ON STRONGLY CONTINUOUS ONE-PARAMETER GROUPS OF AUTOMORPHISMS

L.L. STACHÓ

We prove structure theorems for strongly continuous one-parameter groups formed by surjective linear isometries of spaces of bounded N-linear functionals over strictly convex complex Banach spaces. Complete description is given in the case of Hilbert-equivalent norms on the basis of probability arguments. As a consequence, we classify the strongly continuous one-parameter automorphism groups of all infinite-dimensional Cartan factors of Jordan theory.

AMS 2010 Subject Classification: 46G25, 47H60, 17C99

Key words: strongly continuous one-parameter group; N-linear functional; surjective linear isometry; uniformly convex space; Cartan factor; JB*-triple

1. INTRODUCTION, MAIN RESULTS

In 2008, in a finally unpublished draft version of [2], F. Botelho and J. Jamison launched the following conjecture:

CONJECTURE 1.1. 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 oneparameter 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 [10, Theorem 1.1] we have shown the following slightly generalized version of the conjecture for the case with Hilbert spaces:

THEOREM 1.2. Let H^{(1)}, …, H^{(N)}be Hilbert spaces, [U(t) : t∈IR] be a one-parameter 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 we can find

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] are strongly continuous one-parameter groups. Thus, by Stone’s classical theorem [8, 12] there are possibly unbounded selfadjoint operators A_{k} : dom(A_{k}) → H^{(k)} defined on dense linear submanifolds such that U(t)=[ exp(itA_{1})] ⊗ ⋯ ⊗ [ exp(itA_{N})] (t ∈ IR) .

∗ 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} .

At first sight our arguments in [10] 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.

PROPOSITION 1.3. Let X^{(1)}, …, X^{(N)} be uniformly convex Banach spaces. 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 the functions t↦ ∥U(t)(x_{1,t} ⊗⋯⊗x_{N,t})∥ are all continuous. Then, for each index k =1, …, N there exist multipliers ρ_{k}, ρ̃_{k} : IR→ C such that,

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

Actually, we establish 1.3 a bit more generally: for spaces where weak convergence implies norm convergence. Also we shall see that the multipliers can be chosen to be bounded both from above and below from 0 (that is 0 < inf |ρ_{k}|, inf |ρ̃_{k}| and sup |ρ_{k}|, sup |ρ̃_{k}| < ∞). In particular, the conditions in 1.3 hold if the operators U_{k,t} are all isometries, and in this case we can choose even |ρ_{k}| = |ρ̃_{k}| ≡ 1 .

Unfortunately the later considerations in [3, Sections 4-5] leading to probabilistic arguments 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 facts with not much farther generality from the original Hilbert setting of [3]. We do not proceed into this direction.

A perhaps rather aesthetical generalization with equivalent general Banach norms in the Hilbert spaces instead of the ones defined by inner products can be developed due to the amenability of the additive group of IR .

THEOREM 1.4. 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 ( instead of being ⟨·|·⟩_{k} -unitary 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} -self-adjoint operators A_{k} .

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

PROPOSITION 1.5. Any linear automorphism of a bounded circular domain D in a Hilbert space H is necessarily a scalar type operator as being a preserver of some equivalent inner product. Given a strongly continuous oneparameter group U := [U^{t} : t ∈ IR] of linear automorphisms of D , or more generally a bounded strongly continuous one-parameter subgroup of L(H) , there exists an equivalent U -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 proper references even after an inquiry with several colleagues. Therefore we devote the short Appendix (Section 4) to its proof relying upon Banach limits.

In [10] we emphasized the natural connection of Theorem 1.2 with the Jordan theory of bounded symmetric domains during the Introduction of [10]. One of the aims of this note is to work out the first step in this direction: in Section 3, with slight modifications of the proofs in [10], we obtain the full description of the unbounded JB*-derivations of the infinite-dimensional Cartan factors [5,6] of Types 1,2,3.

Recall that, by a classical representation theorem by F Riesz [8], each Type 1 Cartan factor is isometrically isomorphic to some L(H^{(1)}, H^{(2)}) with suitable Hilbert spaces H^{(1)}, H^{(2)} . Theorem 1.2 furnishes the complete description of their unbounded JB*-derivations as operations of the form X ↦ A_{1}X + XA_{2} with suitable possibly unbounded self-adjoint operators A_{k} ∈ L(H^{(k)}) . The next theorem provides the full description of the unbounded JB*-derivations of the Cartan factors of types 2 and 3 which can be represented without loss of generality in the form F_{2} resp. F_{3} below.

