Grid triples of Z 2 type

Henceforth F denotes a J*-triple spanned by a non-nil weighted grid G:={g_w: w∈W} with non-degenerate weight figure W. For short we write λ_ef := [λ ∈ {0,±¹₂,±1} : (e e^∗)f =λf] fore, f ∈G. We use also the direct notation eRf (e, f ∈G, R = ⊤, ⊥,⊢ ,⊣,≈) for the COG relations of the elements of G(defined in terms of (8.2.1)).

Proposition 8.5.1. Let (w_k : k ∈ Z) be an arithmetic sequence in W and suppose u, v ∈W with u−v = N(w₁−w₀). Then with the notations e := g_u, f :=g_v, a_n := g_wn (n∈Z) and F₀ := Span_n∈Za_n, for some ξ ∈C we have

e f^∗|F₀ =ξa_N a^∗₀|F₀ , f e^∗|F₀ =ξa₀ a^∗_N|F₀ (n∈Z) . Proof. By setting σ := sgn(a₀) and ε:= sgn(a₀)sgn(a₁), we may assume

a_k−ℓ+m =σε^ℓ{a_ka_ℓ^∗a_m} (k, ℓ, m∈Z) , e f^∗ :a_n7→ξ_na_n+N , f e^∗ :a_n7→η_na_n−N for suitable coefficients ξ_n, η_n∈C (n∈Z). Thus

{ef^∗{akaℓ∗am}} = {{ef^∗ak}aℓ∗am} − {ak{f e^∗aℓ}^∗am}+{akaℓ∗{ef^∗am}}

σε^ℓ{ef^∗a_k−ℓ+m} = ξ_k{a_k+Na_ℓ^∗a_m} −η_ℓ{a_ka_ℓ−N^∗a_m}+ξ_m{a_ka_ℓ^∗a_m+N} ξ_k−ℓ+mσε^ℓa_k−ℓ+m+N = [ξ_kσε^ℓ−η_ℓσε^ℓ−N +ξ_mσε^ℓ]a_k−ℓ+m+N

ξ_k+ξ_m−ξ_k−ℓ+m = η_ℓε^N (k, ℓ, m∈Z) .

In particular (withℓ:= 0) we haveξ_k+ξ_m−ξ_k+m =η₀ε^N (k, m∈Z). That isξ_k^′+ξ_m^′ =ξ_k+m^′ (k, m∈Z) for the valuesξ_n^′ :=ξn−η0ε^N. It follows by induction thatξ^′_n=nξ₁^′ (n ∈Z) and hence the sequence (ξ_n: n∈ Z) is arithmetic. On the other hand (withk =ℓ =m=:n) also ξ_n=η_nε^N (n ∈Z). Thus for some α, β ∈C,

ξn=nα+β , ηn =ε^N(nα+β) (n ∈Z).

Since u−v =N(w₁−w₀), we have λ_ge−λ_gf =N(λ_ga1 −λ_ga₀) = 0 (g ∈G). Thuse ≈f and therefore λ_eee e^∗ = λ_{f f}f f^∗ [on the base space if we consider partial J*-triples]. It follows

[e f^∗, f e^∗] ={ef^∗f} e^∗−f {ee^∗f}^∗ =λf ee e^∗−λeff f^∗ =λf fe e^∗−λeef f^∗ = 0 . This means that {ef^∗{f e^∗a_n}} = {f e^∗{ef^∗a_n}} i.e. η_n{ef^∗a_n−N} = ξ_n{f e^∗a_n+N} or η_nξ_n−Na_n=ξ_nη_n+Na_n (n∈Z). Thus

η_nξ_n−N =ξ_nη_n+N (n∈Z)

ε^N(nα+β)[(n−N)α+β] = (nα+β)ε^N[(n+N)α+β]

n²|α|²+n[−N|α|²+ 2Reαβ] + [−N αβ+|β|²] =

= n²|α|²+n[N|α|²+ 2Reαβ] + [−N βα+|β|²]

which is possible only if α = 0. Therefore ξ_n = ξ₀ and η_n = η₀ = ε^Nξ₀ for every index n∈Z.

Definition 8.5.2. We introduce the notation

For every k, ℓ, m∈Z we have Using these relations, we evaluate the identity

{b_ka_k+t^∗{b_kb_k−t^∗a_i}} = {{b_ka_k+t^∗b_k}b_k−t^∗a_i} −

is independent of the index k and has absolute value 1. Thus

λ_tξ_(k−i)−t=αβ(1 +σ^tτ^t)ξ_tξ_k−i−ξ_(k−i)+t , Q_bka_k+t= Λ_ktc_k−t =ασ^k+tλ_tc_k−t for any i, k, t∈Z. We complete the proof by substituting j :=k−i.

Remark 8.5.6. It is well-known from elementary linear algebra that the bilateral shift T : (z_n: n∈Z)7→(z_n+1 : n ∈Z)

on the sequence space S :={

(z_n: n∈Z) : z₀, z₋₁, z₁, . . .∈C}

has the following spectral property: {

(n^kωⁿ: n∈Z) : ω∈C\ {0}, k= 0,1, . . .}

is a basis in S0

whereS0 :={

z ∈ S : ∃ ppolynomial p(T)z = 0}

. Moreover, each sequence (n^kω^k : n ∈ Z) is an eigenvector of orderk (with eigenvalue ω̸= 0).

