Counterexample to Sunada theorems with CRD

Let Ω and Ω be the following compact topological subspaces ofe C²: Ω :=∪

k=1,2{ω_k,t : 0 ≤t <2π} where ω_1,t:= (1, e^it), ω_2,t := (−1, e^it) Ω :=e ∪

k=1,2{eω_k,t : 0 ≤t <2π} where ωe_1,t:= (e^it/2, e^it), ωe_2,t := (−e^it/2, e^it).

Consider the following (symmetric) continuous Reinhardt domains

D:={f ∈ C(Ω) : |f(ω_1,t)|²+|f(ω_2,t)|² <1, 0≤t <2π}, De :={fe∈ C(Ω) :e |f(eωe_1,t)|²+|f(eωe_2,t)|² <1, 0≤t <2π}.

Notice that Ω = {±1} ×T is the union of two disjoint circles and can be regarded as the border of a cylindric band with middle circle {0} ×T, while Ω =e {(e^it/2, e^it) : t ∈ R} is topologically equivalent to a circle and can be regarded as the border of a M¨obius band with the same middle circle {0} ×T. Conveniently, we can identify the both the spaces C(Ω) and C(Ω) with simple subspaces of couples of continuous functions on the compacte interval [0,2π]. Namely, given φ1, φ2 ∈ C[0,2π], with the functions

f_φ1,φ2(ω_k,t) := φ_k(t), fe_φ1,φ2(ωe_k,t) :=φ_k(t), 0≤t <2π

we have C(Ω) ={f_φ1,φ2 : φ₁, φ₂ ∈ F} and C(Ω) =e {fe_φ1,φ2 : (φ₁, φ₂)∈F}e where F :={(φ₁, φ₂)∈(C[0,2π])² : φ₁(0) = φ₁(2π), φ₁(0) = φ₁(2π)}, Fe:={(φ₁, φ₂)∈(C[0,2π])² : φ₁(0) = φ₂(2π), φ₂(0) = φ₁(2π)}. Hence the mapping

Ue_∗f_φ1,φ2 :=fe_cos(t/2)φ1(t)+sin(t/2)φ2(t),e^it/2[−sin(t/2)φ1(t)+cos(t/2)φ2(t)] (f ∈ F) is a linear isomorphism C(Ω)↔ C(Ω) ande U_∗D=D.e

6.4.1. Proof of Theorem 1.8. There is no linear isomorphism W_∗ :C(Ω)↔ C(Ω)e such that W_∗D=De and W_∗ReC(Ω) = ReC(Ω).e

Proof. Given a discrete complex measure µ := ∑_∞

n=1(a_nδ_ω1,tn +b_nδ_ω1,tn) on Ω, we have

|∫

f dµ|<1 for allf ∈Dif and only if∑_∞

n=1(|a_n|²+|b_n|²)^1/2 ≤1. Hence, with the notation of the previous section, Π_D ={

{ω_1,t, ω_2,t}: 0≤ t <2π}

for the partition associated with the domain D. Similarly, Π_De = {

{eω1,t,ωe2,t} : 0 ≤ t < 2π}

. Suppose indirectly that W_∗ :C(Ω)↔ C(Ω) is a linear isomorphism withe W_∗D=De and W_∗ReC(Ω) = ReC(Ω). Bye Theorem 6.3.8 (applied with the weight functions m = 1_Ω and me = 1_Ωe), the operator W_∗ must have the form

W_∗f(ωe_1,t) = w₁₁(t)f(Tωe_1,t) +w₁₂(t)f(Tωe_2,t),

W_∗f(ωe_2,t) = w₂₁(t)f(Tωe_1,t) +w₂₂(t)f(Tωe_2,t), 0≤t <2π, f ∈ C(Ω)

where(_w11w12

w21w22

)is a unitary matrix for anyt∈[0,2π) and T :Ωe ↔Ω is a mapping with the effect{

{Tωe1,t, Teω2,t}: 0≤t <2π}

={T(S) :e Se∈Π_De}= ΠD ={

{ω1, ω2}: 0≤t <2π} . We can write without loss of generalityTωe_k,t =ω_k,T#(t),k= 1,2, 0≤t <2πwith a suitable permutationT_#: [0,2π)↔[0,2π). Thus given any couple (φ₁, φ₂)∈ F, we have

