Proof of Theorem 3.1.1

Remark 3.6.1. According to Theorem 3.5.13 and in view of Remark 4.7, we can represent each operator valued function t7→Uk,t =κk(t)U_k^t in Theorem 3.3.4 in the form

U_k,t = ^⊕

j∈Jk χ^(k)_j (t) exp(

itA^(k)_j )

(t∈R) (3.6.2)

with suitable families [χ^(k)_j : j ∈ J_k] of continuous functions R → T, respectively not necessarily bounded self-adjoint operators

A^(k)_j :D^(k)_j →H^(k)_j , D^(k)_j dense linear submanifold⊂H^(k)_j for some orthogonal decompositions

H^(k)= ^⊕

j∈Jk H^(k)_j (k= 1, . . . , N).

Thus, with the operatorsM_t^(k) := ^⊕

j∈Jk

χ^(k)_j (t)id_H(k) j

, Ue_k^t:= ^⊕

j∈Jk

exp(

itA^(k)_j )

we can repre-sent our object the semigroup U(·) in Theorem 3.1.1 in the form

U(t) =B^tC^t where B^t :=[ eU₁^t]_∗

⊗ · · · ⊗[ eU_N^t ]_∗

, C^t :=[ M_t⁽¹⁾]_∗

⊗ · · · ⊗[

M_t^(N)]_∗ . According to Lemma 3.4.7, the Hilbert space operator-valued functions t 7→ M_t^(k) respec-tively t7→Ue_k^t are strongly continuous. Recalling the elementary fact [27] that the product of bounded strongly convergent nets of linear operators is strongly convergent, we see that both the operator-valued functions t 7→ B^t, t 7→ C^t are also strongly continuous. Since the restrictions of the operators M_t^(k) are multiples of the identity on the subspaces H^(k)_j (j ∈J_k), the family {

U(t),B^t,C^t: t∈R}

is Abelian. Consequently, for all t, h∈R, C^t+h =U(t+h)B^−t−h =U(t)U(h)B^−tB^−h =U(t)B^−tU(h)B^−h =C^tC^h.

Lemma 3.6.3. Assume (as in the setting described in 3.6.1) that we have the orthogonal Thus the statement of the lemma is fulfilled with the choice

a^(k)_j 6.5. Finish of the proof of Theorem 3.1.1

In the setting established in 3.6.1, it suffices to see that in 3.6.2 we can also write U_k,t =

⊕

Chapter 4

Algebraic classification of bounded circular domains

Throughout this chapter let D be a bounded circular domain in a complex Banach space E. In contrast with the fact that the complete holomorphic classification of bounded domains of general type seems to be hopeless, in 1976 Kaup-Upmeier [54] proved that for bounded circular domains holomorphic equivalence is the same as linear equivalence. They achieved this result by a systematic study of the group G:= Aut(D) of all biholomorphic automorphisms ofD, which makes it possible to give further refinements of this statement:

they showedE can be equipped with a partial J*-structure (E, E_D,{..^∗.}_D) (for definitions see Section 1.10) such that, by writing GL(D) :={α inverible∈ L(E) :α(D) = D},

G= GL(D)· {exp[(a− {xa^∗x}_D)∂/∂x] :a∈ED}, G(0) =D∩ED.

It would be a remarkable step, also with a possible independent interest in theoretical physics, characterizing algebraically those partial J*-triples which arise canonically from the biholomorphic automorphism group of some bounded circular domain. This will be our next aim to which first we introduce the following concepts.

Definition 4.1.1. We say that a partial J*-triple (E, E₀,{..^∗.}) is

geometric if all the vector fields (a−{xa^∗x})∂/∂x(a∈E₀) are complete in some bounded circular domain in E that is if E0⊂ED and {..^∗.}={..^∗.}_DE×E0×E for some D⊂E;

hermitian if all the operatorsa a^∗(= [E∋x7→{aa^∗x}]) are hermitian that is if a a^∗ ∈Her(E,∥.∥) :={

A ∈ L(E) : ∥exp(iτ A)∥= 1 (τ ∈R)}

(a∈E₀);

weakly commutative if {{xa^∗x}b^∗x}={xa^∗{xb^∗x}} (a, b∈E₀, x∈E);

positive if Sp(a a^∗)≥0 (a∈E₀) for the spectra of the box-operators;

JB*-based if the J*-triple (E₀,{..^∗.}|E₀³) is a JB*triple satisfying axioms 1.10(JB*1-4).