Corollary 8.5.7. There exist ω₁, ω₂ ∈C\ {0} and δ ∈R such that ξ_n= αβ

2 (1 +iδ)ω₁ⁿ+ αβ

2 (1−iδ)ω₂ⁿ , λ_n= (ω₁ω₂)ⁿ (n∈Z) . The following alternatives hold

i) στ = 1, δ= 0 and |ω₁|=|ω₂|= 1, ii)στ = 1, δ= 0 and ω₂ =ω₁⁻¹,

iii) στ =−1, δ ∈R arbitrary and ω1 =−ω2, |ω1|= 1.

Proof. By Proposition 8.5.5, ξ_j+2t−αβ(1 +σ^tτ^t)ξ_tξ_j+t+λ_tξ_j ≡0. Thus with the notation of the previous remark,

[T^2t−αβ(1 +σ^tτ^t)ξtT^t+λt

](ξn: n∈Z) = 0 (t ∈Z).

Letω₁, ω₂ denote the roots of the polynomial z²−αβ(1 +στ)ξ₁z+λ₁. Therefore we have only the possibilities

1)στ = 1 , ξn = (A+Bn)ωⁿ with ω ∈ {ω1, ω2},

2)στ = 1 , ξ_n =Aω₁ⁿ+Bω₂ⁿ with ω₁ ̸=ω₂ , A, B ̸= 0 , 3)στ =−1, ξ_n =Aϱⁿ+B(−ϱ)ⁿ with ϱ∈ {ω: ω² =λ₁} for some A, B ∈C.

Case 1. We show that necessarily A =αβ, B = 0, and λ_t=ω^2t.

The condition ξ₀ =αβ implies A=αβ. Hence the relation ξ_−n= (στ)ⁿξ_n=ξ_n means (αβ−nB)ω⁻ⁿ = (αβ+nB)ωⁿ (n ∈Z).

Thus we must have ω⁻¹ = ω and B = −B. That is |ω|² = 1 and B ∈ iR. On the other hand

0 = ξ_n+2t−2αβξ_tξ_n+t+λ_tξ_n =

= [αβ+(n+2t)B]ω^n+2t−2αβ(αβ+tB)[αβ+(n+t)B]ω^n+2t+λt(αβ+nB)ωⁿ=

= [

−αβω^2t−2B(tω^2t)−2αβB²(t²ω^2t) +αβλ_t] ωⁿ+ +B[

−ω^2t−2αβB(tω^2t) +λ_t] nωⁿ .

For fixed t ∈ Z, both the coefficients of ωⁿ and nωⁿ should vanish in the last expression.

Hence B = 0 implies λ_t = ω^2t. From the assumption B ̸= 0, we get the contradiction λ_t =ω^2t+ 2αβB(tω^2t)−2αβB²(t²ω^2t)≡ω^2t+ 2αβB(tω^2t).

Case 2. For any fixed t∈Z,

0 = ξ_n+2t−2αβξ_tξ_n+t+λ_tξ_n =

= ω₁ⁿA[(

1−2αβA)

ω^2t₁ −2αβB(ω₁ω₂)^t+λ_t] + +ω₂ⁿB[(

1−2αβB)ω^2t₂ −2αβA(ω1ω2)^t+λt

] (n∈Z).

By assumptionA, B ̸= 0. Therefore, since the coefficients ofω₁, ω₂ must vanish, λ_t = 2αβB(ω₁ω₂)^t+ (2αβA−1)(ω₁²)^t=

= 2αβA(ω₁ω₂)^t+ (2αβB−1)(ω₂²)^t

for anyt ∈Z. By assumptionω₁ ̸=ω₂ and henceω₁ω₂ ̸=ω₁²,ω₁ω₂ ̸=ω²₂ in this case. Thus the coefficients of the terms (ω₁ω₂) should be the same i.e. A=B. SinceA+B =ξ₀ =αβ, necessarily

A =B =αβ/2 , λ_t = (ω₁ω₂)^t (t ∈Z).

Since |λ_t|= 1, also |ω₁ω₂|= 1. On the other handξ_−n=ξ_n (n∈Z). This means ω₁⁻ⁿ+ω₂⁻ⁿ =ω₁ⁿ+ω₂ⁿ (n ∈Z) .

Since the sequences( ζⁿ)

(ζ ∈C\ {0}) form a linearly independent family, it followsω₁⁻¹ ∈ {ω₂⁻¹, ω₁, ω₂}. However, ω₁⁻¹ ̸= ω⁻₁¹ whence ω₁⁻¹ ∈ {ω₁, ω₂}. Similarly ω₂⁻¹ ∈ {ω₁⁻¹, ω₁}. Since ω1 ̸=ω2, we have the subcases

2.1) ω⁻₁¹ =ω₁ and ω⁻₂¹ =ω₂ with |ω₁|² =|ω₂|² = 1 2.2) ω⁻₁¹ =ω₂ i.e. ω₁ =ω, ω₂ =ω⁻¹ .

