Elementary triples in the base

In [12] one has introduced the concept of elementary JB^∗-triples as the smallest infinite dimensional analogues of the classical finite dimensional Cartan factors. Namely, an el-ementary JB^∗-triple of Type I is an isometric copy of a space c₀(H, K) of all compact operators acting between two Hilbert spaces H, K (regarded as a subtriple of L(H, K) with the mentioned operator triple product). An elementary JB^∗-triple of Type II is an isometric copy of the subtriple c^B₀⁺(H) of some c₀(H, H) consisting of the operators with symmetric matrix with respect to some given orthonormed bases B in H. Similarly, an elementary JB^∗-triple of Type III is a copy of some spacec^B−₀ (H) (the subtriple ofc0(H, H) consisting of the operators with antisymmetric matrix with respect to some orthonormed basisB). ElementaryJ B^∗-triples of Type IV are spin factors (possibly infinite dimensional)

and those of Type V and VI are the 18- and 27-dimensional exceptional Cartan factors.

Thus, in terms of quasigrids, one can characterize elementary JB^∗-triples and irreducible JB^∗-triples admitting a norm-dense quasigrid. Indeed, if we consider an elementary JB^∗ -triple F of Type I, II or III as a subtriple of a c₀(H, K)-space as described above then F ∩r(H, K) is a norm-dense quasigrid in F. (Triples of Types IV, V, VI are quasigrids).

Our next aim will be to extend the result of the proposition to elementary JB^∗-triples of Type I, II, III.

Remark 5.4.1. Recall that if H₀, K₀ are finite dimensional Hilbert spaces then the inner derivations of the typical factor F₁ :=c₀(H₀, K₀) of Type I are the mappings of the form x 7→ iBx+ixA with any couple of self-adjoint operators A ∈ L(K₀, K₀), B ∈ L(H₀, H₀) such that trace(A) = trace(B). The inner derivations of the triples of Type II, resp. III F₂ :=c^B₀⁺(H₀) and F₃ :=c^B₀⁻(H₀)] are the mappings of the formx7→iAx+ixA^T (defined on F₂ and F₃, resp.) with any self-adjoint A ∈ L(H₀) where A^T is the operator whose matrix is the transposeof that of A with respect to the basis B.

It follows immediately that for the norm closure of the inner derivations of the infinite dimensional analogues of the factors F₁, F₂, F₃ (i.e. the spaces c₀(H, K), c^B±₀ with infinite dimensional Hilbert spaces H, K) are the mappings of the formx7→iBx+ixA resp. x7→

iAx+ixA^T (with arbitrary self-adjoint compact operatorsA:H →H and B :K →K).

Lemma 5.4.2. Given a self-adjoint operatorA with finite rank on a Hilbert spaceH, there exists a finite sequence δ₁, . . . , δ_m ∈ Re, and there are orthogonal projections P₁, . . . , P_m with finite rank such that

A=

∑m k=1

δ_kP_k,

∑m k=1

|δ_k| ≤2∥A∥.

Proof. We can write A=

∑n j=1

(λ_jR_j +µ_jS_j) λ₁≥λ₂≥· · ·≥λ_n= 0 =µ_n≥µ_n−1≥· · ·≥µ₁

with projections R₁, S₁, . . . , R_n, S_n to pairwise orthogonal finite dimensional subspaces.

Here∥A∥=λ₁−µ₁. Thus be choice m := 2n−2, δ_k :=λ_k−λ_k+1, P_k :=∑

j≤k

Q_j, δ_k+n−1 :=µ_k−µ_k+1, P_k+n−1 :=∑

j≤k

S_j (k = 1, . . . , n−1) suits our requirements.

Definition 5.4.3. Given aQ-derivation ∆ of the base E₀, we call the uniqueQ-derivation

∆ ofe E with ∆ = ∆e|E₀ the Q-extension of ∆.

