THE JAMISON-BOTELHO CONJECTURE

In 2008, in a finally unpublished draft [2] F. Botelho and J. Jamison launched the following conjecture:

1.1. Conjecture. Let X_{1}, X_{2} be complex Banach spaces. Suppose t ↦ U_{k}^{t} (k = 1, 2) are maps IR → U(X_{k}) := { surjective linear isometries of X_{k}} with the one-parameter group property U_{1}^{t+s} ⊗ U_{2}^{t+s} = [U_{1}^{t} ⊗ U_{2}^{t}][U_{1}^{s} ⊗ U_{2}^{s}] (t, s ∈ IR) and such that for every fixed bounded bilinear functional ϕ : X_{1}×X_{2} → C and for every couple (x_{1}, x_{2}) ∈ X_{1}×X_{2} , the function t ↦ ϕ(U_{1}^{t}x_{1}, U_{2}^{t}x_{2}) is continuous. Then both t ↦ U_{1}^{t} and t ↦ U_{2}^{t} are strongly continuous one-parameter groups.^{∗}

Actually they even outlined a proof for 1.1 under certain additional hypothesis, implying a requirement on the representations of the the elementary forms which turned out to be contradictory due to the fact that we have (κ_{1}x_{1})⊗(κ_{2}x_{2}) = x_{1} ⊗x_{2} whenever κ_{1}κ_{2} = 1 . Using a completely different probabilistic approach, in course of the proof of [3, Theorem 1.1] we have shown the following slightly generalized version of the conjecture for the case with Hilbert spaces:

1.2. Theorem. Let H^{(1)}, …, H^{(N)} be Hilbert spaces. Assume [U(t) : t ∈ IR] is a oneparameter group where U(t) := U_{1,t} ⊗ ⋯ ⊗ U_{N,t} with unitary operators U_{k,t} ∈ U(H^{(k)}) having the following continuity property: t ↦ ∏_{k=1}^{N}⟨U_{k,t}x_{k}|h_{k}⟩ is continuous for every fixed x_{1}, h_{1} ∈ H^{(1)}, …, x_{N}, h_{N} ∈ H^{(N)} . Then there exist functions κ_{1}, …, κ_{N} : IR → TT (:= {κ ∈ C : |κ| = 1}) with ∏_{k=1}^{N} κ_{k} ≡ 1 such that the families [κ_{k}(t)U_{k,t} : t ∈ IR] (k = 1, …, N) are strongly continuous one-parameter groups. Thus, by Stone’s classical theorem [4, 5] there are possibly unbounded self-adjoint operators A_{k} : dom(A_{k}) → H^{(k)} (k = 1, …, N) defined on dense linear submanifolds in the respective spaces such that that U(t)=[ exp(itA_{1})] ⊗ ⋯ ⊗ [ exp(itA_{N})] (t ∈ IR) .

At first sight our arguments in [3] rely heavily upon the Hilbert space structure. In this paper we are going to investigate how far can we get rid of the scalar product. In Section 2 we revise the arguments on adjusted componentwise continuity of functions of type t ↦ x_{1,t} ⊗ ⋯ ⊗ x_{N,t} . The new results extend to to the setting of uniformly convex spaces and we get the following conclusion.

1.3. Proposition. Let X^{(1)}, …, X^{(N)} be uniformly convex Banach spaces ( or a bit more generally: reflexive spaces where weak convergence with convergence in norm implies normconvergence ) . Assume [U(t) : t ∈ IR] is a one-parameter group where U(t) = U_{1,t}⊗⋯⊗ U_{N,t} with invertible bounded linear operators U_{k,t} ∈ L(X^{(k)}) such that sup_{t∈IR} ∥U(t)∥ = sup_{t∈IR} ∏_{k=1}^{N} ∥U_{k,t}∥ < ∞. Then there exist functions ρ_{1}, …, ρ_{N} : IR → C respectively ρ̃_{1}, …, ρ̃_{N} : IR → C such that, for each k = 1, …, N ,

[ρ_{k,t}(t)U_{k,t} : t ∈ IR] is a strongly continuous family,
[ρ̃_{k,t}(t)U_{k,t} : t ∈ IR] is a one-parameter group.

∗ That is U_{k}^{t+s} = U_{k}^{t}U_{k}^{s} (t, s ∈ IR, k = 1, 2) and the Banach space valued functions t ↦ U_{k}^{t}x_{k} are norm-continuous for every fixed x_{1} resp. x_{2} .

Unfortunately the later considerations in [3, Sections 4-5] leading to probabilistic considerations seem to be too closely related with Hilbert space structure. Minor modifications seem possible when replacing L^{2} -estimates with L^{p} -type estimates resulting in perhaps technically interesting results with not much farther generality from the original Hilbert setting of [3]. We do not intend to go into the relevant details in this work. Nevertheless a perhaps rather aesthetical generalization with equivalent general Banach norms in the Hilbert spaces instead of the one by inner products can be developed due to the amenability of the additive group of IR .

1.4. Theorem. In the setting of Theorem 1.2, let |∥ · ∥|_{1}, …, |∥ · ∥|_{N} denote equivalent Banach-norms on the respective spaces H^{(k)} and let each operator U_{k,t} be a surjective |∥ · ∥|_{k} -isometry and instead of being an ⟨·| ⋯⟩_{k} -isometry (as supposed originally). Then there are equivalent inner products ⟨⟨ · | · ⟩⟩_{k} on the respective spaces H^{(k)} such that the conclusions of Theorem 1.3 hold with suitable possibly unbounded ⟨⟨ · | · ⟩⟩_{k} -selfadjoint operators A_{k} .