Case 3. Since |ϱ| =√

|λ₁|= 1, the relations ξ₀ =αβ, ξ_−n= (στ)ⁿξ_n = (−1)ⁿξ_n imply A+B =αβ and

0 = [Aϱ⁻ⁿ+B(−ϱ)⁻ⁿ]−(−1)ⁿ[Aϱⁿ+B(−ϱ)ⁿ] = (A−B)ϱⁿ+ (B−A)(−ϱ)ⁿ for all n ∈Z. This is possible if and only if A= (1 +iδ)αβ/2, B = (1−iδ)αβ/2 for some constant δ∈R. In this case

0 = ξ_n+2t−[1 + (−1)^t]αβξ_tξ_n+t+λ_tξ_n =

= αβ 2

((1 +iδ)ϱⁿ+ (1−iδ)(−ϱ)ⁿ) [

λ_t−(−ϱ²)^t]

(n∈Z) , and hence λ_t= (−ϱ²)^t for any fixedt ∈Z.

Remark 8.5.8. Henceforth we assume W :=Z² ⊕1 and we use the abbreviation g_pq for the term g_(p,q,1).

Lemma 8.5.9. Assume

{g_pig_pj^∗g_pk}= sgn(g_pj)g_p,i−j+k (i, j, k ∈Z, p= 0,1) , {g_iqg_jq^∗g_kq}= sgn(g_jq)g_i−j+k,q (i, j, k, q ∈Z) .

Then also

{g_pig_pj^∗g_pk}= sgn(g_pj)g_p,i−j+k (i, j, k, p ∈Z) . Proof. Taking into account 8.2.8, it suffices to verify

sgn(g_pk)g_p,k±2 ={a_p,k±1g_pk^∗g_p,k±1}=Q_gp,k±1(g_pk) for all p, k ∈Z. We prove this statement by induction. By assumption

sgn(gpk)gp,k+2ζ=Qg_p,k+ζ(gpk) for p= 0,1 with k= 0, ζ= 1 or k= 1, ζ=−1;

sgn(g_jq)g_j+2ε,q =Q_gj+ε,q(g_jq) for any q, j ∈Z and ε=±1 . Thus we can apply the following induction step:

For any j, k ∈Z, ε, ζ ∈ {±1}

sgn(g_pk)g_p,k+2ζ =Q_gp,k+ζ(g_pk) (p=j, j+ε, j+2ε) sgn(g_jq)g_j+2ε,q =Q_gj+ε,q(g_jq) (q=k, k+ζ)

}

⇒ sgn(g_jk)g_j+2ε,k+2ζ =

=Qg_j+2ε,k+ζ(gj+2ε,k).

By setting

a_m :=g_j+mε,k , b_m :=g_j+mε,k+ζ , c_m :=g_j+mε,k+2ζ =Q_bma_m (m= 0,1,2), we have to establish the relation Q_c1c₀ = sgn(c₀)c₂.

Since, for any p, q the subspaces Span_j ∈Zg_pj, Span_i ∈Zg_iq are string triples, we have {a_ka_ℓ^∗a_m}=ασ^ℓ , {b_kb_ℓ^∗b_m}=βτ^ℓ (k, ℓ, m∈Z).

with some α, β, σ, τ ∈ {±1}. Thus we can apply Proposition 8.5.5 and its corollary to the strings (a_n), (b_n), (c_n). It follows in particular

Qb1a2 =λ1c0 , Qb1a0 =λ₋1c2 =λ⁻₁¹ =λ1c2

for some λ₁ ∈ T. Notice that for any g ∈ G(= {g_ij : i, j ∈ Z}), Q²_g = id because g g^∗ =±id. Thus we complete the proof by the argument

Q_c1c₀ = Q_c1(λ₁Q_b1a₂) = λ₁Q_Qb

1a1Q_b1a₂ =

= λ₁(Q_b1Q_a1Q_b1)Q_b1a₂ =λ₁Q_b1Q_a1a₂ =

= λ1sgn(a2)Qb1a0 = sgn(c0)c2

since sgn(a₂) = sgn(g_j+2ε,k) = sgn(g_jk) = sgn(g_j,k+2η) = sgn(c₀).

Remark 8.5.10. G_p :={g_pk : k ∈Z}and G^′_q:={g_jq : j ∈Z} in 8.5.9 are weighted grids with weight figure Z. Therefore (cf. 8.2.8) we can define an equivalent non-nil weighted grid G^′ :={g_pq^′ : p, q ∈Z} such that

{g^′_pig_pj^{′ ∗}g_pk^′ }

= sgn(g^′_pj)g_p,i−j+k , {g^′_iqg_jq^{′ ∗}g_kq^′ }

= sgn(g_jq^′ )g_i−j+k,q (p, q, i, j, k ∈Z) by means of the following double recursion:

g^′_pq =g_pq (p, q = 0,1), g^′_p,k+1 :=Q_g′

pkg^′_p,k−1 (p= 0,1; k >1), g_p,k^′ ₋₁ :=Q_g′

pkg_p,k+1^′ (p= 0,1; k <0), g^′_ℓ+1,q :=Q_g′

ℓ,qg_ℓ^′_−1,q (q∈Z; ℓ >1), g^′_ℓ−1,q :=Q_g′

ℓ,qg_ℓ+1,q^′ (q∈Z; ℓ <0).

Corollary 8.5.11. There exists an equivalent non-nil weighted grid {g_kℓ^′ : k, ℓ∈Z} such

that {

g^′_ug_v^′∗g_w^′ }

= sgn(g_xy^′ )g^′_u−v+w (u, v, w ∈Z² lie in one straight line) .

