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} ].

REFERENCES

[1] B. BEAUZAMY, Introduction to to Banach Spaces and their Geometry (2nd rev. edition), North Holland Mat. Studies 68, Elsevier Science Publishers, Amsterdam–New York, 1991.