The above result is an immediate consequence of the following fact.

1.5. Proposition. Given a linear automorphism of a bounded circular domain D in a Hilbert space H , it is necessarily a scalar type operator as being a preserver of some equivalent inner product. Moreover given a strongly continuous one-parameter group [U^{t} : t ∈ IR] of linear automorphisms of D , or more generally a bounded strongly continuous one-parameter group of linear operators of H , there exists an equivalent [U^{t} : t ∈ IR] - invariant inner product on H .

Though it is likely that Proposition 1.5 appeared already in the literature of Banach space geometry, as far we have not found a proper reference even after an inquiry with several colleagues. Therefore we devote the short Section 3 to its proof relying upon Banach limits.

In [3] we emphasized the natural connection of Theorem 1.2 with the Jordan theory of bounded symmetric domains during the Introduction of [3]. One of the aims of this note is to work out the first step in this direction: in Section 4, with slight modifications of the proofs in [3], we obtain the full description of the unbounded JB*-derivations of the infinite-dimensional Cartan factors [6,7] of Types 1,2,3.

Theorem 1.6. Let H be a Hilbert space and let X ↦ X^{T} denote a transposition ^{∗} on L(H)

Recall that, by a classical representation theorem by F Riesz [4], each Type 1 Cartan factor is isometrically isomorphic to some L(H^{(1)}, H^{(2)}) with suitable Hilbert spaces H^{(1)}, H^{(2)} . Therefore Theorem 1.2 furnishes the complete description of its unbounded JB*-derivations as operations of the form X ↦ A_{1}X + XA_{2} with suitable possibly unbounded selfadjoint operators A_{k} ∈ L(H^{(k)}) . By taking into also account that the surjective linear isometries of a spin factor (Cartan factor of Type 4) are automatically unitary with respect to the underlying scalar product and also that the remaining exceptional Cartan factors (of Type

∗ There exists an orthonormed basis \{e_{j} : j ∈ J\} in H such that ⟨X^{T}e_{j}|e_{ℓ}⟩ = \overline{⟨Xe_{j}|e_{ℓ}⟩} for all j, ℓ ∈ J .

5,6) are finite dimensional (16 resp. 27 dimensions), hence the JB*-derivations of the enveloping JB*-triple in the Friedmen-Russo type Gelfand-Neimark theorem [8] are available for any JB*-triple.

MULTILINEAR FUNCTIONALS ON UNIFORMLY SMOOTH SPACES

Assumption. X^{(1)}, …, X^{(N)} smooth convex Banach spaces Remark. 1) [1, p.207] X is uniformly smooth if and only if X^{∗} is uniformly convex (rotond), X is uniformly convex if and only if X^{∗} is uniformly smooth.

2) [1, p.198] In a uniformly convex space weak convergence plus convergence of norms implies norm (strong) convergence.

3) [1, p.196] Uniformly convex spaces (hence also uniformly smooth spaces) are reflexive.

Lemma 1. Let ϕ^{(k)}, ψ^{(k)} ∈ [X^{(k)}]^{∗} with ∥ϕ^{(k)}∥ = ∥ψ^{(k)}∥ = 1 (k = 1, …, N) . Then ϕ^{(1)} ⊗ ⋯ ⊗ ϕ^{(N)} = ψ^{(1)} ⊗ ⋯ ⊗ ψ^{(N)} if and only if there exist κ_{1}, …, κ_{N} ∈ TT such that ψ^{(k)} = κ_{k}ϕ_{k} (k = 1, …, N) .

Proof. The part 0 ≠ ϕ^{(1)} ⊗ ⋯ ⊗ ϕ^{(N)} = ψ^{(1)} ⊗ ⋯ ⊗ ψ^{(N)} of the assumption implies

kerϕ^{(k)} = kerψ^{(k)} (k = 1, …, N).

(Proof: Due to the reflexivity of the spaces X^{(k)} , the are unit vectors x_{k} with ϕ^{(k)}(x_{k}) = 1 (k = 1, …, N) . Given any index ℓ, consider any vector y ∈ ker ψ^{(k)} . With the tuple \overline{x} := (x_{1}, …, x_{k−1}, y, x_{k+1}, …, x_{N}) we have 0 = ψ^{(1)} ⊗ ⋯ ⊗ ψ^{(N)}(\overline{x}) = ϕ^{(1)} ⊗ ⋯ ⊗ ϕ^{(N)}(\overline{x}) = ϕ^{(k)}(y) that is y ∈ ker ϕ^{(k)} showing that ker ψ^{(k)} ⊂ ker ϕ^{(k)} . Similarly also ker ϕ^{(k)} ⊂ ker ψ^{(k)} .) Hence ψ^{(k)} ∈ Cψ^{(k)} for each index. Since ∥ϕ^{(k)}∥ = ∥ψ^{(k)}∥ = 1 it follows even ψ^{(k)} ∈ TT ϕ^{(k)} . Qu.e.d.

