ∘ c) ∘ a.

A tripotent e in a JB^{∗}-triple Z is said to be minimal if P_{1}(e)Z = Ce, and we let Min(Z) be the set of them. Clearly e = 0 lies in Min(Z) and is an isolated point there. If e ∈ Min(Z) and e ≠ 0 then ∥e∥ = 1 and by the Peirce multiplication rules we have {euv} ∈ Z_{1}(e) = Ce for all u, v ∈ Z_{1/2}(e). Therefore we can define a sesquilinear form, called the Levi form, ⟨·, ·⟩_{e}: Z_{1/2}(e) × Z_{1/2}(e) → C by

{euv} = ⟨v, u⟩_{e} e,  u, v ∈ Z_{1/2}(e).  (1)

It is known [10] that ⟨·, ·⟩_{e} is positive definite hence it defines a scalar product in Z_{1/2}(e) whose norm, called the Levi norm and denoted by | · |_{e}, satisfies

|u|_{e}^{2} ≤ ∥u∥^{2},  u ∈ Z_{1/2}(e)  (2)

that is, we have the continuous inclusion (Z_{1/2}(e), ∥ · ∥) ↪ (Z_{1/2}(e), | · |_{e}). We do not know whether (Z_{1/2}(e), | · |_{e}) is complete, but in case of completeness the open mapping theorem yields that the norms | · |_{e} and ∥ · ∥ are equivalent in Z_{1/2}(e). We shall see below that the completeness assumption is satisfied whenever Z is a special Cartan factor. To simplify the notation, we shall omit the subindex e in both the Levi form and the Levi norm if no confusion is likely to occur.

2. Manifolds of tripotents in JB^{∗}-triples

Let Z be a JB^{∗}-triple and Tri(Z) the set of all tripotents in Z endowed with its topology as a subspace of Z. Fix any non zero tripotent e_{0} ∈ Tri(Z) and denote by M the connected component of e_{0} in Tri(Z). Then all tripotents e ∈ M have the same rank as e_{0} and Aut^{∘}(Z) acts transitively on M which is an Aut^{∘}(Z)-invariant real analytic manifold whose tangent space at a point e is

T_{e}M = i A(e) ⊕ Z_{1/2}(e).

For z = iv + u ∈ i A(e) ⊕ Z_{1/2}(e) we set w: = \frac{i}{2}v + 2u and k_{z}: = w e − e w. Then ([15] p. 25) a local chart of M at e in a suitable neighbourhood V × U of (0, 0) in iA(e) × Z_{1/2}(e) is given by

z ↦ f(z): = exp k_{z}(e)  (3).

As in [1], we define a projector E: Z → i A(e) by E(e): = \frac{1}{2}(Q^{2}(e) − Q(e)). Note that for z ∈ Z one has Q^{2}(e)z ∈ Z_{1}(e) and Q(e)z = (Q^{2}(e)z)^{#}, hence E(e)z ∈ i A(e), and E(e) is the projector onto the iA(e)-subspace. In particular, E(e) is Aut^{∘}(Z)-invariant

E(g(e))g(z) = gE(e)z,  g ∈ Aut^{∘}(Z),  z ∈ Z.

Let 𝔇(M) be the Lie algebra of all real analytic vector fields on M and define an affine connection ∇ on M by

(∇_{X} Y)_{e}: = (E(e) + P_{1/2}(e))(Y′_{e}X_{e}),  e ∈ M,  X, Y ∈ 𝔇(M).

Then ∇ is a torsionfree Aut^{∘}(Z)-invariant affine connection on M.

For each e ∈ M and z = iv + u ∈ i A(e) ⊕ Z_{1/2}(e) we let γ_{e,z}: IR → M denote the curve γ_{e,z}(t): = exp tk_{z}(e). Clearly we have γ_{e,z}(0) = e and γ̇_{e,z}(0) = z ∈ T_{e}M. By [1] lemma 2.6 we have E(e)k_{z}(z) = 0 for all z ∈ T_{e}M. And by [1] th. 2.7, if e has finite rank then γ_{e,z} is a ∇- geodesic if and only if either v = 0 or u = 0.

