As usuallyN,Z,Q,R,Care the standard notations for the sets of natural numbers, integers, rational numbers, reals and complex numbers, and we shall write D :={ζ ∈C : |ζ| <1} and T:=∂D={κ∈C: |κ|= 1}for the open unit disc and the unit circle, respectively.
Throughout the whole workE denotes a complex Banach space with norm∥.∥and open unit ball B(E) := {x ∈ E : ∥x∥ < 1}, furthermore we shall write SpanS for the closed linear hull of any setS ⊂E. In the sequelDwill stand for an arbitrarily given domain (open connected subset) in E. A continuous mapping f :D→ Ee into another complex Banach space is holomorphic if all its directional derivatives fv′(a) := limζ→0ζ−1[
f(a+ζv)−f(a)] exist in complex sense. In this case all the iterated directional derivatives fv′ ··· ′1···vn(a) exist as well, and each mapping
f(n)(a)v1· · ·vn:=fv′ ··· ′
1···vn(a)
is a bounded symmetricn-linear form En→Ee and all the expansions f(a+v) :=
∑∞ n=0
1
n!f(n)(a)vn (a ∈D, v∈E)
converge uniformly on the balls{x∈D: ∥x−a∥< ϱ}with radiusϱ <supy∈D∥y−a∥. A holomorphic map isbiholomorphicif it is injective with open range and holomorphic inverse.
We regard the classDof all Banach space domains as a category where, forD1, D2∈D, the family of all morphismD1→D2 coincides with Hol(D1,D2) :={
holomorphic mapsD1→D2
}. Accordingly the holomorphic automorphismsof the domain D form the set
Aut(D) :={
biholomorphic maps D↔D} . We shall write dD for the Carath´eodory pseudodistance onD that is
dD(p, q) := sup{
arth|ϕ(q)|: ϕ∈Hol(U,D), ϕ(p) = 0}
(p, q ∈D).
Recall [38, 102] that (D, dD) is a complete metric space whenever the domainDis bounded and we have the universal contraction property dD2(
f(p), f(q))
≤ dD1(p, q) (p, q ∈ D1) if D1, D2 ∈D and f ∈Hol(D1, D2).
Except for Section 2.3, vector fields appear only in the context of Banach space domains.
In this trivial case one could identify a vector field defined on any subset S ∈ E with a function S → E. Nevertheless we apply the notation and interpretation followed by the majority in our references: given a point pin the domain D⊂E, the set of the vectors at p has the form TpD = {
v ∂/∂z|z=p : v ∈ E}
with the directional derivations v∂/∂z|z=p : ψ 7→ψ′(p)v defined on the class of all holomorphic functions mapping some neighborhood of p into some complex Banach space. Any holomorphic vector field on D is represented with a unique function f ∈ Hol(D, E) in the form f(z)∂/∂z : D∋p 7→ f(p)∂/∂z|z=p. In particular the Poisson product has the form
[f(z)∂/∂z, g(z)∂/∂z]
=(
f′(z)g(z)−g′(z)f(z))
∂/∂z.
A curvex:I →Ddefined on an open real interval around 0 is called anintegral curveof the vector field X :=w(z)∂/∂z starting from the pointp∈D, ifx(0) =p andx′(t) =w(
x(t)) (t ∈ I). It is a standard consequence of the Piccard-Lindel¨of Theorem that, given a holomorphic vector fieldXonDalong with any pointp, there is a unique maximal integral curve of X starting from p. For computational reasons (see e.g. [38]) the point with parametert∈Rof this maximal integral curve is denoted by [exp(tX)](p). The flow phase mapping exp(tX) is called the t-exponential of the vector field X. In general, the vector field X is semicompletein a setS ⊂D if for each point p∈S the maximal integral curves of X starting from p are defied on the whole R+. Analogously X is complete in S if its maximal integral curves starting from S are defined on the whole real lineR. We shall use the standard notation
aut(D) := {
complete holomorphic vector fields on D} .