Lemma 2. Let ϕ_{j}^{(k)} ∈ [X^{(k)}]^{∗} with ∥ϕ_{j}^{(k)}∥ = 1 (j ∈ J , k = 1, …, N) and suppose

ϕ_{j}^{(k)} ⊗ ⋯ ⊗ ϕ_{j}^{(N)}(x_{1}, …, x_{N}) → ϕ^{(k)} ⊗ ⋯ ⊗ ϕ^{(N)}(x^{(1)}, …, x^{(N)})

for any (x^{(1)}, …, x^{(N)}) ∈ X^{(1)} × ⋯ × X^{(N)} . Then there are constants [κ_{j}^{(k)} : j ∈ J , k = 1, …, N] in TT such that

∥κ_{j}^{(k)}ϕ_{j}^{(k)} − ϕ^{(k)}∥ → 0 (k = 1, …, N).

Proof. For any index j ∈ J let ε_{j} := ∑_{k=1}^{N} inf_{μ∈T T} ∥μϕ_{j}^{(k)} − ϕ^{(k)}∥. By the compactness of TT we can choose μ_{j}^{(1)}, …, μ_{j}^{(N)} ∈ TT such that ε_{j} := ∑_{k=1}^{N} ∥μ_{j}^{(k)}ϕ_{j}^{(k)} − ϕ^{(k)}∥. We show ε_{j} → 0 as follows: in the contrary case ε := lim sup_{j∈J} ε_{j} > 0 there would be a subnet j_{ν} ν ∈ N along with functionals ψ^{(k)} of norm one and constants μ^{(k)} ∈ TT such that μ_{j}^{(k)} → μ^{(k)}, ∥ϕ_{j}^{(k)} − ψ^{(k)}∥ → 0 (k = 1, …, N) . Then, however, we would have the contradiction (to the previous Lemma)

[μ^{(1)}ψ^{(1)}] ⊗ ⋯ ⊗ [μ^{(1)}ψ^{(1)}] = ϕ^{(1)} ⊗ ⋯ ⊗ ϕ^{(N)}, ∃ k μ^{(k)}ψ^{(k)} ≠ ϕ^{(k)}.

Hence it follows immediately that the choice

κ_{j}^{(k)} := μ_{j}^{(k)} for k < N and κ_{j}^{(k)} := \overline{μ^{(1)} ⋯ μ^{(N−1)}}

suits the requirements of the lemma. Qu.e.d.

Proposition. Suppose the functionals ϕ_{t}^{(k)} ∈ [X^{(k)}]^{∗} (t ∈ IR, k = 1, …, N) are of norm one and the functions

t ↦ ϕ_{t}^{(k)} ⊗ ⋯ ⊗ ϕ_{t}^{(N)}(x_{1}, …, x_{N})

are continuous for all fixed (x^{(1)}, …, x^{(N)}) ∈ X^{(1)} × ⋯ × X^{(N)} . Then there are functions t ↦ κ_{t}^{(k)} (k = 1, …, N) from IR → TT such that

∏_{k=1}^{N} κ_{t}^{(k)} = 1 (t ∈ IR), t ↦ κ_{t}^{(k)}ϕ_{t}^{(k)} is norm-continuous (k = 1, …, N).

Proof. It suffices to establish the statement locally (i.e. for any τ ∈ IR there is an open interval I_{τ} around τ where the statement holds and we may assume τ = 0 without loss of generality). Since the spaces X^{(k)} are reflexive, we can choose unit vectors x^{(k)} such that ϕ_{0}^{(1)}(x^{(1)}) = ⋯ = ϕ_{0}^{(N)}(x^{(N)}) = 1 . By their aid we define the interval I_{0} as the connected component of the open set {t ∈ IR : ϕ_{t}^{(1)}(x^{(1)}), …, ϕ_{t}^{(N)}(x^{(N)}) ≠ 0}. For the parameters t ∈ I_{0} , we define the functions t ↦ κ_{t}^{(k)} as

κ_{t}^{(k)} := \overline{ϕ_{t}^{(k)}(x^{(k)})}/|ϕ_{t}^{(k)}(x^{(k)}| (k < N), κ_{t}^{(N)} := \overline{κ_{t}^{(1)} ⋯ κ_{t}^{(N−1)}}.

To establish the norm continuity of the functions t ↦ κ_{t}^{(k)}ϕ_{t}^{(k)} (on I_{0} ) we consider a convergent sequence t_{n} → t_{0} in I_{0} and show that (∗) ∥κ_{t_{n}}^{(k)}ϕ_{t_{n}}^{(k)} − κ_{t_{0}}^{(k)}ϕ_{t_{0}}^{(k)}∥ → 0 . In view of Lemma 1, there are convergent sequences μ_{n}^{(k)} → 1 (k = 1, …, n) in TT such that ∥μ_{t_{n}}^{(k)}ϕ_{t_{n}}^{(k)} − ϕ_{t_{0}}^{(k)}∥ → 0 (k = 1, …, n) . Fix any index k. To prove (∗) it suffices to show that (∗∗) κ_{t_{n}}^{(k)}/μ_{t_{n}}^{(k)} → 1 . Observe that κ_{t}^{(k)}ϕ_{t}^{(k)}(x_{k}) = |ϕ_{t}^{(k)}(x_{k})| for any t ∈ I_{0} . Therefore

\frac{κ_{t_{n}}^{(k)}ϕ_{t}^{(k)}}{μ_{t_{n}}^{(k)}} = \frac{κ_{t_{n}}^{(k)}ϕ_{t}^{(k)}(x_{k})}{μ_{t_{n}}^{(k)}ϕ_{t_{n}}^{(k)}(x_{k})} = \frac{|ϕ_{t_{n}}^{(k)}(x_{k})|}{μ_{t_{n}}^{(k)}ϕ_{t_{n}}^{(k)}(x_{k})} = \frac{|μ_{t_{n}}^{(k)}ϕ_{t_{n}}^{(k)}(x_{k})|}{μ_{t_{n}}^{(k)}ϕ_{t_{n}}^{(k)}(x_{k})} → \frac{|ϕ_{t_{0}}(x_{k})|}{|ϕ_{t_{0}}(x_{k})|} = 1.