Lemma 5.4.4. Let H, K be infinite dimensional Hilbert spaces with an orthonormed basis B in H. For k = 1,2,3 let E^k be a partial JB*-triple with base

E₀¹ :=c₀(H, K), E₀² :=c^B₀⁺(H), E₀³ :=c^B−₀ (H),

respectively, and in each case let Q^k be the quasigrid of all finite rank operators in E₀^k. SupposeA∈r(H, H)is a self-adjoint operator with trace (A) = 0. Then theQ^k-extensions

∆ek of the derivations

∆₁ :E₀¹ ∋a7→iaA, ∆₂ :E₀² ∋a7→iAa+iaA^T, ∆₂ :E₀³ ∋a7→iAa+iaA^T, where ·^T denotes operator transposition with respect to B, satisfy the norm-estimate

e∆_k≤4M∥A∥

whenever ∥{xa^∗x}∥ ≤M∥x∥²∥a∥ (a∈E₀^k, x∈E^k, k= 1,2,3).

Proof. Let us write the operator A in the form A=

∑m j=1

δ_jP_j,

∑m j=1

|δ_j| ≤2∥A∥

with orthogonal projections Pj ∈ r(H, H) and coefficients δj ∈ R. Choose a finite dimen-sional subspace H₀ in H such that N := dim(H₀) ≥ rank(P_j) (j = 1, . . . , n) and let P₀ denote the orthogonal projection of H ontoH₀. Consider the derivations

∆_1,j :E₀¹ ∋a7→ia(

P_j− rank(P_j) N P₀

) ,

∆_2,j :E₀² ∋a7→iP_ja+iaP_j^T,

∆_3,j :E₀³ ∋a7→i (

P_j +rank(P_j) N P₀

) a+ia

(

P_j +rank(P_j) N P₀

)T

for any index j. Observe that

∆_k =

∑n j=1

iδ_j∆_k,j (k= 1,2,3) due to the assumption trace(A) = ∑n

j=1δ_jrankP_j = 0. Thus, by the previous lemma, it suffices to check that each ∆_k,j is a Q^k-derivation of E^k₀ whose Q^k-extension ∆e_k,j to E^k satisfies the norm estimate e∆k,j≤4M.

Let us fix an index j arbitrarily and write

P :=P_j, r := rank(P).

Remark that r ≤ N. Let us also fix orthonormed bases {e₁, . . . , e_r} and {f₁, . . . , f_N} in

are partial isometries belonging to the quasigrid Q¹. Indeed, an immediate calculation shows thatI_i1,...,irI_i^∗

with the partial isometry J := whence for the Q³-extension toE³ we obtain

∆e_3,j = 2i

Remark 5.4.5. By changing the roles of the spacesK and H in the treatment of the case of E¹ above, we can see that also that the Q¹-extension∆e^′₁ of a Q¹-derivation

∆^′₁ :E₀¹ ∋a 7→iBa (B self-adjoint∈r(K, K), trace(B) = 0) satisfies the norm estimate e∆^′₁≤4M∥B∥.

Lemma 5.4.6. With the notations of Lemma 5.4.4 and Remark 5.4.5, we have the norm estimates

4M(∥A∥+∥B∥) ≥ ∥∆₁+ ∆^′₁∥ ≥max{∥A∥,∥B∥}, 4M∥A∥ ≥ ∥∆₂∥ ≥ ∥A∥,

4M∥A∥ ≥ ∥∆₃∥ ≥ ∥A∥ even if trace(A),trace(B)̸= 0.²

Proof. For the cases trace(A),trace(B) = 0, the upper estimates are already established.

Each of the quasigridsc^B±₀ (H)∩r(H, H),r(H, K) contains partial isometries of arbitrarily large rank whenever the underlying Hilbert spaces H, K are infinite dimensional. Hence, for k = 1,2,3, any Q^k-derivation of E^k can be perturbed by a Q^k-derivation of the form εI I^∗ of arbitrarily small norm in a manner such that the perturbed derivation have the forma7→iB^′a+iaA^′ with trace(A^′) = trace(B^′) = 0. Therefore the upper norm estimates extend also to the case. Next we proceed to the lower estimates.