2.1 Theorem. Let Z , M and e ∈ M be as above. Let z ∈ T_{e}M with z = iv+u ∈ iA(e)⊕Z_{1/2}(e). Then the projector E satisfies

E(γ_{e,z}(t))γ̇_{e,z}(t) = 0,  E(γ_{e,z}(t))γ̈_{e,z}(t) = 0,  t ∈ IR.

Proof. To simplify notation, we write g_{t}: = exp tk_{z} ∈ Aut^{∘}(Z). Then γ(t) = g_{t}(e) and we have

\begin{aligned}γ̇(t) & = (exp tk_{z})k_{z}(e) = g_{t}k_{z}(e) = g_{t}(z), \\ γ̈(t) & = (exp tk_{z})k_{z}^{2}(e) = g_{t}k_{z}^{2}(e) = g_{t}k_{z}(z). \end{aligned}

and using the Aut^{∘}(Z)-invariance of E(e) and the above mentioned property E(e)k_{z}(z) = 0, we get

\begin{aligned}E(γ(t))γ̇(t) & = E(g_{t}(e))g_{t}(z) = g_{t}E(e)z ∈ g_{t}E(e)Z_{1/2}(e) = 0 \\ E(γ(t))γ̈(t) & = E(g_{t}(e))g_{t}(k_{z}(z)) = g_{t}E(e)k_{z}(z) = 0. \end{aligned}

as we wanted to see.

3. The quotient manifold

Let Z be a JB^{∗}-triple and Tri(Z) the set of all tripotents in Z endowed with its topology as a subspace of Z. Denote by M the connected component of a fixed e_{0} ∈ Tri(Z).

3.1 Definition. For e, e′ ∈ M we set e ∼ e′ if and only if e and e′ have the same k-Peirce projectors for k = 0, 1/2, 1.

This notion extends to tripotents the classical relation of unitary equivalence between projections. Indeed, if e, e′ are two partial isometries in Z = L(H), then e ∼ e′ means that their initial projections p = e^{∗}e and p′ = e^{′∗}e′ are unitarily equivalent and that so are their final projections q = ee^{∗}, q′ = e′e^{′∗}. We let M: = M/_{∼} be the quotient space, π: M → M is the canonical map and e: = πe is the class of e ∈ M. Note that equivalent elements have the same k-Peirce subspaces for k ∈ {0, 1/2, 1}, which allows us to define Z_{k}(e) and P_{k}(e) in an obvious manner.

It has been proved in [10] cor. 4.2 that M is a real analytic manifold whose tangent space at e = πe is T_{e}M = Z_{1/2}(e), a local chart being given in a neighbourhood U of 0 ∈ Z_{1/2}(e) by

ψ: u ↦ ψ(u): = exp k_{u}(e),  u ∈ U.  (4)

Note that Z_{1/2}(e) inherits a complex structure from the JB^{∗}-triple Z and that we have by the Peirce multiplication rules

ψ′(0)h = 2(h e − e h)e = 2{hee} = h, h ∈ Z_{1/2}(e),

that is ψ′(0) = Id . Hence, if ψ′(0) = ∂ψ(0)+\overline{∂}ψ(0) is the decomposition of ψ′(0) into its complexlinear and complex-conjugate linear parts, we have \overline{∂}ψ(0)_{|Z_{1/2}(e)} = 0 and M is a holomorphic manifold. Note that M is Aut^{∘}(Z)-invariant. It was also proved in [10] that the local expression of the differential map dπ_{e}: T_{e}M → T_{e}M with respect to the charts (3) and (4) is the identity map on Z_{1/2}(e) ⊂ T_{e}M

dπ_{e}(0, u) = u,  u ∈ Z_{1/2}(e),  e ∈ M.