Proof. As we have seen, a non-nil weighted grid G^′ has the required property whenever Q_ug_v = sgn(g_v)g_2u−v for all u, v ∈ Z². By Lemma 8.5.9, we may assume without loss of generality that

{g_pig_pj^∗g_pk}= sgn(g_pj)g_p,i−j+k, {g_iqg_jq^∗g_kq}= sgn(g_jq)g_i−j+k,q (p, q, i, j, k∈Z).

Since, in any case sgn(g_pq) = sgn(g_p+2,q) = sgn(g_p,q+2) (p, q ∈Z), we can write sgn(g_pq) =αµ^pν^qκ^p·q (p, q ∈Z)

whereα := sgn(g₀₀),µ:= sgn(g₁₀)sgn(g₀₀),ν := sgn(g₀₁)sgn(g₀₀) andκ:=∏₁

k,ℓ=0sgn(g_kℓ).

Given any x, y, q ∈ Z, we can apply Proposition 8.5.5 and its corollary to the strings a_p :=g_p,y , b_p :=g_p,y+q , c_p :=g_p,y+2q (p∈Z). Since

{a_ka_ℓ^∗a_m}=α_yqσ_yq^ℓ a_k−ℓ+m, {b_kb_ℓ^∗b_m}=β_yqτ_yq^ℓ (k, ℓ, m∈Z)

where α_yq := sgn(a₀), β_yq := sgn(b₀), σ_yq := sgn(a₀)sgn(a₁) and τ_yq := sgn(b₀)sgn(b₁), it follows in particular

Q_gx+p,y+qg_xy = Q_bx+pa_x = sgn(a_x)λ_−p,yqc_x+2p = sgn(g_x,y)ω⁻_1,yq^p ω_2,yq^−p g_x+2p,y+2q =

= sgn(gxy)Ω^p_y,qgx+2p,y+2q (p∈Z)

with some constants Ω_y,q ∈ T for fixedy, q ∈Z. Analogously, by arguing with the strings a^′_q :=gxq, b^′_q :=gx+p,q,c^′_q :=gx+2p,q (q∈Z), we get

Q_gx+p,y+qg_xy = sgn(g_xy)(Ω^′_x,p)^−qg_x+2p,y+2q (q∈Z) with some Ω^′_x,p∈T for fixed x, p. Necessarily

(Ω_y,q)^p = (Ω^′_x,p)^q (x, y, p, q ∈Z) .

Here Ω_y,q = (Ω_y,q)¹ = (Ω^′_0,1)^q (y, q ∈ Z). Similarly Ω^′_x,p = (Ω_0,1)^p (x, p ∈ Z). Hence Ω_0,1 = Ω^′_0,1 and, by denoting this common value by Ω,

Q_x+p,y+qg_xy = sgn(g_xy)Ω^pqg_x+2p,y+2q (x, y, p, q ∈Z). Forζ ∈T, consider the non-nil weighted grid

G_ζ :={ζ^pqg_pq : p, q ∈Z} (ζ ∈T) . Since

{(ζ^(x+p)(y+q)g_x+p,y+q)(ζ^xyg_xy)^∗(ζ^(x+p)(y+q)g_x+p,y+q)}

=

=ζ(x+2p)(y+2q)−2pq{g_x+p,y+qg_xy^∗g_x+p,y+q}=

= (ζ⁻²Ω)^pqζ(x+2p)(y+2q)sgn(g_xy)g_x+2p,y+2q (x, y, p, q ∈Z),

by taking a square root ζ ∈ {ω : ω² = Ω}, the non-nil weighted grid G^′ :=G_ζ suits our requirements.

Definition 8.5.12. Henceforth we shall use the notation u∧v := det

(u₁u₂ v₁v₂

)

=u1v2−u2v1 for u:= (u1, u2), v := (v1, v2)∈Z².

Lemma 8.5.13. Suppose {g_ug_v^∗g_w} = sgn(g_v)g_u−v+w whenever the vectors u, v, w(∈ Z²) lie on one straight line. Then

{g_z+ug_z^∗g_z+w}= sgn(g_z)ξ_u∧vg_z+u+v = (u, v, z ∈Z²) where ξ_n := sgn(g₀₁)sgn(g₀₀)

[g_n1 g^∗₀₁ g_n0 g^∗₀₀ ]

(n∈Z) . Proof. Observe that

{g_z+ug_z^∗g_z+v} =

[ g_z+u g^∗_z g_z+u+v g_z+v^∗

]{g_z+u+vg_z+v^∗g_z+v}=

= S_z(u, v)g_z+u+v for any z, u, v ∈Z² where

S_z(u, v) := sgn(g_z+v)

[ g_z+u g_z^∗ g_z+u+v g^∗_z+v

] . Since the triple product is symmetric in the outer variables,

S_z(u, v) = S_z(v, u) (z, u, v ∈Z²) .

If v ∈Z² and the constant θ∈R is such that θv ∈Z, then for some integers k, n we have

whenever z, u, v, θv ∈ Z. Hence, as it is well-known from elementary linear algebra of determinants, the functionalsS_z satisfies

Theorem 8.5.14. LetEbe a J*-triple spanned by a non-nil weighted gridG={g_pq:p, q∈Z}

over the non-degenerate weight figure Z²⊕1 (notation see 8.5.8). If ∏₄

k=1sgn( g_u(k)

)= 1 whenever the vectorsu⁽¹⁾, . . . , u⁽⁴⁾ form a parallelogram then there exists an equivalent non-nil weighted grid G^′ :={g^′_pq : p, q ∈Z} of E along with a constant ω ∈ T∪R\ {0} such

that {

g_z+u^′ g_z^′∗g^′_z+v}

=[1

2ω^u∧v+1

2ω^−u∧v]

sgn(g^′_z)g_z+u+v^′ (z, u, v ∈Z²).

