Pure algebraic extension with weighted grids

= 0 whenever

∑n k=1

αkak a^∗_k|E0 = 0.

Thus inner derivations of E₀ admit unique inner derivative extensions to E.

In view of Proposition 5.4.7 the main result of this section is immediate.

Theorem 5.4.8. A partial J*-triple whose base is ac₀ direct sum of a family of elementary JB^∗-triples has IDEP.

Definition 5.4.9. Let us call a derivation of a partial JB*-triple aσ-inner derivation if it is the norm limit of some sequence of inner derivations.

As a by-product of the proof of Theorem 5.4.8, we also obtain the following result.

Corollary 5.4.10. Let (E, E₀,{..^∗.}) be a partial JB*-triple whose base E₀ is an elemen-taryJB^∗-triple with infinite rank or finite dimensions³. Then anyσ-inner derivation ofE₀ admits a unique σ-inner derivative extension to E.

5.5 Pure algebraic extension with weighted grids

The aim of this section is a direct approach to the IDEP property in an elegant topology free algebraic context. HenceforthKdenotes a commutative field of characteristics 0 with unit 1 and involution (calledconjugation). As a natural extension of the complex case, given a vector spaceF overK, a subspace F₀ ofF and a 3-variable operation (x, a, y)7→ {xa^∗y} mapping F×F0×F toF such that{F0F0∗F0} ⊂F0, the algebraic structure (F, F0,{..^∗.})

3In other words the elementary JB^∗-tripleE0is not isomorphic to any infinite dimensional spin factor or to anyc0(H, K) with finite dimensionalH and infinite dimensionalK.

is called a partial J*-triple (overK) with base space F₀ if {xa^∗y} is symmetric bilinear in x, y∈E, conjugate linear in a∈F and satisfies the Jordan identity (equivalent to 1.10(J)) (J^′) [a b^∗, c d^∗] ={ab^∗c} d^∗−c {da^∗b}^∗ (a, b, c, d∈F)

where, for anyx∈F and f ∈F0, the symbolx f^∗ denotes the linear operator (x f^∗)y:=

{xf^∗y}(y∈F). In particular, the base spaceF₀with the restricted triple product{..^∗.} |F₀³ is a Jordan triple in the sense of Meyberg [65]. Notice that the vector fields

(V) [ a+∑

k(bk c^∗_k)z− {za^∗z}]

∂/∂z (a, b1, c1, . . . , bn, cn∈F0, n= 1,2, . . .) form a 3-graded Lie algebra if and only if the axiom

{{za^∗z}b^∗z}={za^∗{zb^∗z}} (a, b∈F₀, z∈F)

of weak commutativity holds. A natural but somewhat stronger version of the IDEP property in this pure algebraic setting can be the following: the 3-graded Lie algebra of vector fields of type (V) associated with a finite dimensional weakly commutative complex partial J*-triple is unambiguously determined by its restriction to the base space, because the operation R : L 7→ L|F for L∈ Lin-hull_KF₀ F₀^∗ is injective. The key tool of our approach will be the concept of weighted grids motivating later Chapter 8.

Definition 5.5.1. By a weighted grid in a partial J*-triple (F, F₀,{..^∗.} (over the field K) we mean a linearly independent subsetG:={g_w : w∈W}of F₀ indexed with a figureW contained in some K-real vector space (that over the field ReK :={ξ ∈K : ξ =ξ}) such that

{g_ug_v^∗g_w} ∈Kg_u−v+w (u, v, w ∈W)

with the conventiong_z := 0 forz ̸∈W. Notice: grids in Neher’s sense [68] can be regarded as weighted grids (see Chapter 8 for the case K =C). A typical example for a weighted grid is the set of the real N ×N-matrices u_kℓ :=[

δ_ikδ_jℓ]N

i,j=1 with a unique non-vanishing entry 1 which can be regarded as {g_w : w ∈W}with W :={(e_k, e_ℓ) : 1≤k, ℓ≤N} and g_(ek,eℓ) :=u_kℓ with the unit vectors e_k := [δ_ik]^N_i=1.

We give a self-contained exposition on weighted grids over general fields in Appendix A.2.