Suppose now that the JB^{∗}-triple Z is a J^{∗}-algebra of operators Z ⊂ L(H, K). Then each tripotent e ∈ Tri(Z) is a partial isometry operator e: H → K, and we can consider its initial and final projections p: = e^{∗}e ∈ P(H) and q: = ee^{∗} ∈ P(K) respectively. Note that all tripotents in the same class e have the same initial and the same final projection, therefore we have a mapping

Φ: M → P(H) × P(K)  e ↦ (e^{∗}e, ee^{∗})  (∗)

By [1], th.3.3, both P(H) and P(K) are real analytic manifolds in the local charts

\begin{matrix}x ∈ Z_{1/2}(p)_{s} ↦ f_{1}(x): = exp k_{x}(p) \\ y ∈ Z_{1/2}(q)_{s} ↦ f_{2}(y): = exp k_{y}(q) \end{matrix}

where Z_{1/2}(p)_{s} and Z_{1/2}(q)_{s} stand for self-adjoint part of the corresponding Z_{1/2} spaces. And we have the following 3.2 Theorem. Let Z be a J^{∗}-algebra, and let e ∈ Tri(Z) and M be a tripotent and the corresponding quotient manifold. Then the mapping (∗) is a real analytic diffeomorphism. Proof. Consider the diagram

M  ⟶^{Φ}  P(H) × P(K)
π ∘ exp↑  ↑exp
Z_{1/2}(e) ⟶^{F} Z_{1/2}(p)_{s} × Z_{1/2}(q)_{s}.

For z close to the origin in Z_{1/2}(e), all the tripotents f(z): = exp k_{z}(e) are partial isometries whose initial and final projections are (F_{1}(z), F_{2}(z)) = (f(z)^{∗}f(z), f(z)f(z)^{∗}). Write log: = exp^{−1}. Then the expression of Φ with respect to the chosen local charts is

Φ(z) = (log F_{1}(z), log F_{2}(z)).

Since (F_{1}(0), F_{2}(0)) = (p, q), we have Φ(0) = (log p, log q) = (0, 0). Also for h ∈ Z_{1/2}(e) we have

\begin{aligned}f′(0)h & = \frac{d}{dz}_{|_{z=0}}(exp k_{z}(e))h = \frac{d}{dz}_{|_{z=0}}k_{z}(e)h \\ & = 2(h e − e h)e = 2\{hee\} − 2\{ehe\} = h \end{aligned}

that is f′(0) = Id in Z_{1/2}(e) hence log′(0) = Id . Besides,

\begin{aligned}F′_{1}(0)h & = f(0)^{∗}f′(0)h + (f′(0)h)^{∗}f(0) = e^{∗}h + h^{∗}e \\ F′_{2}(0)h & = f(0)(f′(0)h)^{∗} + (f′(0)h)f(0)^{∗} = eh^{∗} + he^{∗}. \end{aligned}

The chain rule yields Φ′(0)h = (e^{∗}h + h^{∗}e, eh^{∗} + he^{∗}) A little calculus gives that

Φ′(0): Z_{1/2}(e) → Z_{1/2}(p)_{s} × Z_{1/2}(q)_{s}

is a bijection, whence by the inverse mapping theorem Φ is a diffeomorphism.

Recall that a vector field X on M is a map X: M → T M, thus X_{e}, the value of X at e ∈ M, satisfies X_{e} ∈ T_{e}M ≈ Z_{1/2}(e). We let 𝔇(M) be the Lie algebra of holomorphic vector fields on M. Let Y′_{e} be the Fréchet derivative of Y ∈ 𝔇(M) at e. Then Y′_{e}X_{e} ∈ Z_{1/2}(e) ⊂ Z and by the above, it makes sense to take the projection P_{1/2}(e)Y′_{e}X_{e} ∈ Z_{1/2}(e) ≈ T_{e}M. 3.3 Definition. We define a connection ∇ on M by

(∇_{X}Y)_{e}: = P_{1/2}(e)Y′_{e}X_{e}, X, Y ∈ 𝔇(M).