Remark 4.1.2. It is well known [48],[70],[38] Prop.7.10.(c) that given a bounded cir-cular domain D ⊂ E, by taking the infinitesimal Carat´eodory norm ∥v∥ = ∥v∥_D :=

sup{

|φ^′(0)v| : φ ∈ Hol(D,D), φ(0) = 0}

on E, the triple (E, ED,{..^∗.}_D) becomes a weakly commutative Hermitian partial J*-triple. Furthermore, if we have E = E₀ then the weak commutativity is an algebraic consequence of the Jordan identity 1.10(J) (see

[10]). In view of the fact that the piece Sym(D) = D∩E_D is asymmetricbounded circular domain, the considerations in Kaup’s fundamental paper [51] establishing the equivalence between the categories of JB*-triples and the Banach spaces with symmetric unit ball along with convexity of the Harish-Chandra realization of bounded symmetric domains, it readily follows that (E, E_D,{..^∗.})_D with the norm ∥.∥_D is JB*-based as well. Thus any geometric partial J*-triple is linearly equivalent (just by renorming the space E) to a JB*-based weakly commutative hermitian partial J*-triple.

The crucial argument in [51] leading to the convexity of a bounded circular symmetric domain and hence giving rise to the category of JB*-triples is the following observation:

in a geometric partial J*-triple we have inf^a∈E0

∥a∥=1∥ {aa^∗a} ∥ ̸= 0 and 0≤Sp(a a^∗|E₀)⊂ 1

2Ω_a+ 1

2Ω_a (a∈E₀) (4.1.3)

where Ω_a:={0}∪Sp(a a^∗|C₀(a)) and C₀(a) is the smallest a a^∗-invariant subspace con-taining a. The J*-triple (C0(a),{..^∗.}|C0(a)) is isomorphic to that of the function space C₀⁽⁰⁾(Ω_a) :={φ∈C₀(Ω_a) :φ(0) = 0} with max-norm and the triple product {φψ^∗χ}=φψχ.

Definition 4.1.4. Positive JB*-based weakly commutative hermitian partial J*-triples will be called partial JB*-triples. Thus a continuous operation {..^∗.} : E×E₀×E → E with a closed complex subspace E0 is a partial JB*-triple product if for all a, b∈ E0 and x, y ∈E we have

(P1) {xa^∗y}is symmetric bilinear in x, y and conjugate-linear in a, (P2) ia a^∗ ∈Der(E,{..^∗.}),

(P3) {{xa^∗x}b^∗x}={xa^∗{xb^∗x}};

(P4) {aa^∗a} ∈E₀ with ∥{aa^∗a}∥=∥a∥³,

(P5) a a^∗ ∈Her(E,∥.∥)₊ :={A∈Her(E,∥.∥) : Sp(A)≥0}.

Notice that, by polarization in a, axiom (P2) is equivalent to the Jordan identity 1.10(J).

Furthermore the family Her(E,∥.∥)₊ in (P5) can be characterized by the norm estimate A∈Her(E,∥.∥)+ ⇐⇒ ∥exp(ζA)∥ ≤1 whenever Re(ζ)≤0.

Our main result, whose proof along with some independently interesting considerations will occupy the rest of the chapter, is the following theorem.

Theorem 4.1.5. Up to linear Banach space isomorphisms, geometric partial J*-triples and partial JB*-triples are the same. Namely

(i) geometric partial J*-triples are positive;

(ii) given a partial JB*-triple (E, E₀,{..^∗.}), there exists a neighborhood of the origin in E such that, for any bounded circular domainB contained in it and left invariant by the operators exp(ia a^∗) (a∈E₀), the figure

D:= ∪

c∈E0

[exp(

(c− {xc^∗x})∂/∂x)]

(B) (4.1.6)

is a bounded circular domain with (c− {xc^∗x})∂/∂x∈aut(D) (c∈E₀).