Theorem 5.5.2. Let (F, F₀,{..^∗.}) be a partial J*-triple over K whose base F₀ is spanned by a finite weighted gridG:={gw : w∈W}of non-nil elements {gwgw∗gw} ̸= 0 (w∈W).

Then the restriction ∑

u,v∈Wγ_uvg_u g_v^∗ 7→∑

u,v∈Wγ_uvg_u g^∗_v|F₀ is injective.

Proof. It is well-known thata b^∗ = 0 whenevera, bare two orthogonal generalized non-nil tripotents that is

a b^∗ = 0 ⇐⇒ {aa^∗b}= 0 ⇐⇒ {bb^∗a}= 0 (0̸={aa^∗a} ∈Ka, 0̸={bb^∗b} ∈Kb).

Notice: the proof in [71] stated for tripotents {aa^∗a}=a, {bb^∗b}=b in weakly commuta-tive complex partial J*-triples applies in general. Define

∆(W) := {u−v : u, v ∈W, g_u⊥/g_v}, Ld := ∑

u−v=d

Kg_u g^∗_v (d∈∆(W)).

Observe that, for d ∈ ∆(W) and u, v ∈ W with u−v = d, either g_u g_v^∗ = 0 on the whole space F with g_u ⊥g_v or the operator g_u g_v^∗ does not vanish onF₀ (since g_u⊥/g_v ⇒ {g_ug_v^∗g_v} = [g_v g_v^∗]g_u ̸= 0) and maps each subspace Kg_w into Kg_w+d. Moreover, as a deep consequence of the basic structure theory presented in the Appendix, by Corollary A.2.13 we have the linear dependence

g_u1 g^∗_v1 g_u1 g_v^∗₁ (

u₁−v₂=u₂−v₂=d∈∆(W), u₁, v₁, u₂, v₂∈W) .

Actually one can also see that necessarily g_u+d⊥/g_u whenever u ̸= u+d ∈ W, but we do not need this argument. Therefore the operator spaces are 1-dimensional for d ̸= 0. That is, by writing⊕ for algebraic direct sum of subspaces, we have

Lin-hull_KF₀ F₀^∗ = ∑

u,v∈W

K g_u g^∗_v = ⊕

d∈∆(W)

Ld= L0 ⊕ ⊕

0̸=d∈∆(W)

KL_d with suitable, on F₀ non-vanishing operatorsL_d :Kg_w →Kg_w+d (w∈W).

LetA:=∑

u,v∈W αuvgu g_v^∗ and supposeA|F0 = 0. Since the restricted operatorsLd|F0

(d ∈ ∆(W)\ {0}) form a linearly independent system (as weighted shifts into different directions), necessarily A ∈ L0 = ∑

u∈WKg_u g^∗_u. By Corollary A.2.15 we may assume that the grid figure W is the set of columns of the Peirce matrix [

πij

]

i,j∈I (see A.2.3 and A.2.7) of a Cartan factor. Concerning Peirce matrices of Cartan factors, it is shown in [69]

(or see [90] Appendix) that there is a directed connected graph P in W (a path with at most one junction) such that ∆₀ := {u−v : (u, v) edge ofP} is a root base for ∆(W) in the sense that each d∈∆(W) is a positive or negative integer linear combination from

∆0. Hence the set of all vertices of P (denoted also by P) is an Q-linear (rational) basis in Span_QW and given any w ∈ W and for any v, w ∈ W there exists a finite sequence u₁, v₁, . . . , u_n, v_n∈P such that u−v =∑

k(u_k−v_k) where each term (u_k, v_k) or (v_k, u_k) is an edge of P. According to Remark A.2.7, {gu, gv, gw, gz} is always a triangle, quadrangle or diamond wheneveru−v+w−z = 0 andu̸=v, zforu, v, w, z ∈W. Thereforegiven any point w∈W\P, there exists a finite sequencew₀, u₁, v₁, . . . , w_n−1, u_n−1, v_n−1, w_n such that w0 ∈P, wn =w, ui, vi ∈P and {wi−1, ui, vi, wi} is a triangle, quadrangle or diamond with w_i =w_i−1+ (u_i −v_i) (i = 1, . . . , n). By Proposition A.2.12 it follows that each operator g_w g_w^∗ (w ∈ W) is a linear combination from D0 := {g_p g_p^∗ : p ∈ P}. In particular A = ∑

p∈Pαepgp g_p^∗ with suitable constants αep ∈ K. Since Cartan factors are Hermitian, (π_uv)

u,v∈W = S( π_vu)

u,v∈WS⁻¹ for some diagonal matrix S (cf. [90] Lemma 3.7). Since the columns {π_•p : p ∈ P} are linearly independent, hence the rows {π_p• : p ∈ P} and consequently the operators {g_p g_p^∗|F : p ∈ P} are also linearly independent. Therefore A|F = 0 implies αe_p = 0 (p∈P).