It is a matter of routine to check that ∇ is an affine connection on M, that it is Aut^{∘}(E)- invariant and torsion-free, i. e.,

T (X, Y): = ∇_{X}Y − ∇_{Y}X − [XY] = 0, X, Y ∈ 𝔇(M).

3.4 Theorem. Let Z, M and M respectively be a JB^{∗}-triple, a connected component of the manifold of tripotents and the associated quotient manifold. Then the geodesics of the connection ∇ in M are the curves g(t): = π exp tk_{u}(e), t ∈ IR, where e ∈ M and u ∈ Z_{1/2}(e) ⊂ T_{e}M. Proof. Let e ∈ M and u ∈ Z_{1/2}(e) be given. We have γ̇(t) ∈ T_{γ(t)}M = i A[γ(t)] ⊕ Z_{1/2}[γ(t)]. By (2.1) the velocity γ̇(t) and the acceleration γ̈(t) have null projection onto the subspace i A(γ(t)). As previously stated dπ|_{Z_{1/2}[γ(t)]} = Id and therefore d^{2}π|_{Z_{1/2}[γ(t)]} = 0. Thus

ġ(t) = dπ_{g(t)}γ̇(t) = γ̇(t).

Moreover

d^{2}π_{g(t)}(γ̇(t), dπ_{g(t)}γ̇(t)) ∈ d^{2}π_{g(t)}(Z_{1/2}γ(t), Z_{1/2}γ(t)) = 0.

But then

g̈(t) = d^{2}π_{g(t)}(γ̇(t), dπ_{g(t)}γ̇(t)) + dπ_{g(t)}γ̈(t) = dπ_{g(t)}γ̈(t) = γ̈(t).

Since γ satisfies (∇γ̇ γ̇)_{t} = 0, we have

(∇ġ ġ)_{t}: = P_{1/2}[g(t)] g̈(t) = P_{1/2}[γ(t)] γ̈(t) = (∇γ̇ γ̇)_{t} = 0,

which shows that g is a geodesic for the connection in M.

Conversely, let g be a geodesic in M and set e: = g(0), u: = ġ(0) ∈ T_{e}M. Take any representative e ∈ g(0) and define a curve γ: IR → M by γ(t): = exp tk_{u}(e). Then γ is a geodesic in M and clearly γ(0) = e ∈ e, γ̇(0) = u ∈ Z_{1/2}(e). By the previous argument, the curve x(t): = πγ(t) is a geodesic in M which has the same initial conditions as the given geodesic g. Since geodesics are uniquely determined by their initial conditions, x = g.

Recall that for a tripotent e, the Peirce reflection on Z with respect to e is the involution s_{e}: = Id − 2P_{1/2}(e). Clearly s_{e}(e) = e and s_{e}M = M for any e ∈ M. Besides s_{e} is compatible with the equivalence ∼, that is f ∼ f′ implies s_{e}(f) ∼ s_{e}(f′). Hence we can define an involution (in fact a holomorphic symmetry) on M by s_{e}(f): = πs_{e}(f). Every geodesic g_{e,u} is invariant under s_{e} and geodesically symmetric points are s_{e}-symmetric.

4. Manifolds of minimal tripotents.

In this section we shall consider a JB^{∗}-triple Z, the manifold Min(Z) ⊂ Tri(Z) of all minimal tripotents in Z and the connected component M of a fixed element e ∈ Min(Z). Via the Levi form, the minimality condition will allow us to introduce a Kähler manifold structure in the quotient M.