Remark 4.1.7. Theorem 4.1.5 is new even for finite dimensions. Though there is no proof in the literature, the conjecture that the category of all partial JB*-triples coincides with that of the partial J*-triples of the form (E, E_D,{..^∗.}_D), seems very likely on the basis of the construction in Theorem 4.1.5(ii). In genaral, for the domainD in 4.1.5(ii) we trivially have E_D = CSym(D) ⊃ E₀. However, given any partial JB*-triple (E, E₀,{..^∗.}), one can expect that, with some holomorphically rigid admissible figure B, (4.1.6) furnishes a domainD with CSym(D) =E₀.

Example 4.1.8. LetE :=C³ with its canonical scalar product⟨x|y⟩:=∑₃

k=1x_ky_k. Define {xa^∗y}:= ¹₂⟨x|a⟩y+¹₂⟨y|a⟩xforx, y ∈E anda∈E₀ :=C× {(1,0,0)}. Then, in 4.1.6, the figures B =B_p :={

x ∈C³ : |x₁|² + (|x₂|^p+|x₃|^p)^2/p < 1}

with 2 ̸=p ∈[1,∞), give rise to pairwise linearly non-isomorphic bounded circular domainsD_p such thatE_D =E₀. On the other hand, forp= 2, we get the symmetric Hilbert ballD2 =B(E) = {x: ⟨x|x⟩<1} with E_D2 =E.

The idea of the proof of part (i) of Theorem 4.1.5 is the observation that a suitable ultrapower extension [15] of the abelian family {b b^∗ : b ∈ C0(a)} admits convenient joint eigenvectors and its span is linearly homeomorphic to C0(Ω₀) by a mapping which can be factorized through the tensor square of the Gelfand representation of C0(a). With this method we give also a new and Jordan theoretically very simple proof for Kaup’s spectral estimate (4.1.3) for geometric J^∗-triples.

Remark 4.1.9. The analog of (4.1.3) for arbitrary geometric partial J^∗-triples is false:

to every p > 0 the space C² endowed with the triple product {(ζ₁, ξ₁)(α,0)^∗(ζ₂, ξ₂)} :=

α(ζ₁ζ₂(ζ₁ξ₂ +ζ₂ξ₁)· p) defined on C² × (C× {0})× C² is a geometric partial J^∗-triple corresponding to the 2-dimensional Reinhardt domain {ζ, ξ) :|ζ|²+|ξ|^∗2/p <1} (cf. [76], [5] p.162). Here we have Ω_(1,0) ={0,1} and Sp((1,0) (1,0)^∗) = {1, p}.

We begin the proof of part (ii) of Theorem 4.1.5 with some considerations that may have independent interest: We prove that in a weakly commutative partial JB^∗-triple (E, E₀,{..^∗.}) the germs around the unit ball ofE₀ of the mappings exp[(a− {za^∗z})∂/∂z] with a ∈ E₀ generate a Banach-Lie group G. This implies a Cartan-type uniqueness theorem and hence a semidirect product decomposition into a linear and an exponential quadratic part for G — a germ analog of a result of Vigu´e and Isidro [111]. With the aid of this decomposition we show that, given a connected circular neighborhood B of the origin that is invariant under the Jordan automorphisms of E with the property that the integral curves through B are complete in E for all the vector fields (a− {za^∗z})∂/∂z, the orbit∪a∈E0exp[a− {xa^∗x}∂/∂x](B) is the integral manifold through B of the Banach-Lie algebra of vector fields {(a +iL− {xa^∗x})∂/∂x : a ∈ E₀, L ∈ Der(E)}. This shape estimate is a circular generalization of a similar theorem of Panou [70] for bicircular partial JB^∗-triples. This is our main geometrical tool in proving the equivalence of geometric partial J^∗-triples and weakly commutative partial JB^∗-triples.