Otherwise there exists an equivalent non-nil weighted grid G^′ := {g^′_pq : p, q ∈ Z} along with a constant δ ∈R such that

{g_z+u^′ g_z^′∗g^′_z+v}

= Re(

(1 +iδ)i^u∧v)

sgn(g^′_z)g_z+u+v^′ (z, u, v ∈Z²).

All the above operations determine different J*-triple structure.

Proof. Let G^′ be a non-nil weighted grid with the properties described in the previous lemma. For the sake of simplicity, we may assume G = G^′ without danger of confusion.

We know already that we can parameterize the signs of the grid elements as sgn(g_pq) =αµ^pν^qκ^pq (p, q ∈Z)

and the triple product has the form {g_z+ug_z^∗g_z+v}= 1

2sgn(g_z)[

(1 +iδ)ω₁^u∧v + (1−iδ)ω₁^−u∧v]

g_z+u+v

on the grid G with suitable constants δ ∈ R, ω₁, ω₂ ∈ C\ {0}. It is straightforward to verify that

∏4 k=1

sgn(g_vk) =κ^{(v1−v2)∧(v3−v2)} whenever v₄ =v₁−v₂+v₃ . Furthermore we have established that only the following three cases can occur

1)κ= 1 =|ω₁|=|ω₂|, δ = 0; 2)κ= 1 =ω₁ω₂, δ = 0; 3)κ=−1, ω₁=−ω₂∈T. Moreover, since Q_gx+p,y+qg_xy= sgn(g_xy)g_x+2p,y+2q= sgn(g_xy)(ω₁ω₂)^−pqg_x+2p,y+2q (p, q ∈ Z), we have

ω1ω2 = 1 .

Thus, in Case 1)ω2 =ω∈T; in Case 2)ω1, ω2 ∈Rwith ω2 =ω₁⁻¹; in Case 3)ω2 =−ω1 =

±i. Therefore we have actually the following two cases

(i) κ= 1 , ω1 =ω , ω2 =ω⁻¹ , δ= 0 for some ω∈T∪R\ {0} , (ii) κ=−1 , ω₁ =i , ω₂ =−i .

To complete the proof, it suffices to check that the sesqui-trilinear extensions of the oper-ations

{gz+ugzgz+v}^ωα,µ,ν,κ^1,ω2,δ = 1

2αµ^z1ν^z2κ^z1z2[

(1 +iδ)ω₁^u∧v+ (1−iδ)ω⁻₁^u∧v]

gz+u+v

satisfy the Jordan identity whenever the parameters α, µ, ν, κ∈ {±1}, ω₁, ω₂∈C, δ∈R satisfy the relations described in Cases (i),(ii). Moreover it is enough to check the Jordan identity only for the grid elements. Since for the triple product{. . .}:={. . .}^ωα,µ,ν,κ^1,ω2,δ we have {g_ag_b^∗{g_xg_y^∗g_z}},{{g_ag_b^∗g_x}g_y^∗g_z},{g_x{g_bg_a^∗g_y}^∗g_z},{g_xg_y^∗{g_ag_b^∗g_z}} ∈ Cg_{a−b+x−y+z}, we have to prove that

D^ω_α,µ,ν,κ^1,ω2,δ(a, b, x, y, z) = 0 (a, b, x, y, z ∈Z²) where the function D_α,µ,ν,κ^ω1,ω2,δ is defined by the relation

1

4(αµ^b1νb2κ^b1b2)(αµ^y1νy2κ^y1y2)D_α,µ,ν,κ^ω1,ω2,δ(a, b, x, y, z)ga−b+x−y+z :=

:={g_ug_v^∗{g_xg_y^∗g_z}} − {{g_ug_v^∗g_x}g_y^∗g_z}+{g_x{g_vg_u^∗g_y}^∗g_z} − {g_xg_y^∗{g_ug_v^∗g_z}}

for { }:={ }^ωα,µ,ν,κ^1,ω2,δ. Let a, b, x, y, z ∈Z² be arbitrarily fixed.

Case (i). We can write {gugvgz}^ωα,µ,ν,κ^1,ω2,δ =[1

2ω^{(u−v)∧(w−v)}+1

2ω^{−(u−v)∧(w−v)}]

αµ^v1ν^v2gu−v+w

with suitable 0̸=ω∈T∪R. Therefore, in this case D_α,µ,nu,1^ω,1/ω,0 (a, b, x, y, z) =

= (

ω(x−y)∧(z−y)+ω−(x−y)∧(z−y))(

ω(a−b)∧(x−y+z−b)+ω−(a−b)∧(x−y+z−b))

−

−(

ω^{(a−b)∧(x−b)}+ω^{−(a−b)∧(x−b)})(

ω^{a−b+x−y)∧(z−y)}+ω^{−(a−b+x−y)∧(z−y)}) + +(

ω^{(b−a)∧y−a)}+ω^{−(b−a)∧(y−a)})(

ω^{(x−b+a−y)∧z−b+a−y)}+ω^{−(x−ba−y)∧(z−b+a−y)})