Fix a point e ∈ M and consider the Levi form ⟨·, ·⟩ on Z_{1/2}(e). Note that all tripotents in the same class e have the same Levi form, that ⟨·, ·⟩ is Aut^{∘}(Z)-invariant and that the Levi norm satisfies |u| ≤ ∥u∥, u ∈ Z_{1/2}(e). In case (Z_{1/2}(e), | · |) is complete the open mapping theorem yields that the norms | · | and ∥ · ∥ are equivalent in Z_{1/2}(e). We shall see below that the completeness assumption is satisfied whenever Z is a complex special Cartan factor. In case of completeness, M is a Riemann manifold in the metric

g_{e}(X, Y): = ⟨X_{e}, Y_{e}⟩, X, Y ∈ 𝔇(M), e ∈ M.

Remark that g is hermitian, i.e. we have g_{e}(iX, iY) = g_{e}(X, Y), and that it has been defined in algebraic terms, hence it is Aut^{∘}(Z) -invariant. Moreover, ∇ is compatible with the Riemann structure, i. e.

Xg(Y, W) = g(∇_{X}Y, W) + g(Y, ∇_{X}W), X, Y, W ∈ 𝔇(M).

Therefore ∇ is the only Levi-Civita connection on M. On the other hand ∇ is hermitian, i. e.,

∇_{X}(iY) = i∇_{X}Y X, Y ∈ 𝔇(M),

hence it is the only hermitian connection on M. Thus the Levi-Civita and the hermitian connection are the same in this case, and so ∇ is the Kähler connection on M.

Now we compute the sectional curvature of M. Fix any e ∈ M and let us temporarily simplify the notation by setting P: = P_{1/2}(e): Z → Z_{1/2}(e) ⊂ Z. Thus the Fréchet derivative of P at e, denoted by P′(e), is an element of L(Z, L(Z, Z)) which will be identified in the usual way to B(Z × Z, Z), the space of continuous Z-valued bilinear mappings. Recall that the Ricci tensor of ∇ is defined by

R(X, Y)W = ∇_{X} ∇_{Y} W − ∇_{Y} ∇_{X} W − ∇_{[X, Y]}W, X, Y, W ∈ 𝔇(M).

As in [2] p.55, where only differential calculus and properties of the Peirce projectors are used, we have

[R(X, Y) W ]_{e} = P′(e)(X_{e}, P′(e)(Y_{e}, W_{e})) − P′(e)(Y_{e}, P′(e)(X_{e}, W_{e}))

where P′(e) ∈ B(Z × Z, Z). Thus the value of the Ricci tensor at e ∈ M depends on the values X_{e}, Y_{e}, W_{e}, but not on the vector fields X, Y, W. So for x, y ∈ T_{e}M ≈ Z_{1/2}(e), we define a bounded linear operator R_{e}(x, y): Z_{1/2}(e) → Z_{1/2}(e) by

R_{e}(x, y)w: = P′(e)(x, P′(e)(y, w)) − P′(e)(y, P′(e)(x, w)), w ∈ Z_{1/2}(e). (5)

We shall now compute P′(e)(x, u) for x ∈ Z_{1/2}(e) and u ∈ Z. Consider a smooth curve c: t ↦ c(t) ∈ M with c(0) = e and ċ(0) = x, and the 1/2-Peirce projector P (t) = 2(c(t) c(t)−Q^{2}(c(t))). Taking the Fréchet derivative with respect to t at t = 0, we get

\frac{1}{2}P′(e)(x, u) = (x e + e x)u − 2\{x\{eue\}e\} − 2\{e\{xue\}e\}.

Using the multiplication rules and the expression of the scalar product in Z_{1/2}(e), we obtain

for u ∈ Z_{1}(e)  P′(e)(x, u) = −λx where u = λe
for u ∈ Z_{1/2}(e)  P′(e)(x, u) = 2{xeu} − 2⟨u, x⟩e
for u ∈ Z_{0}(e)  P′(e)(x, u) = 2{exu}

which replaced in (5) yields after some calculus

R_{e}(x, y)w = ⟨w, y⟩x − ⟨w, x⟩y + {xyw} − {yxw}.