W_∗f_φ1,φ2 =fe_ψ1,ψ2 for some (ψ₁, ψ₂)∈Fe with ψ_k(t) =∑2

ℓ=1w_kℓ(t)φ_ℓ( T_#(t))

if k = 1,2 and 0 ≤ t < 2π. The assumption W_∗ReC(Ω) = ReC(Ω) means that thee functions ψ₁, ψ₂ are real valued whenever φ₁, φ₂ are real valued in the above formula. By considering the particular cases (φ⁽¹⁾₁ , φ⁽¹⁾₂ ) := (1Ω,0) and (φ⁽²⁾₁ , φ⁽²⁾₂ ) := (0,1Ω) we see that there are continuous functions ψ_kℓ : [0,2π]→Rsuch that

ψkℓ(t) =wkℓ(t), 0≤t <2π, k, ℓ= 1,2; (ψ11, ψ21),(ψ12, ψ22)∈Fe. This is impossible for the following reasons. The matrices (

ψ_kℓ(t))2

k,ℓ=1, 0≤t≤2π are orthogonal with real entries. Thus necessarily

ψ₂₁(t) = ε(t)ψ₁₂(t), ψ₂₂(t) = −ε(t)ψ₁₁(t), ψ₁₁(t)²+ψ₁₂(t)² = 1, 0≤t≤2π for some function ε : [0,2π] → {±1}. The connectedness of the interval [0,2π] along with the continuity of the functions ψ_kℓ entails that actually ε(t) ≡ const, say ε(t) ≡ ε₀. However, since (ψ₁₁, ψ₂₁),(ψ₁₂, ψ₂₂)∈Fe, we also have the boundary conditions

ψ₁₁(0) = ψ₂₁(2π), ψ₂₁(0) =ψ₁₁(2π), ψ₁₂(0) =ψ₂₂(2π), ψ₂₂(0) =ψ₁₂(2π).

Hence ψ11(0) =ψ21(2π) = ε0ψ12(2π) = ε0ψ22(0). On the other hand ψ11(0) = −ε0ψ22(0).

Thus necessarily ψ₂₂=ψ₂₂(0) = 0. Similarly,ψ₂₁(0) =ψ₁₁(2π) =−ε₀ψ₂₂(2π) = −ε₀ψ₁₂(0) and ψ₂₁(0) = ε₀ψ₁₂ implying ψ₂₁(0) = ψ₁₂(0) = 0. These conclusions contradict the fact that (

ψ_kℓ(0))2

k,ℓ=1 ̸= 0.

Chapter 7

Fine structure of partial JB*-triples of continuous Reinhardt domains

In classical finite dimensional complex analysis, Reinhardt domains are popular test objects for conjectures. The considerations of this chapter have two main aims:

(1) describe of all possible holomorphic equivalences of CRDs¹;

(2) we test two open problems on bounded circular domains in the setting of CRDs concerning bidual embedding and inner derivation extension property (IDEP).

Ad (1). Since CRDs are bounded circular domains, on the basis of Chapter 4, this can be done via the description of the partial JB*-triples associated with CRDs. As a starting point, in Section 7.1 we develop an integral representation for their triple product.

Then their restriction to the symmetric part will be refined in Section 7.2 by the aid of our Banach–Stone-type theorems. We shall see that the symmetric part of a CRD (being a CRD itself) is a continuous mixture of finite dimensional Euclidean balls, essentially more involved than direct sums of topological products of balls. Recall that in Chapter 6 we already established matrix representations for the linear isomorphisms between two symmetric CRDs. Based upon the Lie theory of hermitian operators in the dual space, in in Section 6 we extend these matrix representations to a Banach-Stone type theorem on the isomorphisms of general Banach lattice normed commutative C*-algebras. This result includes implicitly the description of all the possible linear isomorphisms between CRSs because the convex hull of any CRD can be regarded as the unit ball of some lattice norm in a commutative C*-algebra and linear isomorphisms preserve convex hulls.