THEOREM 1.6. Let H be a Hilbert space with a fixed orthonormed basis E = \{e_{j} : j ∈ J\} and let X ↦ X^{T}, \overline{X} denote the operator transposition resp. conjugation associated to E .^{∗} Assume [U(t) : t ∈ IR] resp. [V(t) : t ∈ IR] are one-parameter groups of surjective linear isometries of F_{2} := {X ∈ L(H) : X = X^{T}} resp. F_{3} := {Z ∈ L(H) : Z = −Z^{T}} such that all the functions t ↦ ⟨[U(t)X]x|y⟩ resp. t ↦ ⟨[V(t)Z]x|y⟩ are continuous for all fixed x, y∈H and X ∈ F_{2} resp. Z ∈ F_{3} . Then

U(t)X = [exp(itA)]X[exp(it\overline{A})] (X ∈ F_{2}, t ∈ IR),
V(t)Z = [exp( itB)]Z[exp(it\overline{B})] (Z ∈ F_{3}, t ∈ IR) .

Thus the possibly unbounded JB*-derivations of F_{3}, F_{3} are of the form X ↦ AX + X\overline{A} resp. Z ↦ BZ + Z\overline{B} for some possibly unbounded self-adjoint operators A, B on H .

Since the surjective linear isometries of a spin factor (Cartan factor of Type 4) are automatically unitary with respect to the underlying Hilbert space scalar product. and since the exceptional Cartan factors (of Type 5,6) are finite dimensional (16 resp. 27 dimensions), Theorem 1.6 completes the description of the possibly unbounded derivations of any Cartan factor. One may intend to apply this result to get the description of the possibly unbounded JB*- derivations for all JB*-triples^{∗∗} via the Gelfand-Neimark type theorem by Friedmen-Russo [3]. This latter asserts that any JB*-triple be embedded into a suitable ℓ^{∞} -direct sum of Cartan factors as a weak*-dense norm-closed submanifold. We finish by raising the following two related open problems:

Problems 1.7. a) Describe the pointwise continuous (resp. weakly continuous) one-parameter groups of surjective linear isometries for any JB*- triple.

∗ The matrices of X^{T} resp. \overline{X} are the transpose resp. conjugate of the matrix of X with respect to E : ⟨X^{T}e_{j}|e_{ℓ}⟩ = ⟨Xe_{ℓ}|e_{j}⟩ and ⟨\overline{X}e_{j}|e_{ℓ}⟩ = \overline{⟨Xe_{j}|e_{ℓ}⟩} for all j, ℓ ∈ J . ∗∗ Banach spaces with holomorphically symmetric unit ball. They were introduced and axiomatized algebraically by means of a three-variable product in 1983 by Kaup [7]. For an elementary introduction see e.g. [6].

b) Describe the pointwise weak*-continuous one-parameter groups of surjective linear isometries for all JBW*-triples (JB*-triples with predual ) .

c) (Non-linear issue of 1.7). Describe the pointwise continuous ( resp. pointwise weakly-, resp. weak*-continuous in the JBW*-case ) one-parameter groups of holomorphic automorphisms of the unit ball of any JB*-triple. In particular, in the case of the unit ball of a Hilbert space, are they all of the form exp ([a−⟨x|a⟩x+iAx]\frac{∂}{∂x}) with suitable vector a and a possibly unbounded self-adjoint operator A.

2. ADJUSTED CONTINUITY OF TENSOR MAPS

Henceforth, throughout the whole paper, let X^{(1)}, …, X^{(N)} be reflexive Banach spaces with duals denoted by [X^{(k)}]^{∗}, …, [X^{(N)}]^{∗} where weak convergence along with convergence in norm entails norm convergence. That is, by assumption, for each fixed index k = 1, …, N and for any net [x_{j} : j ∈ J] in X^{(k)} we have

(2.1) ∥x_{j} −x∥ → 0 whenever ∥x_{j}∥→∥x∥ and ⟨x_{j} −x, ϕ⟩→0 (ϕ∈[X^{(k)}]^{∗}).