4.2 Joint eigenvectors of box operators

Throughout this section let (EE₀,{..^∗.}) be a geometric partial J^∗-triple and assume thatD is a bounded circular domain inE in which the vectors fields (b−{zb^∗z})∂/∂z are complete for allb ∈E0. Let us also fixa∈E0arbitrarily. We denote byT the Gelfand representation [51], [[38] Th.10.38] of C0(a), i.e. TC0(Ω₀(a) → C^∼ 0(a) is a topological isomorphism such that

T(φχψ) ={T(φ)T(χ)^∗T(ψ)} (φ, χ, ψ ∈ C0(Ωa)), T(ξ) = a where ξ(ω) := √

ω (ω ∈ Ω₀) and C0(Ω_a) := {φ ∈ C(Ω_a) : φ(0) = 0}. Recall [51] that Ωa ≥ 0 and that {b b^∗ : b ∈ C0(a)} is a commutative family of bounded E-hermitian operators. Define L(a) := Span{b b^∗ : b∈ C0(a)}.

Lemma 4.2.1. L(a) =C0(a) C0(a)^∗ and there exists a linear homeomorphismL:C0(Ω_a)→ L(a) such that

L(φψ) =T(φ) T(ψ)^∗ (φ, ψ∈ C0(Ω_a)). (4.2.2) Proof. Fixa ∈E₀ arbitrarily and let

D:={

ϕ∈ C0(Ω_a) : ϕ vanishes in a neighborhood of 0 ∈Ω_a} . We may define L₀(ϕ) := T(ϕ/ξ) T(ξ)^∗ (ϕ∈ D). It is well-known [11] that

T(p) T(q)^∗ =T(pq/ξ) T(ξ)^∗ for p, q ∈ P :={odd polynomials of ξ).

Givenφ, ψ∈ D, we can find sequences (p_n), (q_n) inP tending uniformly toφ/ξ2 andψ/ξ2, respectively. Then

L0(φψ) = T(φψ) T(ξ)^∗ = lim

n T(ξ³pnqn) T(ξ)^∗ = lim

n T(ξ²pn T(ξ²qn)^∗ =

= T(φ) T(ψ)^∗

Hence ∥L₀(φ)∥=∥T(ϕ^1/2) T(ϕ^1/2)^∗∥ ≤M∥ϕ∥ (ϕ∈ D+) where M := sup{∥T(φ) T(ψ)^∗∥ :

∥φ∥ = ∥ψ∥ = 1} < ∞. Decomposing the functions of D into linear combinations from D₊, it follows ∥L₀∥ ≤ M. By the density of D in C0(Ω_a) there is a unique continuous linear extension L : C0(Ω₀) → L(a) of L₀ satisfying (4.2.2). On the other hand every ϕ ∈ C0(Ω_a) can be written in the form ϕ = φψ for some φ, ψ ∈ C0(Ω_a). Hence with d:= max{∥T∥,∥T⁻¹∥} we get

d· ∥L(ϕ)∥ ≥ sup

∥χ∥=1

∥L(ϕ)T(χ)∥=

= sup

∥χ∥=1∥T(φ)T(ψ)^∗T(χ)}∥ ≥ sup

∥χ∥=1

1

d∥φψχ∥= 1 d∥ϕ∥.

Thus L is a linear homeomorphism. In particular the range of L is a closed subspace of L(a) and ran(L) = L{ϕψ: φ, ψ∈ C0(Ω_a)}=T(C0(Ω_a)) T(C0(Ω_a))^∗ =C0(a) C0(a)^∗.

Lemma 4.2.3. Let F be a Banach space and A a separable linear subspace of L(F) con-sisting of commuting operators and let α₀ ∈ A. Then, to every approximate eigenvalue λ₀ of α₀, there exist a sequence (x_n) in F and a continuous linear functional Λ on A such that λ₀ = Λ(α₀) and

∥x_n∥ →1,∥αx_n−Λ(α)x_n∥ →0 (n → ∞, α ∈ A).

Proof. Everyλ∈Aacts onℓ^∞(N, F) by (x_n)7→(αx_n) and hence also onFe:=ℓ^∞(N, F)/M where M :={(x_n)∈ℓ^∞(N, F) : lim_nx_n= 0}. Denote this operator by α. Thene Ae:={eα : α ∈ A} is a commutative subspace of L(Fe). It suffices to show that the operators in Ae admit a joint eigenvector in the λ0-eigenspace ofαe0.