Case of E¹. Given a couple of unit vectors u ∈ H and v ∈K, we have −i(∆₁+ ∆^′₁) (v⊗u^∗) =v⊗(Au)^∗+ (Bv)⊗u^∗,∥v⊗u^∗∥= 1. Choosing first uto be an eigenvector ofA with eigenvalue of highest module and v from the kernel of B (by assumption A, B have finite rank), we have ∥(∆₁+ ∆^′₁)(v⊗u^∗)∥=∥(Au)⊗v^∗∥ =∥Au∥ · ∥v∥=∥A∥. Therefore

∥∆₁+ ∆^′₁∥ ≥ ∥A∥. Choosingvto be an eigenvector ofB with eigenvalue of highest module and u from the kernel ofA, similarly we get∥∆₁+ ∆^′₁∥ ≥ ∥B∥.

Case of E². If u, v ∈H are unit vectors and u is an eigenvector of A with eigenvalue λ∈ {±∥A∥} and v belongs to the kernel of A then

∆₂(u⊗v^∗+v⊗u^∗) =iλ(u⊗v^∗ +v⊗u^∗).

Hence immediately ∥∆₂∥ ≥ ∥A∥.

Case of E³. A similar argument to the one used for E² with u⊗v^∗ −v⊗u^∗ instead of u⊗v^∗+v⊗u^∗.

Proposition 5.4.7. A partial J*-tripleE := (E, E₀,{..^∗.})whose base E₀ is an elementary JB^∗-triple has IDEP.

Proof. If E₀ itself is a quasigrid, the statement is already contained in Proposition 5.1.3.

If E₀ is not a quasigrid then E₀ is isometrically isomorphic to some space of the form c₀(H, K) or c^B±₀ (H) with infinite dimensional underlying Hilbert spaces H, K. Thus we may assume thatE₀ coincides with one of the mentioned operator spaces. These cases are considered in Lemmas 5.4.4, 5.4.6. In any case, letQdenote the quasigrid of all finite rank elements of E₀. Furthermore, let Dresp. D0 be the families of all Q-derivations ofE resp.

the base space E₀. According to the norm estimates in 5.4.6, the Q-extension T(∆) :=[ e∆∈ D: ∆e|E0 = ∆]

(T ∈ D0)

2For any operator c ∈ r(H, K) we have trace(C^∗c) = trace(cc^∗). Since (c c^∗)a = cc^∗a+ac^∗c)/2 (a, c∈c0(H, K)), it follows that the mapping [c0(H, K)∋a7→iBa+iaA] witha∈r(H, H),B∈r(K, K) is anr(H, K)-derivation if and only if trace(A) = trace(B).

is a bijective linear operationD0 ←→ Dsuch that∥T∥ ≤4M and∥T⁻¹∥ ≤1, (The inverse T⁻¹ : ∆e 7→ ∆e|E₀ is obviously contractive). Therefore the operation T admits a unique linear bicontinuous extension

T :D0 ←→ D

where D0 resp. D denote the closures of the linear submanifolds D0 resp. D in the spaces Der(E0) resp. Der(E) with respect to operator norm. Since the quasigridQis norm dense in E₀, and since {ia a^∗ : a ∈Q} ⊂ D0, by the continuity of the mapping a 7→ a a^∗ we have

a a^∗ = lim

Q∋a1→aaia^∗_i = lim

Q∋ai→aT(ai a^∗|E0) = T( lim

Q∋ai→aaia|E0) =

= T(a a^∗|E0) (a∈E0).

Hence, by the linearity of the mappingT,

∑n k=1

αkak a^∗_k =T (∑ⁿ

k=1

αkak a^∗_k|E0

)

= 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.

In document Dissertation for the title DSc (Pldal 74-80)