Notice [1] that uniformly convex spaces are reflexive satisfying (2.1), and a Banach space is uniformly convex if and only if its duals uniformly smooth (and vice versa, via reflexivity).

As usually, for the natural coupling between X^{(k)} and [X^{(k)}]^{∗} we write ⟨x, ϕ⟩(:= ϕ(x)). Given any family x^{(k)} ∈ X^{(k)}, ϕ^{(k)} ∈ [X^{(k)}]^{∗} (k = 1, …, N) , x^{(1)} ⊗ ⋯ ⊗ x^{(N)} resp. ϕ^{(1)} ⊗ ⋯ ⊗ ϕ^{(N)} will denote the elementary N -linear functionals (ψ^{(1)}, …, ψ^{(N)}) ↦ ∏_{k=1}^{N}⟨x^{(k)}, ψ^{(k)}⟩ resp. (y^{(1)}, …, y^{(N)}) ↦ ∏_{k=1}^{N}⟨y^{(k)}, ϕ^{(k)}⟩. Remark that

(2.2)

0 ≠ x^{(1)} ⊗ ⋯ ⊗ x^{(N)} = y^{(1)} ⊗ ⋯ ⊗ y^{(N)} if and only if

x^{(k)} =ρ_{k}y_{k} (k =1, …, N) with ∏_{k=1}^{N} ρ_{k} =1 for some ρ_{1}, …, ρ_{N} ∈C .

It is well-known [9] that the tensor coupling

⟨x^{(1)} ⊗ ⋯ ⊗ x^{(N)}, ϕ^{(1)} ⊗ ⋯ ⊗ ϕ^{(N)}⟩ := ∏_{k=1}^{N}⟨x^{(k)}, ϕ^{(k)}⟩

admits a bounded N -linear extension to the projective tensor product of the spaces X^{(k)} with the injective tensor product of the dual spaces, and the latter is a closed subspace in B(X^{(1)}, …, X^{(N)}) := { bounded N -linear functionals X^{(1)} ×⋯ X^{(N)} → C} with its natural sup -norm. We shall speak of weak convergence in X^{(1)} ⊗ ⋯ ⊗ X^{(N)} in the sense of the tensor coupling. In particular x_{j}^{(1} ⊗ ⋯ ⊗ x_{j}^{(N)} ⟶^{w} x^{(1)} ⊗ ⋯ ⊗ x^{(N)} if ∏_{k} ϕ^{(k)}(x_{j}^{(k)} → ∏_{k} ϕ^{(k)}(x^{(k)} for fixed ϕ^{(k)} ∈ [X^{(k)}]^{∗} (k = 1, …, N) .

LEMMA 2.3. Let [e_{j}^{(k)} : j ∈ J ] be nets of unit vectors in the respective spaces X^{(k)} such that

e_{j}^{(1)} ⊗ ⋯ ⊗ e_{j}^{(N)} ⟶^{w} e^{(1)} ⊗ ⋯ ⊗ e^{(N)}

where e^{(k)} ∈ X^{(k)} (k = 1, …, N) are also vectors with norm 1 . Then there are nets of constants [κ_{j}^{(k)} : j ∈ J ] (k = 1, …, N) in TT := {κ∈ C : |κ|=1} such that

∏_{ℓ=1}^{N} κ_{j}^{(ℓ)} = 1, ∥κ_{j}^{(k)}e_{j}^{(k)} − e^{(k)}∥ → 0 (k = 1, …, N).

Proof. Let TT_{0}^{N} := {(κ_{1}, …, κ_{N}) ∈ TT^{N} : ∏_{k} κ_{k} = 1}, and for any index j ∈J define ε_{j} resp. (κ_{j}^{(1)}, …, κ_{j}^{(N)}) as the minimal value resp. a minimum

location of the continuous function Δ_{j}(κ_{1}, …, κ_{N}) := ∑_{k=1}^{N} ∥κ_{k}e_{j}^{(k)} − e^{(k)}∥ on

the compact domain TT_{0}^{N} . We have to show ε_{j} → 0 . As a consequence of the compactness of TT along with the weak compactness of the closed unit ball in each X^{(k)} , we can choose a subnet [j_{ν} : ν ∈ N ] of J along with vectors y^{(k)} of norm ≤ 1 such that

(2.4) ε_{j_{ν}} →lim sup_{j} ε_{j} , ⟨κ_{j}^{(k)}e_{j_{ν}}^{(k)}−y^{(k)}, ϕ^{(k)}⟩→0 for all ϕ^{(k)} ∈[X^{(k)}]^{∗}

for k =1, …, N . By passing to j_{ν} -limits we have

y^{(1)} ⊗ ⋯ ⊗ y^{(N)} = e^{(1)} ⊗ ⋯ ⊗ e^{(N)}.