Ad (2). The first problem we treat takes its origin in a work of Panou [71] where it is shown that every inner derivation of the Jordan-triple associated with the symmetric part of a finite-dimensional bounded circular domain admits a unique extension to an inner derivation of the partial Jordan triple associated with the whole domain. Though it is natural to expect that the analog holds in general Banach spaces, the only known infinite-dimensional results concern domains with nearly atomic symmetric part [82]. On the basis of Theorem 7.1.7 along with the fine structure description of the Jordan triple product

1For definition see 6.1.6

associated with a symmetric CRD in Section 7.3 we can establish immediately that the partial Jordan triple of a CRD has the IDEP extension property of inner derivations.

The second question we solve for CRDs is the mentioned open problem of the Jordan structure of second dual of a partial JB*-triple. First, in Section 7.4 we refine Theorem 7.1.7 into a natural extension of the results for symmetric CRDs whose proofs there relied upon some bidual considerations. By proceeding the opposite way, in Section 7.5 we apply the fine structure description obtained in Section 7.4 along with function representations ofC0(Ω)^′′ spaces to establish that the Jordan triple product associated with a CRD admits a separately weak*-continuous bidual extension which can be regarded as the canonical Jordan triple of some not necessarily unique CRD.

In course of carrying out this program, we achieved a result of independent interest which we present separately in Appendix A.3: a short direct proof for a multilinear analog of Riesz’ representation theorem which seems to be new in the sense that only an implicit version is available in the literature hidden in a paper by Villanueva [112].

7.1 Integral formula for the triple product

As previously, throughout the whole chapter Ω denotes an arbitrarily fixed locally compact space,E :=C0(Ω) is the space of all continuous functions Ω→Cvanishing at∞equipped with the spectral norm∥f∥_∞:= max|f|and Dwill stand for an arbitrarily fixed bounded CRD in E. Our primary aim is to describe the partial JB*-triple E_D:= (E, E_D,{..^∗.}_D) normed with the equivalent norm ∥.∥_D whose unit ball ball the convex hull B := co(D).

It is convenient to study this problem in a somewhat more general setting.

Conventions 7.1.1. Throughout the whole chapter let E := (E, E₀,{. . .}) denote a fixed partial JB*-triple over E(

:=C0(Ω))

equipped with a lattice norm ∥.∥ and having the Reinhardt property

(R) Ψ⊂Aut(E) where Ψ :={ψ·: ψ ∈ C(Ω), |ψ|= 1} with ψ· denoting the multiplication operator C0(Ω) ∋f 7→φf and

Aut(E) :=

{

L∈L(E) : LE₀⊂E₀, L{xay}={(Lx)(La)^∗(Ly)} (a∈E₀, x, y∈E) }

denoting the group of all {..^∗.}-automorphisms.

Notice that the canonical partial JB*-tripleE_D of any CRDDhas property (R). Recall also [11] that Aut(E_D) contains the injective linear transformations L :E →E such that LD = D. Moreover, as a consequence of Theorem 4.1.5(ii), from Lemma 7.1.2 below applied to E with a suitably small ∥,∥_∞-ball V :={x∈E :∥x∥_∞< δ}, we know also that

E can be regarded as a subtriple inE_D of some CRD D⊂E. Namely, by using henceforth the abbreviation

L(a) :=a a^∗

without danger of confusion in any partial J*-triple under consideration, we have the following. a bounded group of linear mappings, V₁ is a bounded Ψ-invariant circular domain in F. Given any a∈ F0 and ψ ∈Ψ, since ψ ∈Aut(F, F0,{. . .}), we haveL(ψ⁻¹a) = ψ⁻¹L(a)ψ. it only remains to prove that ψ∪

a∈F0 with respect to the variable t yields

ψE₀ ⊂E₀ , ψ{xa^∗y}={(ψx)a^∗y}−{

x(ψa)^∗y }

+{xa^∗(ψy)} (7.1.3) for all bounded continuous functionsψ : Ω→C. In particularE0 is a closed ideal inC0(Ω) regarded as a commutativeC^∗-algebra with the pointwise product of functions. Therefore necessarily

E0 =C0(Ω0) :={f ∈ C0(Ω) : f(Ω\ΩD) = 0} with the open set Ω₀ :={ω ∈Ω : ∃ a∈E₀ a(ω)̸= 0}.