−

−(

ω^{(a−b)∧(z−b)}+ω^{−(a−b)∧(z−b)})(

ω^{(x−y)∧(a−b+z−y)}+ω^{−(x−y)∧(a−b+z−y)}) for ω∈T∪R\ {0} and a, b, x, y, z ∈Z². Since

ω^d+ω^−d=ω^d+ω^−d (ω∈T∪R\ {0} , d∈R) , the identity D^ω,1/ω,0_α,µ,nu,1(a, b, x, y, z) = 0 holds. Namely, by setting

A := (x−y)∧(z−y) =x∧z−y∧z−x∧y , B := (a−b)∧(x−b) =a∧x−b∧x−a∧b , C := (b−a)∧(y−a) =b∧y−a∧y+a∧b , D := (a−b)∧(z−b) =a∧z−b∧z−a∧b , we have

(a−b)∧(x−y+z−b) = B+C+D , (z−y)∧(a−b+x−y) = −A−C−D , (x−b+a−y)∧(z−b+a−y) = A−B+D ,

(x−y)∧(a−b+z−y) = A−B−C .

Hence indeed

(ω^{(x−y)∧(z−y)}+ω^{−(x−y)∧(z−y)})(

ω^{(a−b)∧(x−y+z−b)}+ω^{−(a−b)∧(x−y+z−b)})

−

−(

ω^{(a−b)∧(x−b)}+ω^{−(a−b)∧(x−b)})(

ω^{(z−y)∧(a−b+x−y)}+ω^{−(z−y)∧(a−b+x−y)}) + +(

ω^{(b−a)∧(y−a)}+ω^{−(b−a)∧(y−a)})(

ω^{(x−b+a−y)∧(z−b+a−y)}+ω^{−(x−ba−y)∧(z−b+a−y)})

−

−(

ω^{(a−b)∧(z−b)}+ω^{−(a−b)∧(z−b)})(

ω^{(x−y)∧(a−b+z−y)}+ω^{−(x−y)∧(a−b+z−y)})

=

= (ω^A+ω^−A)(ω^B+C+D+ω^−B−C−D)−(ω^B+ω^−B)(ω^−A−C−D +ω^A+C+D) + +(ω^C+ω^−C)(ω^A−B+D+ω^−A+B−D)−(ω^D+ω^−D)(ω^A−B−C+ω^−A+B+C) = 0.

Case (ii). With someδ∈R we can write {g_ug_vg_w}^ωα,µ,ν,κ^1,ω2,δ = Re[

(1 +iδ)i^{(u−v)∧(w−v)}]

αµ^v1ν^v2(−1)^v1v2g_u−v+w . Therefore, by settingγ := (1 +iδ)/2, with the same terms A, B, C, D as above

D_α,µ,ν,^i,−i,δ₋₁ = 4Re(γi^{(x−y)∧z−y)})Re(γi^{(a−b)∧(x−y+z−b)})−

−4Re(γi^{(a−b)∧(x−b)})Re(γi^{(a−b+x−y)∧(z−y)}) +

+(−1)^{(b−a)∧(y−a)}4Re(γi^{(b−a)∧(y−a)})Re(γi^{−(x−b+a−y)∧(z−b+a−y)})−

−4Re(γi^{(a−b)∧(z−b)})Re(γi^{(x−y)∧(a−b+z−y)}) =

= 2Re[

γi^A(γi^B+C+D+γi^−B−C−D)−γi^B(γi^A+C+D+γi^−A−C−D) + +γi^−C(γi^A−B+D+γi^−A+B−D)−γi^D(γi^A−B−C +γi^−A+B+C)]

=

= 2Re[

γ²(i^A+B+C+D−i^A+B+C+D+i^A−B−C+D−i^A−B−C+D) + +|γ|²(i^{A−B−C−D} −i^{−A+B−C−D}+i^{−A+B−C−D}−i^−A+B+C+D)]

=

= 2|γ|²Re(i^{A−B−C−D} −i^−A+B+C+D) = 0 .

Remark 8.5.15. The grid triples F_α,µ,ν,1^ω,1/ω,0 with the triple products{..^∗.}^ω,1/ω,0_α,µ,ν,1 are pair-wise non-isomorphic for different parameters ω ∈ T+ ∪ (0,1] where T+ := {ζ ∈ T : Re(ζ),Im(ζ)≥0}. On the other hand, F_α,µ,ν,1^ω,1/ω,0, F_α,µ,ν,1^1/ω,ω,0 and F_α,µ,ν,1^{−ω,−1/ω,0} are isomorphic to each other for anyα, µ, ν ∈ {±1} and ω ∈T∪R\ {0}.

The grid triples F_α,µ,ν,^i,−i,δ₋₁ with triple products {..^∗.}^i,_α,µ,ν,^−i,δ₋₁ are pairwise non-isomorphic for different parameters δ∈[0,∞). On the other hand, F_α,µ,ν,^i,−i,δ₋₁, F_α,µ,ν,^−i,i,δ₋₁ and F_α,µ,ν,^i,−i,−₋^δ₁ are isomorphic to each other for any α, µ, ν ∈ {±1}and δ∈R.