In particular, for x ⊥ y and |x| = |y| = 1 and w = y (orthogonality in the Levi sense) we get the sectional curvature of M

κ_{e}(x, y): = ⟨R_{e}(x, y)y, x⟩ = 1 + 2⟨(x y − y x)y, x⟩.

Using the Schwarz inequality one can easily obtain lower and upper bounds for κ_{e}.

5. Minimal tripotents in special Cartan factors

In this section we prove that, whenever Z is a special Cartan factor and e ∈ Min(Z), the Levi norm in Z_{1/2}(e) is complete. Recall that the special Cartan factors are J^{∗}-algebras of operators Z ⊂ L(H, K) where H and K are complex Hilbert spaces with dim H ≤ dim K. Tripotents in a special Cartan factor Z are precisely the partial isometries in Z and except for a complex number λ ∈ 𝕋, the non zero minimal tripotents are the operators (see [3] for details)

type I type II type III type IV

e = (·, α)β e = (·, α)\overline{α} e = (·, α)\overline{β} − (·, β)\overline{α} e^{2} = 0

where α ∈ H, β ∈ K, ∥α∥ = ∥β∥ = 1
where α ∈ H, ∥α∥ = 1
where α, β ∈ H, α ⊥ β, ∥α∥ = ∥β∥ = 1
where e ∈ Z, ∥e∥ = 1

5.1 Theorem. Let Z and e respectively be a special complex Cartan factor and a non zero minimal tripotent in Z. Then the operator norm and the Levi norm on Z_{1/2}(e) are equivalent; in particular, (Z_{1/2}(e), | · |_{e}) is a Hilbert space. Proof. We make a type by type discussion. Since the proof is essentially the same in all cases, we only consider the type I factors. Remark that for λ ∈ 𝕋, e and λe have the same Peirce projectors hence the same Peirce subspaces; thus in the above expression of the minimal tripotents we may assume λ = 1. For any operator u ∈ Z = L(H, K) we have u ∈ Z_{1/2}(e) if and only if u = 2{eeu} = ee^{∗}u + ue^{∗}e and an elementary calculation shows that in that case u has the form

u = (·, u^{∗}β)β + (·, α)uα, α ⊥ u^{∗}β, β ⊥ uα.

Recall from the definition of the Levi form that we have {euu} = |u|^{2}e. Replacing here the above expression of u and making an elementary computation we get 2{euu} = eu^{∗}u + uu^{∗}e = (∥uα∥^{2} + ∥u^{∗}β∥^{2}) e. Hence 2|u|^{2} = ∥uα∥^{2} + ∥u^{∗}β∥^{2}. But then, using the relation β ⊥ uα we derive

∥u∥^{2} = sup_{∥ξ∥≤1} ∥uξ∥^{2} = sup_{∥ξ∥≤1} ∥(ξ, u^{∗}β)β + (ξ, α)uα∥^{2} =

sup_{∥ξ∥≤1} [|(ξ, u^{∗}β)|^{2}∥β∥^{2} + |(ξ, α)|^{2}∥uα∥^{2}] ≤ ∥u^{∗}β∥^{2} + ∥uα∥^{2} = 2|u|^{2}.

Thus ∥u∥ ≤ \sqrt{2}|u| and so \frac{1}{\sqrt{2}}∥u∥ ≤ |u| ≤ ∥u∥ for all u ∈ Z_{1/2}(e).

Using the above formulas for the minimal tripotents in a special Cartan factor Z and making a type by type discussion, it is easy to show that two tripotents e, e′ ∈ Min(Z) satisfy e ∼ e′ if and only if e′ ∈ e𝕋. Therefore Min(Z) is a fibre space, each fibre being isomorphic to U(1).

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José M. Isidro Facultad de Matemáticas Universidad de Santiago 15706 Santiago de Compostela, Spain. jmisidro@zmat.usc.es