Lemma 7.1.4. {xa^∗y}(ω) = 0 whenever x(ω) = y(ω) = 0.

Proof. By symmetry, it suffices to see the statement for the case x =y. Furthermore, by the continuity of the triple product and since continuous functions vanishing at ω(∈ Ω) can uniformly be approximated with continuous function vanishing on some neighborhood ofω, it suffices to see that{xa^∗x}(ω) = 0 ifx(U) = 0 for some neighborhoodU ⊂Ω of the pointω. Assumeω ∈U open⊂Ω,x∈E, andx(U) = 0. Choose a compact neighborhood V of ω within U and let ϕ : Ω →[0,1] be a continuous function such that ϕ(ω) = 1 and ϕ(Ω\V) = 0. Observe that if c∈ E₀ is a function with c(V) = 0 then ϕx= ϕc= 0 and, by (7.1.3),

{xc^∗x}(ω) =ϕ{xc^∗x}(ω) = 2{(ϕx)c^∗x}(ω)− {x(ϕc)^∗x}(ω) = 0.

Consider any a ∈ E₀. Choose a continuous function ψ : Ω → [0,1] with ψ(V) = 0 and ψ(Ω\U) = 1. Since ψx=x, by the aid of the functionc:=ψa vanishing on V we get

0 = ψ{xa^∗x}(ω) = 2{(ψx)a^∗x}(ω)− {x(ψa)^∗x}(ω) =

= 2{xa^∗x} − {xc^∗x}= 2{xa^∗x}. Corollary 7.1.5. We have

{xa^∗y}= 1

2x{z(ya)^∗z}+ 1

2y{z(xa)^∗z} if z ∈E with xz =x and yz =z . Proof. Suppose xz =x and yz =z. Consider any pointω ∈Ω and apply Lemma 7.1.4 to the functions x_ω :=x−x(ω)z and y_ω :=y−y(ω)z satisfying x_ω(ω) =y_ω(ω) = 0. We get

0 = {x_ωa^∗y_ω}(ω) = {[x−x(ω)z]a^∗[y−y(ω)z]}(ω) =

= {xa^∗y}(ω) +x(ω)y(ω){za^∗z}(ω)−x(ω){za^∗y}(ω)−y(ω){za^∗x}(ω) . Thus, everywhere on Ω,

{xa^∗y}=−xy{za^∗z}+x{za^∗y}+y{za^∗x} . Observe that, by (7.1.3) and sinceyz =y, here we have

xy{za^∗z}=x[

2{ya^∗z} − {z(ya)^∗z}] and similarly yx{za^∗z}=y[2{xa^∗z} − {z(xa)^∗z}]. Therefore

xy{za^∗z} = x{za^∗y}+y{za^∗x} − 1

2{z(xa)^∗z} − 1

2{z(ya)^∗z} , {xa^∗y} = −xy{za^∗z}+x{za^∗y}+y{za^∗x}=

= 1

2x{z(ya)^∗z}+ 1

2y{z(xa)^∗z} . Lemma 7.1.6. The triple product {..^∗.} is positive in the sense

x, a, y ≥0⇒ {xa^∗y} ≥0 .

Proof. Fix 0≤x, y ∈E and 0≤a ∈E₀ arbitrarily. Since functions with compact support are dense in C0-spaces, we may assume

supp(x),supp(y) compact⊂Ω, supp(a) compact⊂Ω₀ . Then we can choose 0≤x₀, x₁, y₀, y₁, z ∈ C0(Ω) with compact support such that

x=x₀+x₁ , y =y₀+y₁ , supp(x₀),supp(y₀)⊂Ω₀ ,

supp(x₁)∩supp(a) = supp(y₁)∩supp(a) =∅ , supp(x)∪supp(y)∪supp(a)⊂ {ζ ∈Ω : z(ζ) = 1} . We have{xa^∗y}=∑₁

k,ℓ=0{x_ka^∗y_ℓ}. Moreover{x₀a^∗y₀} ≥0 for the following reasons. The subtriple (E₀, E₀,{. . .}|E₀³) is a JB*-triple with Ψ⊂Aut(E₀, E₀,{. . .}|E₀³). Therefore it is necessarily the canonical JB*-triple of a bounded symmetric continuous Reinhardt domain in E₀ = C0(Ω₀). However, by [94] Theorem 2, the triple product is negative for non-negative functions for symmetric CRDs. Thus indeed {x₀a^∗y₀} ≥ 0 since a, x₀, y₀ ∈ E₀. On the other hand, by Corollary 7.1.5,