Since ∏_{k} ∥y^{(k)}∥ = ∥y^{(1)} ⊗ ⋯ ⊗ y^{(N)}∥ = ∥e^{(1)} ⊗ ⋯ ⊗ e^{(N)}∥ and ∥y^{(1)}∥, …, ∥y^{(1)}∥≤1 , all the y^{(k)} must be unit vectors. Since also ∥e^{(k)}∥=1 , in view of (2.2) here we even have y^{(k)} = ρ^{(k)}e^{(k)} with suitable constants ρ^{(k)} ∈ TT . Thus, by our basic topological assumption on the equivalence of weak and norm convergence on the unit sphere of the spaces X^{(k)} , from (2.4) it follows

κ_{j_{ν}}^{(k)}e_{j_{ν}}^{(k)} → y^{(k)} = ρ^{(k)}e^{(k)} in norm,

Δ_{j_{ν}} (κ_{j_{ν}}^{(1)}, …, κ_{j_{ν}}^{(N)}) → ∑_{k=1}^{N}∥y^{(k)}−e^{(k)}∥= ∑_{k=1}^{N}|ρ^{(k)}−1|.

By the definition of the terms κ_{j}^{(k)} by means of Δ_{j} , for any fixed tuple (κ_{1}, …, κ_{N}) ∈ TT_{0}^{N} we have ∑_{k} ∥κ_{k}e_{j}^{(k)} − e^{(k)}∥ ≥ ∑_{k} ∥κ_{j}^{(k)}e_{j}^{(k)} − e^{(k)}∥. Since, by taking a convergent subnet [(κ_{j_{ντ}}^{(1)} , …, κ_{j_{ντ}}^{(N)}) : τ ∈ T ] with limit

point (ξ_{1}, …, ξ_{N}) ∈ TT_{0}^{N} , we have e_{j_{ντ}}^{(k)} = [κ_{j_{ντ}}^{(k)} ]^{−1}κ_{j_{ντ}}^{(k)} e_{j_{ντ}}^{(k)} → \overline{ξ_{k}}ρ^{(k)}e^{(k)} . Hence we conclude that

∑_{k=1}^{N} |κ_{k}\overline{ξ_{k}} − 1| = ∑_{k=1}^{N} ∥κ_{k}\overline{ξ_{k}}ρ^{(k)}e^{(k)} − e^{(k)}∥ ≥ ∑_{k=1}^{N} ∥ρ^{(k)}e^{(k)} − e^{(k)}∥

for any (κ_{1}, …, κ_{N}) ∈ TT_{0}^{N} . With the choice κ_{k} := ξ_{k} (k = 1, …, N) we get ρ^{(k)} = 1 (k = 1, …, N) and lim sup_{j} ε_{j} = lim_{ν} Δ_{j_{ν}} (κ_{j_{ν}}^{(1)}, …, κ_{j_{ν}}^{(N)}) = 0 which completes the proof.

PROPOSITION 2.5. Suppose e_{t}^{(k)} ∈ X^{(k)} (t ∈ IR, k = 1, …, N) are unit vectors such that function t ↦ e_{t}^{(1)} ⊗ ⋯ ⊗ e_{t}^{(N)} is weakly continuous. Then one can find a function t ↦ (κ_{t}^{(k)}, …, κ_{t}^{(k)}) from IR to TT_{0}^{N} := {(κ_{1}, … κ_{N}) ∈ TT : ∏_{k} κ_{k} = 1} such that the functions t ↦ κ_{t}^{(k)}e_{t}^{(k)} (k = 1, …, N) are norm-continuous.