Chapter 6

Banach-Stone type theorem for continuous Reinhardt domains

Given two locally compact topological Hausdorff spaces Ω and Ω, the classical Banach–e Stone theorem asserts that any surjective isometry U :C0(Ω)e → C0(Ω) with respect to the spectral norms ∥f∥_∞ = max|f| has the form U f(ω) = u(ω)f(T ω) (f ∈ C0(Ω), ω ∈ Ω) with some bijection T : Ω ↔ Ω and a functione u : Ω→ C with |u| = 1. Our aim in this chapter is to achieve an analogous description if we replace spectral norms with arbitrary Banach lattice norms with respect to the natural pointwise ordering of the functions. We conclude the following main results.

Theorem 6.1.1. Given a complex lattice norm ∥.∥ onC0(Ω), there is a (unique) partition Π of the space Ω into pairwise disjoint finite subsets such that η, ω ∈S for some S ∈Π if and only if [

exp(iA)f]

(ω) =λ_ω,ηf(ω) (f∈C0(Ω)) for some ∥.∥-hermitian operator A and a constant λ_ω,η and the restrictions to members of Π of any ∥.∥-hermitian operator have the form

Af|S =a^A(S)[ f|S

] (f∈C0(Ω), S∈Π) (6.1.2)

with a (unique) family of linear maps a^A(S) : C(S) → C(S). Given any partition member S ∈Π, there exists a(unique) inner product⟨

..⟩

S on the finite-dimensional function space C(S) such that

{f|S : ∥f∥ ≤1}

={

φ∈ C(S) : ⟨ φφ⟩

S ≤1}

(S∈Π). (6.1.3) Theorem 6.1.4. Let U :C0(Ω)e → C0(Ω) be a surjective linear isometry with respect to two complex Banach lattice norms∥.∥^∼and ∥.∥. Write Π,[⟨

..⟩

S: S∈Π]

andΠ,e [⟨

..⟩_∼

Z : Z∈Πe] for the respective partitions and families of inner products associated with these norms by Theorem6.1.1. Then there exists a bijectionT : Π↔Πe along with a family [

u(S) : S∈Π] of surjective linear ⟨

..⟩_∼

T(S) → ⟨ ..⟩

S unitary operators u(S) : C(T(S))→ C(S) such that the sets S and T(S) have always the same cardinalities and

Ufe|S =u(S)fe|T(S) (fe∈C0(Ω), Se ∈Π). (6.1.5)

In course of the proofs we achieve a more detailed description of the partition Π and the inner products ⟨..⟩

S in terms of some geometrical data of the unit ball of the norm

∥.∥. In several steps we follow a remarkably similar pattern to some arguments appearing also in [1, 2, 20, 21, 47] and [79]. A heuristical reason for this fact is that Theorems 6.1.1 and 6.1.4 can be formulated in terms of the atomic part of the dual lattice. However, we have to establish that dual hermitian operators preserve both the atomic and continuous parts which requires new arguments in Section 6.2. It seems also (see Remark 6.3.5) that even earlier results on generalized orthogonal systems [20, 47, 79] in atomic lattices cannot reduce our treatment essentially.

Besides the independent interest for all researchers in Banach space geometry, we are motivated by problems in infinite-dimensional complex analysis concerning generalized Reinhardt domains. A classical Reinhardt domain is an open connected 0-neighborhood in the spaceC^N of all complex N-tuples, being invariant under all coordinate multiplications (z₁, . . . , z_N)7→(λ₁z₁, . . . , λ_nz_N) with |λ₁|, . . . ,|λ_N| ≤1. Regarding C^N as a the complex ordered space of the functions z :{1, . . . , N}→C, this property can be stated as

(CR) f ∈D and |g|=|f| ⇒ g ∈D .