{x₁a^∗y₁}= 1

2x₁{z(y₁a)^∗z}+1

2y₁{z(x₁a)^∗z}= 0

becausex₁a =y₁a= 0. It only remains to see{x₀a^∗y₁} ≥0 (since the proof of{x₁a^∗y₀}= {y₀a^∗x₁} ≥0 is analogous). Define

c:=√ x₀a .

Since supp(c)⊂supp(a)⊂Ω0, we have c∈E0 andcz=c. By Corollary 7.1.5 (applied with y₁ instead ofy and first with c instead both of a and x and then with x₀ instead of x),

{cc^∗y₁} = 1

2c{z(y₁c)^∗z}+1 2y₁{

z(c²)^∗z}

= 1 2y₁{

z(c²)^∗z} , {x₀a^∗y₁} = 1

2x₀{z(y₁a)^∗z}+1

2y₁{z(x₀a)^∗z}= 1

2y₁{z(x₀a)^∗z} because y₁a=y₁c= 0. That is

{x₀a^∗y₁}={cc^∗y₁}= 1

2y₁{z(x₀a)^∗z} .

By assumption, Sp(L(c)) ≥0. However, it is a basic fact about the spectra of multipliers in commutative Banach algebras that

u(Ω) ⊂Sp[

C0(X)∋f 7→uf]

if X open⊂Ω and u∈ C0. In particular, by taking X := Ω\supp(a) andu:={z(x₀a)^∗z} we have

1

2{z(x0)a^∗z}(

Ω\supp(a))

⊂L(c)⊂[0,∞).

Thus{z(x₀a)^∗z} ≥0 on supp(y₁) and hence 2{x₀a^∗y₁}=y₁{z(x₀a)^∗z} ≥0.

We can summarize the results of this section in the following theorem.

Theorem 7.1.7. Let Ω be a locally compact space, E :=C0(Ω) and suppose(E, E₀,{. . .}) is a partial JB*-triple with the Reinhardt property (R). Then there exists an open subset Ω₀ in Ω such that E₀ = {f ∈ E : f(Ω\Ω₀) = 0}. Given any point ω ∈ Ω, there is a Proof. We have established already the relation E0 = C0(Ω0) and the positivity of the triple product in the sense of Lemma 7.1.6. Fixω ∈Ω arbitrarily. According to [112]², the positivity of the bounded 3-linear functional (x, a, y) 7→ {xa^∗y}(ω) implies the existence of a positive Radon measure νω of finite total variation on Ω×Ω0×Ω such that By the inner compact regularity of Radon measures, given any functions x, y ∈ E with compact support and anda ∈E₀, we can choose an increasing sequenceK₁ ⊂K₂ ⊂. . .⊂Ω where 1_Ω denotes the function identically 1 on Ω. Thus with the measure

µ_ω(X) :=ν_ω(Ω×X×Ω) (X Borel⊂Ω)

we have the stated relation for x, y ∈E with compact support. The statement follows by the uniform density of functions with compact support in E.

Remark 7.1.9. It would be tempting to conjecture that every partial JB*-triple satisfying the hypothesis of Theorem 7.1.7 is the canonical JB*-triple of some CRD. However there is a counterexample even in 2 dimensions: Let Ω := {1,2}, Ω₀ := {1} and {xay} :=

[17→x(1)a(1)y(1), 27→x(1)a(1)y(2)/2 +x(2)a(1)y(1)/2]. Then any CRD D overΩ such that the vector fields [

a− {xax}]

∂/∂x are complete in D must be an ellipsoid of the form D={x: |x(1)|²+λ|x(1)|² <1} for some λ >0 with E_D =E ̸=E₀.

2This fact is implicit in [112]. We include a short direct proof in Appendix A.3.

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