Notice that w(z)∂/∂z ∈ aut(D) if and only if the initial value problems x′(t) = w(x(t)) with x(0) = p admit solution on the whole real line with any starting point p ∈ D, and {exp(X) : X ∈aut(D)} ⊂ Aut(D). Moreover, in case D is bounded, Aut(D) can be equipped with a topology (called Upmeier’s topology) making it a Banach-Lie group whose Lie algebra is aut(D) with the Poisson product and the locally uniform convergence.
For the sake of being more self-contained, we recall the familiar basic geometric concepts giving rise to the Jordan structures studied throughout this work. We say that the domain D is circular resp. balanced if TD =D resp. DD = D (i.e. {κx : |κ|= 1, x ∈ D} =D resp. {ζx: |ζ| ≤1, x ∈D}=D). The domainDissymmetricif for any pointp∈Dthere is a mappingσp ∈Aut(D) such thatσp(p) = pand σ′p(p) = −Id. In case Dis bounded,σp is uniquely determined by Cartan’s Uniqueness Theorem and we call it the symmetryofD throughp. Any bounded symmetric domain is biholomorphically equivalent to a bounded circular domain, namely given any point o in a bounded symmetric domain D⊂E, there is a unique biholomorphic mapping Φo : D ↔ Db (called the Harish-Chandra realization of D with origin o) onto a bounded necessarily symmetric balanced domain Db ⊂ E such that Φo(o) = 0 and Φ′o(o) = Id (see e.g. [38]). We know already that this Harish-Chandra realization is necessarily convex in every case. In finite dimensions this fact was a (by far not trivial) byproduct of the Lie-algebraic classification, which could not be extended to
Banach space setting. Jordan algebraic arguments were used to establish it in 1983 (Kaup [51]) in full generality. In Chapter 4 we present an alternative proof based upon arguments from my paper [80] from 1990 applying to general bounded balanced domains. On the other hand, it is also of fundamental importance that, by a theorem of Kaup [48, 54], in any bounded circular domain D⊂E, the orbit Aut(D)0(:={ψ(0) : ψ ∈Aut(D)}) of the origin by its automorphism group is a linear piece ofD, namely there is a (trivially unique) closed complex subspaceE0 ⊂Esuch that Aut(D)0 =D∩E0. The setD∩E0 is necessarily a bounded balanced symmetric domain in E0. We shall call it the symmetric part of D with the notation Sym(D). It is worth to note [55] that actually Sym(D) coincides with the family {
a∈D: ∃S∈Aut(D) S(a) =a, S′(a) =−id}
of all points of symmetries of D.
The tuple (E, E0,{..∗.}) is a partial Jordan*-triple (partial J*-triple for short) if E0 is a closed complex subspace of E and {..∗.}:E×E0×E →E is a continuous real trilinear operation with the property that{xa∗x}is symmetric complex-linear in the outer variables x, y, conjugate-linear in the inner variable a and satisfies the Jordan identity
(J) {ab∗{xc∗y}}={{ab∗x}c∗y} − {x{ba∗c}∗y}+{xc∗{ab∗y}}.