Proof. An analogous argument with the σ -compctness of the real line as in [10, proof of Lemma 2.3] shows: it suffices to see that for any τ ∈ IR there is an open interval I_{τ} around τ where the statement holds. Let us fix τ arbitrary. By choosing a family ϕ^{(k)} ∈ [X^{(k)}]^{∗} (k = 1, …, N) of functionals with ⟨e_{τ}^{(k)}, ϕ^{(k)}⟩ = ∥ϕ^{(k)}∥ = 1 (guaranteed by the Hahn-Banach theorem), define the interval I_{τ} as the connected component of τ in the open set {t ∈ IR : ∏_{k=1}^{N}⟨e_{t}^{(k)}, ϕ^{(k)}⟩ ≠ 0}. For the parameters t ∈ I_{τ} , we define the functions t ↦ κ_{t}^{(k)} as

κ_{t}^{(k)} := |⟨e_{t}^{(k)}, ϕ^{(k)}⟩|/⟨e_{t}^{(k)}, ϕ^{(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_{τ} ) we consider a convergent sequence t_{n} → t_{0} in I_{τ} and show that

(2.6)  ∥κ_{t_{n}}^{(k)}e_{t_{n}}^{(k)} − κ_{t_{0}}^{(k)}e_{t_{0}}^{(k)}∥ → 0 .

In view of Lemma 2.3, there are convergent sequences [μ_{t_{n}}^{(k)} : n = 1, 2, …] (k = 1, …, N) in TT such that ∥μ_{t_{n}}^{(k)}e_{t_{n}}^{(k)} − e_{t_{0}}^{(k)}∥ → 0 (k = 1, …, N) . Fix any index k. To prove (2.6) we have to see that

κ_{t_{n}}^{(k)}/μ_{t_{n}}^{(k)} → κ_{t_{0}}^{(k)} .

Observe that κ_{t}^{(k)}⟨e_{t}^{(k)}, ϕ^{(k)}⟩ = |⟨e_{t}^{(k)}, ϕ^{(k)}⟩| for any t ∈ I_{0} . Therefore

\frac{κ_{t_{n}}^{(k)}}{μ_{t_{n}}^{(k)}} = \frac{κ_{t_{n}}^{(k)}⟨e_{t_{n}}^{(k)}, ϕ^{(k)}⟩}{μ_{t_{n}}^{(k)}⟨e_{t_{n}}^{(k)}, ϕ^{(k)}⟩} = \frac{|⟨e_{t_{n}}^{(k)}, ϕ^{(k)}⟩|}{μ_{t_{n}}^{(k)}⟨e_{t_{n}}^{(k)}, ϕ^{(k)}⟩} =

= \frac{|μ_{t_{n}}^{(k)}⟨e_{t_{n}}^{(k)}, ϕ^{(k)}⟩|}{μ_{t_{n}}^{(k)}⟨e_{t_{n}}^{(k)}, ϕ^{(k)}⟩} ⟶ \frac{|⟨e_{t_{0}}^{(k)}, ϕ^{(k)}⟩|}{⟨e_{t_{0}}^{(k)}, ϕ^{(k)}⟩} = κ_{t_{0}}^{(k)}.

COROLLARY 2.7. If t ↦ x_{t}^{(k)} (k = 1, …, N) are nowhere vanishing functions IR → X^{(k)} with weakly continuous tensor product t ↦ x_{t}^{(1)} ⊗ ⋯ ⊗ x_{t}^{(N)} whose norm t ↦ ∏_{k=1}^{N} ∥x_{t}^{(k)}∥ is also continuous, then there are multipliers κ_{t}^{(k)} ∈ TT (t ∈ IR, k = 1, …, N) making the functions t ↦ κ_{t}^{(k)}∥x_{t}^{(k)}∥^{−1} ∏_{j=1}^{N} ∥x_{t}^{(j)}∥^{1/N}x_{t}^{(k)} norm-continuous.

Proof. A choice in accordance with Proposition 2.5 for the multipliers κ_{t}^{(k)} to the functions t ↦ e_{t}^{(k)} := ∥x_{t}^{(k)}∥^{−1}x_{t}^{(k)} suits the statement.

For the next two lemmas let Z denote a complex topological vector space with separating topological dual (thus for any vector 0 ≠ z ∈ Z there is a continuous linear functional ψ : Z → C with ψ(z) = 1 ).