Remark 8.5.16. Givenδ ∈[0,∞), by settingθ:= arcotgδ, the triple product{..^∗.}^i,_1,1,1,^−i,δ₋₁ of the grid triple E_θ :=F_1,1,1,^i,−i,cotg₋₁ ^θ has the form

{g_ug_v^∗g_w}^i,_1,1,1,^−i,cotg₋₁ ^θ = (−1)^v1v2sin[θ−(u−v)∧(w−v)π/2]

sinθ g_u−v+w . Therefore the scaled operation

{g_ug_v^∗g_w}_θ := (−1)^v1v2sin[(u−v)∧(w−v)π/2−θ]g_u−v+w

is a triple product on E_θ for any θ ∈ (0, π/2]. Thus, by passing to the limit θ ↓ 0, the operation {..^∗.}₀ is also a well-defined non-trivial J*-triple product on the vector space Span_w∈Z2g_w such that (g_w g_w^∗)₀ :={g_wg_w^∗ · }₀ = 0 (w∈Z²).

Chapter A Appendix

A.1 Finite dimensional unitary tensor operators

For the purposes of Step (1) of the proof Theorem 2.4.17, let H⁽¹⁾, . . . ,H^(N) be fixed finite dimensional Hilbert spaces. We are aimed to describe the structure of the linear unitary operators in the space E := B(H⁽¹⁾, . . . ,H^(N)) of N-linear functionals with unit ball denoted by B :=B(E) for short. We shall use the tensorial notations of 2.4.13 along with the dual objects B^∗ :=B(E^∗),

K :={Φ∈∂B:∃!F ∈∂B^∗ ⟨F,Φ⟩= 1}, K^∗ :={F ∈∂B^∗ :∃Φ∈K ⟨F,Φ⟩= 1}. Lemma A.1.1. K^∗ ={δ_e1,...,eN : e₁ ∈∂B(H₁), . . . ,e_N ∈∂B(H^(N))}.

Proof. Since dimE < ∞, B is compact thus for any n-linear functional Φ ∈ ∂B, one can find e₁ ∈ ∂B(H⁽¹⁾), . . . ,e_N ∈ ∂B(H⁽⁾N) with Φ(e₁, . . . ,e_N) = 1. Hence K^∗ ⊂ {δ_e1,...,eN : e_j ∈ ∂B(H^(j)}. On the other hand, every E-unitary operator maps K onto itself and therefore also

U^∗K^∗ =K^∗ for all E-unitary operators. (A.1.2) From the compactness of B it follows K ̸=∅ (indeed: for any smooth norm ∥.∥1 on E,

∅ ̸= {Φ ∈ ∂B : ∥F∥1 ≤ ∥G∥1∀G ∈ ∂B} ⊂ K) whence K^∗ ̸= ∅. That is, for some unit vectors e⁰₁ ∈ H⁽¹⁾, . . . ,e⁰_N ∈ H^(N) we have δ_e⁰

1,...,e⁰_n ∈ K^∗. Now, from (A.1.2) we obtain δ_U1e⁰

1,...,Une⁰_n = (U₁⊗ · · · ⊗U_n)^∗δ_e⁰

1,...,e⁰_n ∈K^∗ whenever the U_j-s are H^(j)-unitary operators.

Thus{δ_e1,...,eN :e_j ∈∂B(H^(j))} ⊃K^∗.

Lemma A.1.3. Let ∆₁ = δ_f1,...,fN, ∆₂ := δ_g1,...,gN and ∆₃ := δ_h1,...,hN where 0 ̸= f_j,g_j,h_j ∈H^(j) (j = 1, . . . , N) and assume ∆₁+∆₂ =∆₃. Then there exists k such that for each j ̸=k we have f_j∥g_j (i.e. f_j and g_j are linearly dependent).

Proof. The statement holds obviously if for some index m, we have f_j∥h_j for all j ̸=m or f_j∥g_j for allj ̸=m. In the contrary casef_k̸∥g_kandf_m̸∥h_mfor some pair of indicesk ̸=m.

We may then suppose k = 1 and m = 2. First we show that in this case we have h₁̸∥f₁. Indeed: from h₁ ̸∥f₁ it follows that introducing the functional Φ :=e ge^∗₁ ⊗g^∗₂ ⊗ · · · ⊗g_N^∗ where eg₁ := g₁ − ∥f₁∥⁻²⟨g₁|f₁⟩f₂, the relations ⟨∆₁,Φe⟩ = ⟨∆₃,Φe⟩ = 0 ̸= ⟨∆₂,Φe⟩ hold.

One can see in the same manner that h₂̸∥ g₂. Since h₁̸∥ f₁, there exists u₁ ∈ H⁽¹⁾ with f₁ ⊥u₁ ̸⊥ h₁ and since h₂̸∥ g₂, one can find u₂ ∈ H⁽²⁾ with g2 ⊥u₂ ̸⊥ h₂. But then the functional Θ :=u^∗₁⊗u^∗₂⊗h^∗₃⊗ · · · ⊗h_N satisfies ⟨∆₁,Θ⟩=⟨∆₂,Θ⟩= 0̸=⟨∆₃,Θ⟩ which is impossible.