Qu.e.d.

Definition. The injective tensor product is the space

X^{(1)} ⊗ ⋯ ⊗ X^{(N)} := { N -linear functionals X^{(1)} × ⋯ × X^{(N)} → C }

with its natural norm. The projective tensor product X^{(1)} ˆ⊗ ⋯ ˆ⊗X^{(N)} is the space of the linear functionals on X^{(1)} ⊗ ⋯ ⊗ X^{(N)} of the form

∑_{n=1}^{∞} δ_{x_{n}^{(1)},...,x_{n}^{(N)}}, δ_{x^{(1)},...,x^{(N)}} : ϕ ↦ Φ(x^{(1)}, …, x^{(N)})

with the norm

∥φ∥ := inf { ∑_{n=1}^{∞} ∏_{k=1}^{N} ∥x_{n}^{(k)}∥ : φ = ∑_{n=1}^{∞} δ_{x_{n}^{(1)},...,x_{n}^{(N)}}}.

Remark. X^{(1)} ˆ⊗ ⋯ ˆ⊗X^{(N)} is a (metric) predual of X^{(1)} ⊗ ⋯ ⊗ X^{(N)} .

Lemma 3. Let t ↦ L_{t} be a mapping IR → L(X^{(1)} ˆ⊗ ⋯ ˆ⊗X^{(N)}) such that all the functions t ↦ L_{t}δ_{x^{(1)},...,x^{(N)}} are continuous and ∥L_{t}∥ ≤ 1 (t ∈ IR) . Then also all the functions t ↦ L_{t}φ (φ ∈ X^{(1)} ˆ⊗ ⋯ ˆ⊗X^{(N)}) are continuous.

Proof. Let φ ∈ X^{(1)} ˆ⊗ ⋯ ˆ⊗X^{(N)} be arbitrarily fixed. By definition we can represent it in the form

φ = ∑_{n=1}^{∞} δ_{x_{n}^{(1)},...,x_{n}^{(N)}}, ∑_{n=1}^{∞} ∏_{k=1}^{N} ∥x_{n}^{(k)}∥ < ∞.

Therefore the statement follows from the observation that the partial sums t ↦ ∑_{n=1}^{ν} L_{t}δ_{x_{n}^{(1)},...,x_{n}^{(N)}} (ν = 1, 2, …) form a uniformly convergent sequence of continuous functions IR → X^{(1)} ˆ⊗ ⋯ ˆ⊗X^{(N)} due to the inequalities

∥ ∑_{n=1}^{ν} L_{t}δ_{x_{n}^{(1)},...,x_{n}^{(N)}} − ∑_{n=1}^{μ} L_{t}δ_{x_{n}^{(1)},...,x_{n}^{(N)}}∥ ≤ ∑_{n=min{μ,ν}}^{max{μ,ν}} ∏_{k=1}^{N} ∥x_{n}^{(k)}∥ → 0 (μ, ν → ∞).

\underline{TYPES 2,3}

C := { complex numbers }, TT := {κ ∈ C : |κ| = 1}. H complex Hilbert space, inner product ⟨.|.⟩, norm ∥.∥, h^{∗} := [H ∋ x ↦ ⟨x|h⟩] Ball(H) := {h ∈ H : ∥h∥ < 1 unit ball, ∂Ball(H) := {h ∈ H : ∥h∥ = 1 unit sphere. g^{∗} ⊗ h^{∗} := [(x, y) ↦ g^{∗}(x)h^{∗}(y)] , g^{∗} ∧ h^{∗} := g^{∗} ⊗ h^{∗} − h^{∗} ⊗ g^{∗} .

F_{2} := { symmetric continuous bilinear functionals H × H → C} F_{3} := { antisymmetric continuous bilinear functionals H × H → C}

Lemma 0. [Stachó 2010, Lemma 2.3]. Suppose F : IR → IP(H) := {TT g : ⟨g|g⟩ = 1} is a continuous mapping with respect to the distance dist(TT g, TT h) := min_{|κ|=|λ|=1} |κg − λh|. Then F(t) = TT h_{t} (t ∈ IR) for some continuous function t ↦ h_{t} ∈ ∂Ball(H) .

Lemma 1. Let t ↦ e_{t} be a function IR → ∂Ball(H) ranging in unit vectors. If for any convergent net t_{ν} → t ∈ IR there exists a constant σ_{∗} ∈ {−1, 1} along with a subnet t_{ν_{α}} such that e_{t_{να}} → σ_{∗}e_{t} then there also exists a function σ : IR → {−1, 1} such that t ↦ σ(t)e_{t} is continuous.