LEMMA 2.8. Assume z_{1}(t), z_{2}(t) are linearly independent vectors in Z for all t ∈ IR and let ρ_{1}, ρ_{2}, μ : IR → C \ {0} be functions making the maps ζ_{1}z_{1}, ζ_{2}z_{2}, μ(z_{1} + z_{2}) continuous. Then the functions ζ_{2}z_{1}, ζ_{1}z_{2} are also continuous.

Proof. Let us fix any point t_{0} ∈ IR . It suffices to see the continuity of ζ_{1}/ζ_{2} at t_{0} . To this aim let us choose a couple ψ_{1}, ψ_{2} of continuous linear functionals Z → C with det[{ψ_{1}z_{1}(t_{0}) ψ_{1}z_{2}(t_{0}) \atop ψ_{2}z_{1}(t_{0}) ψ_{2}z_{2}(t_{0})}] ≠ 0 and define P z := [{ψ_{1}z \atop ψ_{2}z}] (z ∈ Z) . This can be done by the linear independence of z_{1}(t_{0}), z_{2}(t_{0}) . Then, by the continuity of both t ↦ ζ_{k}(t)z_{k}(t) (k = 1, 2) , there is a neighborhood I of t_{0} where det[P ζ_{1}(t)z_{1}(t), P ζ_{2}(t)z_{2}(t)] ≠ 0 (t ∈ I) and the matrix valued function t ↦ L_{t} := [P ζ_{1}(t)z_{1}(t), P ζ_{2}(t)z_{2}(t)]^{−1} is continuous on I . The continuity of t ↦ μ(t)z_{1}(t) + z_{2}(t)] entails the continuity of the function t ↦ L_{t}P μ(t)[z_{1}(t) + z_{2}(t)] = [{μ(t)/ζ_{1}(t) \atop μ(t)/ζ_{2}(t)}] on I whence the continuity of ζ_{1}/ζ_{2} = [μ/ζ_{1}]/[μ/ζ_{2}] at t_{0} is immediate.

LEMMA 2.9. Let t ↦ U_{t} be a map IR → GL(Z) and let ρ : IR → C\{0} resp. λ : IR^{2} → C \ {0} be functions such that all the maps t ↦ ρ(t)U_{t}z (z ∈ Z) are continuous and we have U_{s+t} = λ(s, t)U_{s}U_{t} for all s, t ∈ IR . Then there exists a function ρ̃ : IR → C \ {0} making [ρ̃(t)U_{t} : t ∈ IR] a one-parameter group.

Proof. By induction on n, we see that there are functions λ_{n} : IR^{n} → C \ {0} with

U_{s_{1}+···+s_{n}} = λ_{n}(s_{1}, …, s_{n})U_{s_{1}} ⋯ U_{s_{n}}

for all possible choices. We establish the symmetry of each λ_{n} by showing that the family {U_{t} : t ∈ IR} is Abelian. Since U_{0} = U_{0+0} = λ(0, 0)U_{0}^{2} , necessarily U_{0} = λ(0, 0)^{−1}Id . Hence the identity λ(t, −t)U_{t}U_{−t} = U_{0} implies that U_{−t} is always a multiple of U_{t}^{−1} . Since, for n = 2, 3, … we have U_{nt} = λ_{n}(t, …, t)U_{t}^{n} , the set {U_{q} : q ∈ Q} = ⋃_{m=1}^{∞}{U_{n/m!} : n ∈ ZZ } is Abelian as being the union of an increasing sequence of Abelian families. By the density of the rational numbers within IR , our strong continuity assumption entails that {ρ(t)U_{t} : t ∈ IR} and hence {U_{t} : t ∈ IR} are Abelian.

From the relation U_{nt} ∈ CU_{t}^{n} we see also that any operator of the form ζU_{t} admits arbitrary m -th roots of the form ηU_{t/m} . Hence, for any given t ∈ IR , we can construct a sequence of group homomorphisms g_{t,m} : {n/m! : n ∈ ZZ } → GL(Z) (m = 1, 2, …) with g_{t,m+1} extending g_{t,m} such that g_{t,m}(1) = U_{t} and g_{t,m}(n/m!) ∈ CU_{t/m!}^{n} = CU_{nt/m!} for all m, n. Thus, for any t ∈ IR , there is a homomorphism g_{t} : IR → GL(Z) along with a function ρ̃_{t} : Q → C such that

g_{t}(n/m!) = g_{t,m}(n/m!) = ρ̃_{t}(n/m!)U_{nt/m!} (n = 0, ±1, ±2, …; m = 1, 2, …).