Partial J*-triples and bounded circular domains are closely related: given a bounded cir-cular domain D⊂E, there is a (necessarily unique) partial J*-triple (E, ED,{..∗.})D such that
aut(D) = {
[a− {za∗z}+ℓz]∂/∂z : a∈ED, ℓ∈ L(E), exp(iτ ℓ)D=D (τ ∈R)} In particular above domain D is symmetric if and only if E = ED and in this case D is necessarily convex thus the it can be regarded as the unit ball of some equivalent norm on E. Partial J*-triples with triple product defined on the whole underlying space in all the three variables are simply called J*-triples. Complex Banach spaces with symmetric unit ball are called JB*-triples. The category of JB*-triples (including among others C*-algebras) is axiomatized in terms of J*-triple products completely as follows. A complex Banach space E equipped with an operation {..∗.} : E3 → E is a JB*-triple if for all a, x, y ∈E we have
(JB∗1) {xa∗y} is symmetric bilinear in x, y and conjugate-linear in a, (JB∗2) ∥{aa∗a}∥=∥a∥3,
(JB∗3) exp(
iτ a a∗)= 1 for τ ∈R, (JB∗4) ia a∗ ∈Der(E,{..∗.}),
(JB∗5) Sp( a a∗)
≥0
where Der(E,{..∗.}) stands for the family of all continuous derivations2 of the product {..∗.} and Sp is the customary abbreviation forspectrum.
2In general, by a derivation of a real multilinear operationm:En →Ewe mean a real-linear mapping δ:E→E satisfying the identityδm(x1, . . . , xn) =∑n
k=1m(x1, . . . , xk−1, δxk, xk+1, . . . , xn).
Chapter 2
A projection principle
2.1 Holomorphic projections
Recall that, given a subset S ∈D, a mappingP :D→D is called a projection of D onto a S if ran(P)(
:=P(D))
=S and P|S = Id (i.e. P(x) =x for x∈S).
Proposition 2.1.1. Let P be a holomorphic projection of D with ran(D) = S. Then given any holomorphic vector fieldX =v(z)∂/∂z onD, its projection by P the vector field P′X :=P′(z)w(z)∂/∂z is tangent to S.
Proof. Consider any point a ∈ S. Since the vector field P′(z)w(z)∂/∂z is holomorphic, it admits a unique maximal integral curve x : I → D starting from a where I is some open real interval around zero. It suffices to establish that there exists ε > 0 such that x(t)∈S i.e. P(x(t)) =x(t) for 0≤t < ε. Since the maps P, P′, w,(p, q)7→P′(p)w(q) are holomorphic, there are constants M ∈(1,∞) and δ ∈(0,1) with
∥w(p)∥, ∥P(p)−P(q)∥
∥p−q∥ , ∥P′(p)w(p)−P′(q)w(q)∥
∥p−q∥ < M (2.1.2) whenever the points p, q are contained in the ball a+δB(E) of radius δ around a. By setting ε := 12δ/M2, we are going to construct a sequence x1, x2, . . . : [0, ε] ⊂ In → S of continuous piecewise real-analytic curves such that we have the uniform convergence xn|[0, ε]→→x|[0, ε] that is lim
n→∞max
0≤t≤ε∥xn(t)−x(t)∥= 0.
For any index n, define the pointsakn (k= 0,1, . . .) recursively by a0n:=a, ak+1n :=P(
akn+ 2−nvkn)
where vnk:=w(akn)
for all possible indices k withakn ∈a+δB(E). The points{akn: k ∈Kn} thus constructed serve as breaking points of the analytic polygonxn : [0,2−nsupKn)→S defined by
xn(t) :=P(
akn+ (t−2−nk)vkn)
for 2−nk≤t ≤2−n(k+ 1)
consecutively on the intervals [0,2−n], [2−n,2−n·2], . . .. Notice that, by (2.1.2),
for all n > n0 and t ∈ [0, ε] except for finitely many places. Notice that the function t 7→ ∥xn+1(t)−xn(t)∥ is locally Lipschitzian, with ∥xn+1(0)−xn(0)∥ = ∥a−a∥ = 0.
Therefore, in terms of the solutionφ of the initial value problem d
whence, by passing to the limit with n,
for all t ∈ [0, ε]. In particular the function xe is the solution of the initial value problem e
with ex(0) = 0. By definition, the same initial value problem is fulfilled by the functionx. Hence we conclude thatx(t) =ex(t) =P(
x(t))
∈S (0≤t≤ε).
Corollary 2.1.4. Any integral curve of P′X starting from a point of S ranges in S.