Proof. We can apply [Stachó 2010, Lemma 2.3] as follows. Observe that t ↦ TT e_{t} is necessarily continuous. Indeed, if t_{ν} → t then lim inf_{ν} dist(TT e_{t_{ν}} , TT e_{t}) ≤ lim inf_{ν} min_{σ_{∗}=±1} |e_{t_{ν}} − σ_{∗}e_{t}| = 0 . Thus TT e_{t} = TT h_{t} (t ∈ IR) for some continuous function t ↦ h_{t} ∈ ∂Ball(H) , that is t ↦ κ(t)e_{t} is continuous for some κ : IR → TT . For any point t ∈ IR , we can choose an open interval I_{t} around it such that ⟨κ(s)e_{s}|κ(t)e_{t}⟩ > 0 for all s ∈ I_{t} . Each function σ_{t}(s) := sign Re(κ(s)\overline{|κ(t)}) is continuous on the the interval I_{t} . By the σ -compactness of IR , we can find a sequence ⋯ < τ_{−2} < τ_{−1} < τ_{0} < τ_{1} < τ_{2} < ⋯ such that each interval [τ_{n−1}, τ_{n}] is included in some of member of the family {I_{t} : t ∈ IR}, say [τ_{n−1}, τ_{n}] ⊂ I_{t_{n}} (n = 0, ±1, ±2, …) . The we obtain a function σ suiting the requirements of the lemma by letting σ(t) := σ_{t_{0}}(t) for t ∈ [τ_{−1}, τ_{0}] , and then, recursively for k = 1, 2, …, σ(t) := σ(τ_{−k})σ_{t_{−k}}(τ_{−k})σ_{t_{−k}}(t) for t ∈ [τ_{−k−1}, τ_{−k}] and σ(t) := σ(τ_{k−1})σ_{t_{k}}(τ_{k−1})σ_{t_{k}}(t) for t ∈ [τ_{k−1}, τ_{k}] , respectively,

Corollary 1. If t ↦ ϕ(e_{t}, e_{t}) is continuous for all ϕ ∈ F_{2} then there is a function σ : IR → {−1, 1} such that t ↦ σ(t)e_{t} is continuous.

Proof. Given any vector h ∈ H , with the functional ϕ_{h}(x, y) := h^{∗} ⊗ h^{∗}(∈ F_{2}) we see that the function t ↦ ⟨e_{t}|h⟩^{2} is continuous. Consider a convergent net t_{ν} → t ∈ IR and let t_{ν_{α}}(→ t) be its universal subnet. Since the range {e_{t} : t ∈ IR} is contained in the weakly compact closed unit ball \overline{Ball}(H) , for some vector g ∈ \overline{Ball}(H) we have

⟨e_{t_{να}} |h⟩ → ⟨g|h⟩, ⟨g|h⟩^{2} = ⟨e_{t}|h⟩^{2} (h ∈ H).

Since the mapping h → ⟨g|h⟩ is conjugate-linear, it follows that g ∈ {±e_{t}}. That is, for some σ_{∗} ∈ {±1} we have σ_{∗}e_{t} = weak lim_{ν_{α}} e_{t_{να}} . It is well-known that weak convergence entails norm convergence for nets of unit vectors in Hilbert spaces. Therefore we even have σ_{∗}e_{t} = norm lim_{ν_{α}} e_{t_{να}} whence Lemma 1 applies.

Corollary 1′ . Let t ↦ [e_{t}, f_{t}] be a function IR → { orthonormed 2-frames }. If t ↦ ϕ(ξe_{t} + ηf_{t}, ξe_{t} + ηf_{t}) is continuous for all ϕ ∈ F_{2} and ξ, η ∈ {0, ±1} then there is a function σ : IR → {−1, 1} such that t ↦ σ(t)[e_{t}, f_{t}] is continuous.

Proof. By Corollary 1, for some {−1, 1} -valued functions t ↦ κ(t), t ↦ λ(t), t ↦ μ(t) , the functions t ↦ λ(t)e_{t}, t ↦ κ(t)f_{t} , t ↦ μ(t)2^{−1/2}[e_{t} + f_{t}] are continuous. Therefore also t ↦ ⟨λ(t)|e_{t}⟩ = λ(t)μ(t) and similarly t ↦ κ(t)μ(u) are continuous and necessarily constant with value 1 or −1 . Hence the statement is immediate.

Corollary 2. Suppose t ↦ [e_{t}, f_{t}, g_{t}] is a function IR → { orthonormed 3-frames in H} such that the functions t ↦ ϕ(e_{t}, f_{t}) , t ↦ ϕ(e_{t}, g_{t}) , t ↦ ϕ(f_{t}, g_{t}) are continuous for all ϕ ∈ F_{3} . Then there exists a function σ : IR → {−1, 1} such that t ↦ σ(t)[e_{t}, f_{t}, g_{t}] is continuous.

Proof. Consider a convergent net t_{ν} → t ∈ IR and let t_{ν_{α}}(→ t) be again its universal subnet. By the weak compactness of \overline{Ball}(H) , for some vectors e_{∗}, f_{∗} ∈ \overline{Ball}(H) we have ⟨e_{t_{να}} |h⟩ → ⟨e_{∗}|h⟩, ⟨f_{t_{να}} |h⟩ → ⟨f_{∗}|h⟩, ⟨g_{t_{να}} |h⟩ → ⟨g_{∗}|h⟩ for any fixed vector h ∈ H . Given any couple of vectors x, y ∈ H , with the functional ϕ_{x,y} := x^{∗} ∧ y^{∗}(∈ F_{3}) we see that the function τ ↦ ⟨e_{τ} |x⟩⟨f_{τ} |y⟩−⟨f_{τ} |x⟩⟨e_{τ} |y⟩ is continuous. Hence we conclude that

⟨e_{τ} |x⟩⟨f_{t}|y⟩ − ⟨f_{t}|x⟩⟨e_{t}|y⟩ = ⟨e_{∗}|x⟩⟨f_{∗}|y⟩ − ⟨f_{∗}|x⟩⟨e_{∗}|y⟩ (x, y ∈ H).