Let us fix any Hamel basis H in IR and define the map g : IR → GL(Z) resp. the function ρ̃: IR → C \ {0} as

\begin{matrix}g(q_{1}t_{1} + ⋯ + q_{n}t_{n}) := g_{t_{1}}(q_{1}) ⋯ g_{t_{n}}(q_{n}), \\ ρ̃(t_{1}q_{1} + ⋯ + t_{n}q_{n}) := λ_{n}(q_{1}t_{1}, …, q_{n}t_{n})^{−1}ρ̃_{t_{1}}(q_{1}) ⋯ ρ̃_{t_{n}}(q_{n}) \end{matrix}

for q_{1}, …, q_{n} ∈ Q and t_{1}, …, t_{n} ∈ H. We complete the proof by observing that g is a group homemorphism with g(1) = U_{1} and satisfying the identities g(q_{1}t_{1} + ⋯ + q_{n}t_{n}) = λ_{n}(q_{1}t_{1}, …, q_{n}t_{n})^{−1}ρ̃(q_{1}t_{1} + ⋯ + q_{n}t_{n})U_{q_{1}t_{1}} ⋯ U_{q_{n}t_{n}} .

Proof of Proposition 1.3.

By assumption, for any tuple [x^{(j)}] ∈ X^{(1)} × ⋯ X^{(N)} the map t ↦ U_{1,t}x^{(1)} ⊗ ⋯ ⊗ U_{N,t}x^{(N)} is weakly continuous. An application of Corollary 2.7 establishes the existence of functions ρ_{k,[x^{(j)}]} : IR → C \ {0} such that all the maps t ↦ ρ_{k,[x^{(k)}]}(t)U_{k,t}x^{(k)}(t) are continuous. Let us fix any family 0 ≠ x_{0}^{(j)} ∈ X^{(k)} (j = 1, …, N) and define ρ_{k}(t) := ρ_{k,[x_{0}^{(j)}]} . To complete

the proof that each [ρ_{k}(t)U_{k,t} : t ∈ IR] is a strongly continuous family, it suffices to see that, given any index ℓ with vector x^{(ℓ)} ∈ X^{(ℓ)} which is linearly independent of x_{0}^{(ℓ)} , the function t ↦ U_{ℓ,t}x^{(ℓ)} is continuous. This fact follows immediately from Lemma 2.8 applied with Z := X^{(ℓ)} , z_{1}(t) := U_{ℓ,t}x_{0}^{(ℓ)}(t) , z_{z}(t) := U_{ℓ,t}x^{(ℓ)}(t) , ζ_{1}(t) := ρ_{ℓ}(t) , ζ_{2}(t) := ρ_{ℓ,[y^{(k)}]}(t) , μ(t) := ρ_{ℓ,[u^{(k)}]}(t) where y^{(k)} := (1 − δ_{kℓ})x_{0}^{(k)} + δ_{kℓ}y^{(ℓ)} and u^{(k)} := (1 − δ_{kℓ})x_{0}^{(k)} + δ_{kℓ}y^{(ℓ)} in terms of the Kronecker-delta δ_{kℓ} := [1 for k = ℓ and 0 else ] .

It is a well-known consequence of (2.2) and the Schur lemma that the factorization of a non-vanishing tensor products of linear operators is unique up to constant factors with product one. Hence the semigroup property U(s+ t) = U(s)U(t) entails the existence of functions λ_{k} : IR^{2} → C \ {0} such that U_{k,s+t} = λ_{k}(s, t)U_{k,t}U_{s,t} s, t,∈ IR; k = 1, …, N) . Given any index ℓ ∈ {1, …, N}, by applying Lemma 2.9 again with Z := X^{(ℓ)} , the operators U_{t}:=U_{ℓ,t} and the function ρ := ρ_{ℓ} constructed above, we obtain the existence of a function ρ̃_{ℓ} : IR→ C making the family [ρ̃_{ℓ}U_{ℓ,t} : t∈IR] a one-parameter group.