Postulating (CR) in terms of the order absolute value, we can speak of bounded com-plete Reinhardt domains in complex Banach lattices in a natural manner. In particular a bounded convex complete Reinhardt domain in a normed complex vector lattice is the unit ball of some equivalent lattice norm (a norm with|f| ≤ |g| ⇒ ∥f∥ ≤ ∥g∥).

In 1974 Sunada [97],[98] investigated the structure of bounded classical Reinhardt do-mains from the viewpoint of holomorphic equivalence. He established that holomorphically equivalent bounded Reinhardt domains containing the origin in Cⁿ admit linear equiva-lences which preserve the positive coneRⁿ₊. In the light of later developments, holomorphic equivalence is nothing more than linear equivalence in the category of bounded convex Reinhardt domains in Banach lattices. Indeed, since 1976 we know [54],[11],[107] that holomorphically equivalent bounded circular domains in Banach spaces are linearly iso-morphic. Sunada’s Lie algebraic methods were peculiar to finite dimensions. Motivated by this fact, several concepts of infinite-dimensional Reinhardt domains appeared soon.

The results ranged in contexts of various sequence spaces, separable Banach spaces with unconditional basis and atomic Banach lattices [20],[47],[79],[108],[2],[1] with the common features that they entailed positive linear equivalence from holomorphic equivalence as a consequence of a direct decomposability of the underlying space to so called Hilbert com-ponents. In 2003, inspired by an interesting work of Vigu´e [110] on the possible lack of symmetry of continuous products of discs with different radius, in [94] we introduced the following concept which will play a chief role in this and the next chapter.

Definition 6.1.6. By a CRD(abbreviation for continuous Reinhardt domain) we mean a bounded domain containing the origin and satisfying (CR) in the C*-algebra of all bounded continuous functions over some locally compact topological space or which is the same, in a commutative C*-algebra.

It seems, as far only symmetric CRDs were intensively investigated: in [94] we achieved a rather precise description for them by showing they are some topological mixture of finite-dimensional Euclidean balls, essentially more involved than direct sums of topo-logical products of balls. Later on [43] matrix representations were found for the linear isomorphisms between two symmetric CRDs. To prove these results, we intensively used the Jordan theory of the bidual embedding of symmetric domains. However, the main points of both Sunada’s and Vigu´e’s papers concern the non-symmetric case which we settle in Theorems 6.1.1 and 6.1.4 with completely different tools along with a disproof of the seemingly plausible continuous analog of Sunada’s theorem.

Theorem 6.1.7. There are linearly isomorphic CRDs without admitting a linear isomor-phism which maps real valued functions to real valued functions.

6.2 Hermitian operators in the dual space

Henceforth let Ω be an arbitrarily fixed locally compact topological Hausdorff space and, as usually, letC0(Ω),Cb(Ω) andB(Ω) denote the complex Banach spaces of all continuous func-tions vanishing at infinity, bounded continuous funcfunc-tions and bounded Borel-measurable functions Ω→C, respectively, equipped with the spectral norm∥f∥_∞ := sup|f|. We keep fixed the notations

E :=C0(Ω), D for a CRD inE, ∥.∥for a norm with co(D) = {f ∈E : ∥f∥= 1}. Notice that the convex hull of a CRD is a CRD again¹ whose gauge function is necessarily a lattice norm being equivalent to ∥.∥_∞ (according to [76] Cor.4 of Thm.5.3). Since the norms ∥.∥ and ∥.∥_∞ are equivalent, their continuous linear functionals coincide and the common dual space admits the Riesz representation

C0(Ω)^′ = dM(Ω) ={

dµ: µ∈ M(Ω)}

, dµ:C0(Ω) ∋f 7→∫ f dµ

whereM(Ω) denotes the family of all complex Radon measures with bounded total varia-tion. For any Borel setS ⊂Ω we letM(S) :={