That is we have e_{t}^{∗} ∧ f_{t}^{∗} = e_{∗}^{∗} ∧ f_{∗}^{∗} . It is well-known from classical linear algebra that then for some constants λ_{11}, λ_{12}, λ_{21}, λ_{22} ∈ C we have

e_{∗} = λ_{11}e_{t} + λ_{12}f_{t}, f_{∗} = λ_{21}e_{t} + λ_{22}f_{t}, det[λ_{k,ℓ}] = 1.

In particular e_{∗} ∈ Ce_{t} + Cf_{t} . Similarly (with g_{t_{να}} in place of f_{t_{να}}) , also e_{∗} ∈ Ce_{t} + Cg_{t} and therefore e_{∗} ∈ Ce_{t} that is e_{∗} = λe_{t} with |λ| ≤ 1 . Analogously f_{∗} = μf_{t} and g_{∗} = κg_{t} with |μ|, |κ| ≤ 1 as well. Since e_{t}^{∗} ∧ f_{t}^{∗} = e_{∗}^{∗} ∧ f_{∗}^{∗} , it follows λμ = 1 . Similarly λκ = μκ = 1 . Therefore κ = λ = μ ∈ {−1, 1}. By writing σ_{t} for the common value of κ, λ, μ, we get lim_{ν_{α}} e_{t_{να}} = σ_{t}e_{t} . According to Lemma 1, for suitable functions t ↦ λ(t), t ↦ μ(t), t ↦ κ(t) ranging in {−1, 1} the functions t ↦ λ(t)e_{t}, t ↦ μ(t)f_{t}, t ↦ κ(t)g_{t} are continuous. In particular also both t ↦ [λ(t)e_{t}]^{∗} ∧ [λ(t)f_{t}]^{∗} and t ↦ e_{t}^{∗} ∧ f_{t}^{∗} are continuous entailing the continuity and hence the constant being of the {−1, 1} -valued function t ↦ λ(t)μ(t) . t ↦ [λ(t)e_{t}]^{∗} ∧ [λ(t)f_{t}]^{∗} . Thus μ(t) ≡ λ(t) or μ(t) ≡ −λ(t) . Similarly κ(t) ≡ λ(t) or κ(t) ≡ −λ(t) . In any case t ↦ λ(t)[e_{t}, f_{t}, g_{t}] must be continuous.

Example of prebidual

Let F := L(H^{(1)}, H^{(2)}) be a Cartan factor of type 1, E a weak*-dense subtriple of F .

QUESTION: Is E a TRO (ternary ring of operators)? NO.

Example. Let H^{(1)} = H^{(2)} := H an infinite dimensional complex Hilbert space with conjugation. Consider

E := F_{0} + F_{1} = { compact op.-s in F } + { antisymm. op.-s in F } .

If a, b, c ∈ F_{0} and A, B, C ∈ F_{1} then \{(a+A)(b+B)(c+C)\} = \frac{1}{2}[(a+A)(b+B)^{∗}(c+C)+ (c + C)(b + B)^{∗}(a + A)] ∈ {ABC} + F_{0} because F_{0} is a self-adjoint multiplicative ideal of F . On the other hand, [(a + A)(b + B)(c + C)] = (a + A)(b + B)^{∗}(c + C) ∈ AB^{∗}C + F_{0} . Let H = ⊕_{k=1}^{3}H_{k} with dim(H_{k}) = dim(H) with self-conjugate components H_{k} = \overline{H_{k}} . If A, B, C have the (antisymmetric) matrix forms [α_{jk}I]_{j,k=1}^{3}, [β_{jk}I]_{j,k=1}^{3}, [γ_{jk}I]_{j,k=1}^{3} with respect to the decomposition H = ⊕_{k=1}^{3}H_{k} , then

{ABC} + F_{0} = [ABC] + F_{0} ⟺ {[α_{jk}][β_{jk}][γ_{jk}]} = [[α_{jk}][β_{jk}][γ_{jk}]].

We have no equality e.g. with the matrices A=B:=[ \begin{matrix}0 & 1 & 1 \\ −1 & 0 & 1 \\ −1 & −1 & 0 \end{matrix} ], C:=[ \begin{matrix}0 & 1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ].

BOUNDED STROGLY CONTINUOUS 1-PARAMETER GROUPS IN HILBERT SPACES

Let H be a Hilbert space with scalar product ⟨.|.⟩. Suppose |∥.∥| is an equivalent norm on H that is M^{−1}∥x∥ ≤ |∥x∥| ≤ M∥x∥ (x ∈ H ).

Proposition. Assume U is a surjective isometry H → H with respect to the norm |∥.∥|. Then there exists a U -invariant equivalent scalar product ⟨⟨.|.⟩⟩.^{∗}

Proof. By assumption we have

M^{−1}∥x∥ ≤ |∥x∥| ≤ M∥x∥ , M^{−2}∥x∥ ≤ ∥Ux∥ ≤ M^{2}∥x∥ (x ∈ H, t ∈ IR) .