CARTAN FACTORS OF TYPES 2,3; PROOF OF THEOREM 1.6

Throughout this section let H be an arbitrarily fixed complex Hilbert space. We shall write h^{∗} := [H ∋ x ↦ ⟨x|h⟩] for the dual functionals, Ball(H) := {h ∈ H : ∥h∥ < 1} for the unit ball, ∂Ball(H) := {h ∈ H : ∥h∥ = 1} for the unit sphere and g^{∗}∧h^{∗} := g^{∗}⊗h^{∗}−h^{∗}⊗g^{∗} for the basic antisymmetric functionals, respectively, where g^{∗} ⊗ h^{∗} := [(x, y) ↦ g^{∗}(x)h^{∗}(y)] . The spaces F_{2}, F_{3} in Theorem 1.6 are isomorphic copies of the Banach spaces 𝔉_{2}, 𝔉_{3} of all symmetric resp. antisymmetric continuous bilinear functionals H × H → C . Recall [11] that in both cases k = 2, 3 , the surjective linear isometries of 𝔉_{k} are of the form [U ⊗U]Φ = [(x, y) ↦ Φ(Ux, Uy)] (Φ ∈ 𝔉_{k}) . Therefore the conclusion that, given any of the indices k ∈ {2, 3} with a oneparameter group [U(t) : t ∈ IR] of the form U(t) = U_{t} ⊗ U_{t} such that all the functions t ↦ Φ(U_{t}x, U_{t}y) (Φ ∈ 𝔉_{k}; x, x ∈ H) are continuous, there are functions κ, κ̃ : IR → TT making [κ(t)U_{t} : t ∈ IR] resp. [κ̃(t)U_{t} : t ∈ IR] a strongly continuous family resp. a one parameter group is immediate from Propositions 3.3,3.5 and Corollary 3.4 below even with κ, κ̃ admitting only the values ±1 . Since the results of [10, Sections 4,5] concern only a single underlying Hilbert space, hence we can conclude Theorem 1.6 (without reference to [10, Section 6]).

LEMMA 3.1. [10, Lemma 2.3]. Suppose F : IR → IP(H) := {TT g : ⟨g|g⟩ = 1} is a continuous map 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 3.2. Let t ↦ e_{t} be a function IR → ∂Ball(H) . If for any convergent net t_{ν} → t in 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 σ : IR → {−1, 1} such that t ↦ σ(t)e_{t} is continuous.

Proof. We can apply Lemma 3.1 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 a suitable function 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,

PROPOSITION 3.3. If t ↦ ϕ(e_{t}, e_{t}) with unit vectors e_{t} ∈ H 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 a universal subnet [4] for it. 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 wellknown 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 3.2 applies.

COROLLARY 3.4. Let t ↦ [e_{t}, f_{t}] be a map IR → { orthonormed 2frames }. If the map 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 Proposition 3.3, 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.

PROPOSITION 3.5. Suppose t ↦ [e_{t}, f_{t}, g_{t}] is a function IR → {orthonormed 3-frames } 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 a 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.

PROOF OF PROPOSITION 1.5

Let U := [U^{t} : t ∈ IR] be a strongly continuous one-parameter subgroup of H such that

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

Consider a Banach limit L_{R→∞} 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

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

Since all the functions t ↦ U^{t}x are continuous and bounded, 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 (4.1) entail immediately that ⟨⟨.|.⟩⟩ is also a scalar product on H with

M^{−2}⟨x|x⟩ ≤ ⟨⟨x|x⟩⟩ ≤ M^{2}⟨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^{2}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⟩⟩ .

Acknowledgements. The author was supported by the projects TÁMOP-4.2.2.A-11/1/KONV-2012-0060 Impulse Lasers for Use in Materials Science and Biophotonics and TÁMOP-4.2.2.A-11/1/KONV-2012-0073 Telemedicine Oriented Research in the Fields of Mathematics, Informatics and Medical Sciences of the European Union and co-financed by the European Social Fund.

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L.L. STACHÓ Bolyai Institute University of Szeged Aradi Vértanúk tere 1 H-6720 Szeged Hungary stacho@math.u-szeged.hu