It is clear that Fe₀ := {ex ∈ Fe : αe₀xe = λ₀xe} ̸= 0 and that Fe₀ is left invariant by all e

α ∈ Ae. Let (αn) be a dense sequence in A and for each n ∈ N define an Ae-invariant subspace Fe_n and λ_n ∈ C recursively in the following way: Let λ_n be an approximate eigenvalue of the operatorα_n|Fe_n−1 and letFe_n:={ex∈Fe_n−1 : αe_nex=λ_nxe}. This is possible since the approximate point spectrum of every bounded linear operator on a Banach space is not empty [76] p.310. The only thing we have to verify is that

∩

m

Fe_m ̸= 0.

First we show by induction thatFen̸= 0 (n = 0,1, . . .). AssumeFen−1 ̸= 0. By the definition of λ_n there is a sequence (xe^k) in Fe_n−1 with ∥ex^k∥ = 1 (k ∈ N) and αe_nxe^k → 0 (k → ∞).

Since Fe₀ ⊃. . .⊃Fe_n, we also have αe_jex^k=λ_jex^k (0≤j < n) for all k ∈N. For anyk chose a representing sequence (y_m^k : m ∈ N) in F for ex^k. It follows that for each ℓ ∈ N we can find k(ℓ) such that, by setting z_n,ℓ :=y^k(ℓ)_m(ℓ), we have

| ∥z_n,ℓ∥ −1|< ℓ⁻¹ and ∥eα_jz_n,ℓ −λ_jz_n,ℓ∥< ℓ⁻¹ (0≤j ≤n).

Hence the relation Fe_n ̸= 0 is immediate. We complete the proof by observing that the vector ez ∈ Fe which is represented by the diagonal (z_n,n) of the double sequence (z_n,ℓ) constructed above satisfies ∥ez∥= 1 andαe_jze=λ_jez (j ∈N).

LetU be a non-trivial ultrafilter onNandE^U theU-ultrapower ofEthat isℓ^∞(N, E)/N where N :={(x_n)∈ℓ^∞(N, E) : lim_Ux_n= 0}. The elements of E^U are the cosets (x_n)_U :=

(x_n)+N with the norm∥(x_n)_U∥:= lim_U∥x_n∥((x_n)∈ℓ^∞(N, E)). We regardEas a subspace of E^U by the embedding x 7→ (x, x, . . .)_U. Taking E₀^U :={(a_n)_U : (a_n) ∈ ℓ^∞(N, E₀)}, the canonical extension

{(x_n)_U(a_n)_U^∗(y_n)_U}:= ({x_na_n^∗y_n}_U ((x_n),(y_n)ℓ^∞(N, E); (a_n)∈ℓ^∞(N, E₀)) of the triple product makes (E^U, E₀^U,{∗}U) into a partial J^∗-triple. We denote it also by E^U and write simply {..^∗.} instead of {..^∗.}_U. Note that the vector fields (eb−{

e zeb^∗ez

} )∂ez are complete in the closed set De := {(z_n)_U : z₁, z₂, . . . ∈ D} (the arguments of [15] Th.9 apply with straightforward modifications). Since these vector fields are locally bounded it follows that they are complete also in the interior ofD.e

Since the spectrum of any hermitian operator is real [6], by [76] p.310 it coincides with the approximate point spectrum. Therefore we can summarize the previous results as follows.

Proposition 4.2.4. Let E be a geometric partial J^∗-triple and U a non-trivial ultrafilter onN. ThenE^U is also a geometric partialJ^∗-triple. Givena ∈E₀ andλ ∈Sp(a a^∗)there exists a complex Radon measure µ of bounded variation on Ω_a and 0̸=ex∈E^U such that

λ₀ =

∫

ωdµ(ω) (4.2.5)

{T(φ)T(ψ)^∗ex} =

∫

φψdµ·xe (φ, ψ∈ C0(Ω_a)). (4.2.6)

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