Consider a Banach limit Λ_{N→∞} (for its existence see [??]) on the space c_{b} of all bounded bisequences as functions ZZ = {0, ±1, ±2, …} → C . Thus Λ_{N→∞} is a linear functional c_{b} → C such that

(1)  lim inf_{N→∞} f(N) ≤ Λ_{N→∞}f(N) ≤ lim sup_{N→∞} f(N)  whenever range(f) ⊂ IR .

Since the functions n ↦ U^{n}x are bounded with respect to both equivalent norms ∥.∥ and |∥.∥|, the operation

⟨⟨x|y⟩⟩ := Λ_{N→∞} (\frac{1}{2N + 1} ∑_{n=−N}^{N} ⟨U^{u}x|U^{n}y⟩ dt)

is well-defined for every couple of vectors x, y ∈ H . The sesquilinearity of the product ⟨.|.⟩ along with the inequalities (1) entail immediately that ⟨⟨.|.⟩⟩ is also a scalar product on H with

M^{−4}⟨x|x⟩ ≤ ⟨⟨x|x⟩⟩ ≤ M^{4}⟨x|x⟩ (x ∈ H) .

We complete the proof by showing the U -invariance of the new scalar product as follows. Indeed, we have

⟨⟨Ux|Uy⟩⟩ = Λ_{N→∞} (\frac{1}{2N + 1} ∑_{n=−N}^{N} ⟨U^{n+1}x|U^{n+1}y⟩) =

= Λ_{N→∞} (\frac{1}{2N + 1} [−⟨U^{−N}x|U^{−N}y⟩ + ∑_{n=−N}^{N} ⟨U^{n}x|U^{n}y⟩ + ⟨U^{N+1}x|U^{N+1}y⟩]) =

= Λ_{N→∞} (\frac{1}{2N + 1} ∑_{n=−N}^{N} ⟨U^{n}x|U^{n}y⟩ + O(N^{−1})) =

= Λ_{N→∞} (\frac{1}{2N + 1} ∑_{n=−N+m}^{N+m} ⟨U^{n}x|U^{n}y⟩) =

= ⟨⟨x|y⟩⟩ . Qu.e.d.

∗ That is ∥x∥^{2}/M ≤ ⟨⟨x|x⟩⟩ ≤ M∥x∥^{2} and ⟨⟨Ux|Uy⟩⟩ = ⟨⟨x|y⟩⟩ for some M > 0 and for all x, y ∈ H .

Proposition. Suppose |∥.∥| is an equivalent norm on H and let U := [U^{t} : t ∈ IR] be a strongly continuous one-parameter group of surjective isometries H → H with respect to the norm |∥.∥|. Then there exists a U -invariant equivalent scalar product ⟨⟨.|.⟩⟩.^{∗}

Proof. Since |∥.∥| is an equivalent norm on H and since all the operators U^{t}, U^{−t} are its isometries, for some constant M ∈ [1, ∞) we have

M^{−1}∥x∥ ≤ |∥x∥| ≤ M∥x∥ , M^{−2}∥x∥ ≤ ∥U^{t}x∥ ≤ M^{2}∥x∥ (x ∈ H, t ∈ IR) .

Consider a Banach limit L_{R→∞} (see [??]) on the space C_{b}[0, ∞) of all bounded continuous functions [0, ∞)→ C . Thus L_{R→∞} is a linear functional C_{b}[0, ∞)→ C such that

(1′)  lim inf_{R→∞} f(R) ≤ L_{R→∞}f(R) ≤ lim sup_{R→∞} f(R)  whenever range(f) ⊂ IR .

Since the functions t ↦ U^{t}x are continuous and bounded with respect to both equivalent norms ∥.∥ and |∥.∥|, the operation

⟨⟨x|y⟩⟩ := L_{R→∞} (\frac{1}{2R} ∫_{−R}^{R} ⟨U^{t}x|U^{t}y⟩ dt)

is well-defined for every couple of vectors x, y ∈ H . The sesquilinearity of the product ⟨.|.⟩ along with the inequalities (1’) entail immediately that ⟨⟨.|.⟩⟩ is also a scalar product on H with

M^{−4}⟨x|x⟩ ≤ ⟨⟨x|x⟩⟩ ≤ M^{4}⟨x|x⟩ (x ∈ H) .

We complete the proof by showing the U -invariance of the new scalar product as follows. Given any parameter s ∈ IR , we have

|∫_{I} ⟨U^{t}x|U^{t}y⟩ dt| ≤ M^{4}s∥x∥^{2} whenever I is an interval of length |s| .

Therefore

⟨⟨U^{s}x|U^{s}y⟩⟩ = L_{R→∞} (\frac{1}{2R} ∫_{−R}^{R} ⟨U^{t+s}x|U^{t+s}y⟩ dt) =

= L_{R→∞} (\frac{1}{2R} ∫_{−R+s}^{R+s} ⟨U^{t}x|U^{t}y⟩ dt) =

= L_{R→∞} (\frac{1}{2R} [∫_{−R}^{R} + ∫_{−R+s}^{−R} + ∫_{R}^{R+s}] ⟨U^{t}x|U^{t}y⟩ dt) =

= L_{R→∞} (\frac{1}{2R} ∫_{−R+s}^{R+s} ⟨U^{t}x|U^{t}y⟩ dt + O(R^{−1})) =

= L_{R→∞} (\frac{1}{2R} ∫_{−R+s}^{R+s} ⟨U^{t}x|U^{t}y⟩ dt) =

= ⟨⟨x|y⟩⟩ . Qu.e.d.