µ∈ M(Ω) :µ(U) = 0 forU Borel⊂Ω\S} . In the sequel we shall use the notations

∥dµ∥∗ := sup

f∈D|∫

f dµ|, µ∈ M(Ω) and D_∗ :={dµ: µ∈ M(Ω), ∥dµ∥∗ <1} for the dual norm of∥.∥ and its open unit ball, respectively. To simplify formulas, in later calculations we shall write∫

f Ldµ instead of the operator form [Ldµ]f whenever Lis any

1Proof. AssumeC ⊂ C0(Ω) is an bounded open convex set satisfying (CR) and letf, g ∈ C0(Ω) with f ∈C and|g| ≤ |f|. For k = 1,2 define fk :=Tk(f, g) whereT1, T2 :{

(ζ1, ζ2)∈ C² : |ζ1| ≥ |ζ2|}

→C are the continuous transformationsTk(ζ1, ζ2) :=ζ2+ (−1)^ki[ζ2/|ζ1|]

√

|ζ1|²− |ζ2|²for (ζ1, ζ2)̸= (0,0) and Tk(0,0) := 0. Then we havef1, f2∈ C0(Ω) and|f1|=|f2|=|f|. Property (CR) of C impliesf1, f2∈C.

On the other handg=¹₂f1+¹₂f2∈ ¹₂C+¹₂C=C.

self-mapping of dM(Ω). Furthermore we writeϕinstead ofM_ϕ. With these conventions we have the handsome formal identity [ϕ dµ]f =∫

f ϕ dµfor anyf ∈ C0(Ω). Given a bounded linear functional Φ on C0(Ω), we introduce the formal notations ∫

SΦ and supp(Φ) for the value µ(S)(= ∫

Sdµ) respectively the support of the unique measure µ∈ M(Ω) satisfying Φ = dµ. Observe that ω ∈ supp(Φ) if and only if for every neighborhoodU of the point ω there is a function f ∈ C0(Ω) vanishing outsideU and such that Φ(f)̸= 0. In particular supp(gΦ) = supp(g)∩supp(Φ) if g ∈ Cb(Ω) where supp(g) := closure{ω : g(ω) ̸= 0}. We shall denote with δ_ω the measure with unit mass supported on {ω} and 1_S stands for the indicator function of a Borel subset S in Ω (that is ∫

f dδ_ω = f(ω) for f ∈ B(Ω) and 1_S(η) = [

1 if η ∈ S, 0 else]

). Notice that 1_{ω}Φ = M_1{ω}Φ = (∫

{ω}Φ) dδ_ω for any Φ∈ C0(Ω)^′. For later use we remark the following.

Lemma 6.2.1. Assume F is a subspace inC0(Ω)^′ such that f F ⊂F, f ∈ C0(Ω). We have dim(F)≥n if and only if there are functionals 0̸= Φ₁, . . . ,Φ_n∈ F with pairwise disjoint support. If dim(F)<∞ then F = dM(S) for some set S ⊂Ω with #S = dim(F).

Proof. Define S := ∪

Φ∈F supp(Φ) and consider a sequence ω₁, . . . , ω_m ∈ S. There are open sets U₁, . . . , U_m ⊂ Ω with pairwise disjoint closures such that ω_k ∈ U_k. We can choose Ψ₁, . . . ,Ψ_m ∈ F and g₁, . . . , g_m ∈ C0(Ω) with ω_k ∈ supp(Ψ_k), g_k(Ω\U_k) = 0 and

∫ g_kΨ_k = Ψ_k(g_k) ̸= 0. Then the functionals Φ_k := g_kΨ_k have pairwise disjoint supports and hence they are are linearly independent. Thus dim(F)≥#S≥sup{

m:∃Φ₁, . . . ,Φ_m∈ F \ {0} supp(Φ_k)∩supp(Φ_ℓ) (k̸=ℓ)}

. Suppose #S < ∞. By choosing m to be maximal withS={ω₁, . . . , ω_m}, we see thatF ⊂ {Φ : supp(Φ) ⊂S}= dM(S) =∑m

k=1Cdδ_k. Definition 6.2.2. We extend slightly the concept ofhermitian operatorsin Banach spaces.

Following the terminology established in [1], given a bounded domainGin a Banach space E, we say that a continuous linear map A : E → E is G-hermitian if exp(itA)G =G for all t ∈ R. We shall write Her(G) for the family of all G-hermitian operators. Though this notation does not contain any explicit hint to space E and the underlying norm, the domainG itself determines E up to norm equivalence unambiguously.