Proposition A.1.4. Set r_j = dimH^(j)(j = 1, . . . , N) and let U ∈ L(E) be fixed so that U|B ∈aut(B). Then one can chooseH^(j)-unitary operatorsU_j such that U =U₁⊗· · ·⊗U_n. Proof. It is enough to prove the statement only forE-unitary operators lying in a suitable neighborhood of id_E as it is well-known (see e.g. [32]). To do this, fix ε >0 such that the functionalsF :=δ_e1,...,eN,Fe :=δ_ee1,...,eeN, G:=δ_f1,...,fN,Ge :=δ_ef

1,...,efN(∈E^∗) fulfill

∃k e_k⊥ee_k,f_k ⊥ef_k and ∀j̸=k e_j∥ee_j, f_j∥ef_j (A.1.5) whenever we have

F −F , Ge −Ge ∈K^∗,∥F −Fe∥=∥G−Ge∥=√

2 , ∥F −G∥,∥Fe−Ge∥< ε, (A.1.6)

∥e_j∥=∥ee_j∥=∥f_j∥=∥ef_j∥= 1 (j = 1, . . . , N). (A.1.7) A valueε >0 with the above properties in fact exists. Otherwise there would be a sequence F_m := δ_e^m

1,...,e^m_N, Fe_m := δ_ee^m

1 ,...,ee^m_n, G_m =:= δ_f^m

1 , . . . ,f_N^m, Ge_m := δ_efm

1 ,...,ef_N^m (m = 1,2, . . .) satisfying (A.1.6),(A.1.7) for ε = _m¹ but without property (A.1.5). For a suitable index subsequence [

m_s]_∞

s=1 and for some unit vectors e_j,ee_j,f_j,ef_j we have e^m_j ^s → e_j, ee^m_j ^s →ee_j, f_j^ms → fj, ef_j^ms → efj (s → ∞, j = 1, . . . , N). Then the limits F,F , G,e Ge satisfy F = G, Fe = G,e ∥F −Fe∥ = ∥G−Ge∥ = √

2 and the contrary of (A.1.5). At the same time we also have F −F , Ge −Ge∈ K^∗ because of the figure K^∗ is closed. Thus by Lemma A.1.3,

∃!k0 ∀j̸=k0 ej∥eej. Since ∥F −Fe∥=√

2, hence ∥ek0 −eek0∥=√

2 i.e. ek0 ⊥eek0. Similarly

∃!ℓ₀ f_ℓ0 ⊥ef_ℓ0 and ∀j̸=ℓ₀ f_j∥ef_j. Since (A.1.5) does not hold, necessarily k₀ ̸=ℓ₀. However, the relations F =G, Fe=Ge entail k₀ =ℓ₀.

Now assume ∥U − id_E∥ < ε. Fix orthonormed basis {e^k_j : j = 1, . . . , r_k} ⊂ H^(k) (k = 1, . . . , N), and let us write the functional U^∗δ_e¹

1,...,e^N₁ in the formU^∗δ_e¹

1,...,eⁿ₁ =δ_f¹

1,...,f₁ⁿ

(cf. Lemma A.1.1) where f₁^k is a fixed unit vector in H^(k) (k = 1, . . . , N). It follows from the choice of ε that for arbitrary indexk, the singleton {f₁^k} can be continued to an orthonormed basis{f_j^k : j = 1, . . . , r_k} of H^(k) in a unique way so that we have

U^∗δ_e1

1,...,e^k₁⁻¹,e^k_j,e^k+1₁ ,...,e^N₁ =δ_f1

1,...,f₁^k−1,f_j^k,f₁^k+1,...,f₁ⁿ (j = 1, . . . , r_k).

Set I0 :={1, . . . ,1, j,1, . . . ,1) : k= 1, . . . , N; j= 1, . . . , rk}, I1 :=×ⁿk=1{1, . . . , rk} and let a family I ⊂ I₁ of multiindices be called thick if ∀i∈I ∀i^′∈I₁ i^′≤i ⇒ i^′ ∈ I. Observe that for any i:= (i₁, . . . , i_N)∈I₁, there is a unique complex number κ_i ∈T such that

U^∗δ_e¹

i1,...,eⁿ_in =κ_iδ_f¹

i1,...,f_inⁿ. (A.1.8)

Indeed: if not, we could find a minimal multiindex i∈ I₁ (w.r.t ≤) not satisfying (A.1.8). linearity of the mapping U, (A.1.8) yields the statement of the proposition immediately.

Assume I₁ \I ̸= ∅ and let j be minimal element of I₁ \ I. Observation: ∀i ∈ I₁

Throughout this section we use the notations and concepts established at the beginning of 5.5. In particular (F, F₀,{..^∗.}) denotes an arbitrarily fixed partial J*-triple over K. Definition A.2.1. An element 0̸= e ∈F₀ is a tripotent with sign λ∈ K if {ee^∗e} =λe.

Remark that the sign of a tripotent is unambiguously determined. We shall write sgn(e) :=

[λ∈K: e³ =λe]. Clearly, weighted grids consist of (signed) tripotents.

Lemma A.2.2. Supposee is a tripotent in F₀ with λ := sgn(e)̸= 0. Then λ∈ ReK and F₀ =⊕²k=0

{x∈F₀ : (e e^∗)x= (λ/2)x} .

Proof. It is well-known [10] that for any fixedc∈F₀ the J*-tripleF₀becomes a commutative Jordan algebra when equipped with the c-product x•^c y:={xc^∗y}. Hence, for any a∈ coefficientsand we reserve the notationλ_ab, λ_bafor them. Notice that weighted grids consist of pairwise compatible tripotents.