∗ That is ∥x∥^{2}/M ≤ ⟨⟨x|x⟩⟩ ≤ M∥x∥^{2} and ⟨⟨U^{t}x|U^{t}y⟩⟩ = ⟨⟨x|y⟩⟩ for some M > 0 and for all x, y ∈ H .

Proposition. In general, if [U^{t} : t ∈ IR] is a strongly continuous one-parameter group of bounded linear operators on a Banach space E such that M := sup_{t} ∥U^{t}∥ < ∞, then there exists an equivalent norm ∥|.|∥ on E such that M^{−1}∥.∥ ≤ ∥|.|∥ ≤ M with respect to which all the operators U^{t} are isometries.

Proof. By the aid of any Banach limit L_{R→∞} , a similar argument as in the previous proof shows that the functional

∥|x|∥ := L_{R→∞} (R^{−1} ∫_{−R} ∥U^{t}x∥ dt) (x ∈ E)

is a norm suiting the requirements of the proposition.

Remark. It is easy to construct a bounded one-parameter group [U^{t} : t ∈ IR] even on the 4-dimensional space Mat(2, 2, C) which cannot be of the form U^{t}X = exp(itA_{1})X exp(itA_{2}) .

Example. Let Π_{1} : X ↦ \frac{1}{2}(X + X^{T}) , Π_{2} : X ↦ \frac{1}{2}(X − X^{t}) denote the bicontractive projections onto the symmetric and antisymmetric parts of Mat(2, 2, C) , respectively. Clearly, the operators

U^{t}X := Π_{1}X + e^{it}Π_{2}X (X ∈ Mat(2, 2, C))

form a bounded (necessarily norm continuous) one-parameter group in L(Mat(2, 2, C)). Suppose U^{t}X = U_{1}XU_{2} where U_{k}^{t} = exp(itA_{k}) . Given any parameter t ∈ IR , since U^{t} preserves the manifold ran(Π_{1}) of symmetric matrices, necessarily U_{1}^{t} = U_{2}^{−t} = exp(ta)[{1 0 \atop 0 1}] for some a ∈ C .^{∗} Hence, with the antisymmetric matrix Y := [{0 −1 \atop −1 0}] we get the contradiction e^{it}Y = U^{t}Y = exp(ta)Y exp(−ta) = Y .

Example. Let E := Mat(2, 2, C) . Fix any real number ω ≠ 0 and, in terms of the entries, for any (X = [x_{jk}] and t ∈ IR define

[U^{t}X]_{jk} := x_{jk} if (j, k) ∉ {(1, 2), (2, 1)} , [U^{t}X]_{12} := e^{iωt}x_{12}, [U^{t}X]_{21} := e^{iωt}x_{21}.

Clearly [U^{t} : t ∈ IR] is a bounded (necessarily norm continuous) one-parameter subgroup of L(E) . The assumption (∗) U^{t}X = U_{1}^{t}XU_{2}^{t} (t ∈ IR, X ∈ E) leads to the following contradiction. Given any parameter t ∈ IR with e^{iωt} ≠ ±, by considering the matrices X := \left[{1 0 \atop 0 0}\right] , \left[{0 0 \atop 0 1}\right] , \left[{0 1 \atop 0 0}\right] , \left[{0 0 \atop 1 0}\right] , with the non-zero entries in (∗) we get the contradictory system of equations 1 = [U_{1}^{t}]_{11}[U_{2}^{t}]_{11} = [U_{1}^{t}]_{22}[U_{2}^{t}]_{22} resp. e^{iω} = [U_{1}^{t}]_{11}[U_{2}^{t}]_{22} = [U_{1}^{t}]_{22}[U_{2}^{t}]_{11} (implying indeed [U_{2}^{t}]_{11} = 1/[U_{1}^{t}]_{11} , [U_{2}^{t}]_{22} = 1/[U_{1}^{t}]_{22} and hence [U_{1}^{t}]_{11}/[U_{1}^{t}]_{22} = [U_{1}^{t}]_{22}/[U_{1}^{t}]_{11} = e^{iω} with the consequence [U_{1}^{t}]_{11}/[U_{1}^{t}]_{22} = 1 = e^{2iω} ≠ (±1)^{2} ).

∗ Proof: We have \frac{d}{dt}|_{t=0}U_{1}^{t}XU_{2}^{t} = A_{1}X + XA_{2} for the matrices X = [{1 0 \atop 0 1}], [{0 0 \atop 0 1}], resp. X = [{0 1 \atop 1 0}]. Hence, for the entries we get [A_{1}]_{11} = −[A_{2}]_{11} , [A_{1}]_{22} = −[A_{2}]_{22} , [A_{1}]_{12} = [A_{2}]_{12} = [A_{1}]_{21} = −[A_{2}]_{21} = 0 , resp. [A_{1}]_{11} = −[A_{2}]_{22} , [A_{1}]_{22} = −[A_{2}]_{11} . By setting a := [A_{1}]_{11} , this means that A_{1} = a[{1 0 \atop 0 1}], A_{2} = −a[{1 0 \atop 0 1}].

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