Lemma 6.2.3. Let L ∈ Her(D_∗), ω ∈ Ω and f : Ω → R a bounded Borel function such that f(ω) = 0. Then the operator Le:=f L1_{ω}+ 1_{ω}Lf is D_∗-hermitian. We have Le = 0 if f Ldδω = 0. Otherwise the constant λ:=∫

{ω}Lf²Ldδω is strictly positive and exp(itL)e dδ_ω = cos(λ^1/2t)dδ_ω+iλ^−1/2sin(λ^1/2t)f Ldδ_ω (t ∈R).

Proof. The multiplication operators with the bounded real valued Borel functions f and 1_{ω} are D_∗-hermitian. The commutator of two D_∗-hermitian operators is i-times a D_∗ -hermitian operator and hence Le = −[f,[1_{ω}, L]]∈ Her(D_∗). By writing δefor the unique measure with deδ =f Ldδ_ω, direct calculation yields that Ledδ_ω = deδ and Ledeδ =λdδ_ω. In particular the two-dimensional subspace spanned by {dδ_ω, dδe} is L-invariant and fore allt ∈Rwe have

exp(itL)e dδ_ω = {

dδ_ω+itdeδ if λ= 0, cos(λ^1/2t)dδω+iλ^−1/2sin(λ^1/2t)deδ if λ̸= 0.

Since Le ∈ Her(D_∗), the orbit {exp(itL)e dδ : t ∈ R} must be bounded. This is possible only if λ= 0 andf Ldδ_ω = deδ= 0 or if λ >0.

Corollary 6.2.4. Given a D_∗-hermitian operatorL and a point ω ∈Ω, we have

∥gLdδ_ω∥²_∗ =⟨

gg⟩(L,ω) (

g ∈ B(Ω)) in terms of the sesquilinear form ⟨

gh⟩(L,ω) According to [94], multiplication operators with Borel functions of module 1 are ∥.∥_∗ -isometries. In particular ∥gLdδω∥_∗ = ∥|g|Ldδω∥_∗. Thus we may assume without loss characterization [7] of hermitian operators establishes ∫

{ω}Ldδ_ω = ∆_ω(Ldδ_ω)∈R. Then

Recall that the unit ball D of the norm ∥.∥ is a bounded open neighborhood of the origin with respect to the natural maximum-norm ∥.∥∞ in C0(Ω). Therefore we can fix a natural numberN_D such that

N_D⁻¹∥.∥_∞≤ ∥.∥ ≤N_D∥.∥_∞. (6.2.6)

Lemma 6.2.7. Given a D_∗-hermitian operator L and a point ω ∈ Ω, the support of the

Recall that Radon measures with finite total variation admit a unique decomposition into atomic and continuous part. That is M(Ω) = Mat(Ω)⊕ Mc(Ω) where Mat(Ω) :=

Theorem 6.2.8. Any D_∗-hermitian operator L preserves the subspaces dMat(Ω) and dMc(Ω) of C0(Ω)^′.

Lemma 6.2.9. Given any operator L∈Her(D_∗), the matrix a^(L) =[

a^(L)_ηω]

η,ω∈Ω , a^(L)ηω :=∫

{η}Ldδ_ω (6.2.10)

indexed with the points of the space Ω has at most N_D⁴ non-zero entries in every column [a^(L)ηω

]

η∈Ω respectively every row [ a^(L)ηω

]

ω∈Ω. Furthermore a^(L) is self-adjoint with respect to the inner product

Thus an application of Corollary 6.2.4 with the indicator function f := 1_{η} yields 1_{η}Ldδ_ω²

Remark 6.2.13. By the Alaoglu-Bourbaki theorem, the unit ball D_∗ of the dual norm

∥.∥_∗ is weak*-compact in C0(Ω)^′ = dM(Ω). We have not applied this property during the considerations in Section 3. Therefore the statements in 6.2.3-6.2.12 hold when ∥.∥_∗ denotes any lattice norm on dM(Ω) with respect to the natural ordering and D_∗ is the unit ball of ∥.∥_∗. By [76] Cor.4 of Thm.5.3, the norm ∥.∥_∗ is necessarily equivalent to ∥.∥₁ even in this more general setting and the constantN_D in (6.2.6) can be replaced with some N without reference to a ball in the predual.

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