Journal of Lie Theory Volume 14 (2004) 215–226 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

Asymptotic Products and Enlargibility of Banach-Lie Algebras

Daniel Beltiţă

Communicated by K.-H. Neeb

Abstract. The paper provides a “standard” proof of a local theorem on enlargibility of Banach-Lie algebras. A particularly important special case of that theorem is that a Banach-Lie algebra is enlargible provided it has a dense locally finite subalgebra. The theorem is due to V. Pestov, who proved it by techniques of nonstandard analysis. The present proof uses a theorem concerning enlargibility of asymptotic products of contractive Banach-Lie algebras. 2000 MSC: Primary 22E65; Secondary 17B65, 46B08 Keywords: asymptotic product; enlargible Banach-Lie algebra

1. Introduction

The present paper provides a “standard” proof of an enlargibility criterion (see Corollary 4.4 below) discovered by V. Pestov in [Pe88] and [Pe92], where it is proved by techniques of nonstandard analysis. A particularly important special case of this criterion is the fact that a real Banach-Lie algebra is enlargible (i.e., it is the Lie algebra of some Banach-Lie group) whenever it has a dense locally finite subalgebra; see Corollary 1 in [Pe88] and Remark 4.5 below. The latter fact turns out to play an essential role in connection with the very existence of groups corresponding to pseudo-restricted Lie algebras, eventually leading to a class of Banach-Lie groups which possess quite natural complex homogeneous spaces (see [Be02]). Other interesting applications of the enlargibility criterion of [Pe88] and [Pe92] can be found in [Pe93].

The main ingredients in the present proof of the aforementioned criterion are a slight sharpening of a result in [GN01] concerning quotient Banach-Lie groups, the notion of enlargibility radius (see Definition 2.1 below), and the asymptotic products. Thus, these three ingredients allow us to give a new example of the general principle that the use of nonstandard analysis is equivalent to the ultraproduct approach (see e.g., page 27 in [HM83]). We note that the idea of ultraproduct had been previously exploited in connection with the Lie algebras. See e.g., [Fr82] (or §20 in [BS01]), as well as [CGM].

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

The structure of the paper is as follows. In §2 we introduce the notion of enlargibility radius of a Banach-Lie algebra with respect to a closed subalgebra (Definition 2.1), and prove an important property in the case when the subalgebra is actually an ideal (Theorem 2.6).

In §3 we are concerned with lower estimates for the enlargibility radius of an asymptotic product of Banach-Lie algebras (Theorem 3.1). As a consequence, we get an enlargibility criterion for asymptotic products (Corollary 3.9).

In §4, we use Theorems 2.6 and 3.1 to prove lower estimates for the enlargibility radius of a Banach-Lie algebra 𝔤 in terms of a local system of closed subalgebras whose union is dense in 𝔤 (Theorem 4.3). As a special case (see Corollary 4.4), we then recover the Local Theorem on Enlargibility of [Pe88] and [Pe92].

We now introduce some notation and terminology. For any Banach-Lie group G, we denote its Lie algebra by L(G). If φ: G → H is a homomorphism of Banach-Lie groups, then L(φ): L(G) → L(H) denotes the corresponding homomorphism of Banach-Lie algebras. By contractive Banach-Lie algebra we mean a real Banach-Lie algebra 𝔤 equipped with a fixed norm ∥·∥ which defines the topology of 𝔤 and has the property that ∥[x, y]∥ ≤ ∥x∥ · ∥y∥ for all x, y ∈ 𝔤. In this case, for every R > 0 we denote B_{𝔤}(0, R) = {x ∈ 𝔤 | ∥x∥ < R}.

Now let J be a directed set and {𝔤_{j}}_{j∈J} a family of Banach spaces over 𝕂 ∈ {ℝ, ℂ}. Then

ℓ^{∞}({𝔤_{j}}_{j∈J}) := {x = (x_{j})_{j∈J} ∈ ∏_{j∈J} 𝔤_{j} | ∥x∥ := sup_{j∈J} ∥x_{j}∥ < ∞}

(with componentwise defined addition and scalar multiplication) is in turn a Banach space over 𝕂, and

c_{0}({𝔤_{j}}_{j∈J}) := {(x_{j})_{j∈J} ∈ ℓ^{∞}({𝔤_{j}}_{j∈J}) | lim_{j∈J} ∥x_{j}∥ = 0}

is a closed subspace of ℓ^{∞}({𝔤_{j}}_{j∈J}). Thus the quotient

𝔤 := ℓ^{∞}({𝔤_{j}}_{j∈J})/c_{0}({𝔤_{j}}_{j∈J})

is a Banach space over 𝕂 (with the quotient norm), and we call it the asymptotic product of the family {𝔤_{j}}_{j∈J}. If moreover 𝔤_{j} is a contractive Banach-Lie algebra for each j ∈ J, then it is clear that ℓ^{∞}({𝔤_{j}}_{j∈J}) is a contractive Banach-Lie algebra (with componentwise defined bracket) and c_{0}({𝔤_{j}}_{j∈J}) is a closed ideal of ℓ^{∞}({𝔤_{j}}_{j∈J}), so that the asymptotic product 𝔤 has a natural structure of contractive Banach-Lie algebra.

For later reference, we now recall a simple result which allows us to compute the norms of elements in asymptotic products.

Lemma 1.1. Let J be a directed set and {𝔙}_{j∈J} a family of Banach spaces over 𝕂 ∈ {ℝ, ℂ}. Denote 𝔙̃ = ℓ^{∞}({𝔙_{j}}_{j∈J}), 𝔙̃_{0} = c_{0}({𝔙_{j}}_{j∈J}) and 𝔙 = 𝔙̃/𝔙̃_{0} . Then for all ã = (a_{j})_{j∈J} ∈ 𝔙̃ the norm of ã + 𝔙̃_{0} ∈ 𝔙 can be computed by

∥ã + 𝔙̃_{0}∥ = lim sup_{j∈J} ∥a_{j}∥ = inf_{i∈J} sup_{i≤j∈J} ∥a_{j}∥.

Proof. The proof of Proposition A.6.1 (at page 343) in [ER00] extends word by word.

Also for later reference, we state a well-known property of the Baker-Campbell-Hausdorff series H(·, ·).

Lemma 1.2. For every ε > 0 there exists δ ∈ (0, ε) such that for every contractive Banach-Lie algebra 𝔤 and every x, y ∈ B_{𝔤}(0, δ) we have

∥H(x, y)∥ < ε.

Proof. Use e.g., Lemma 1 and formula (13) in no. 2 in §7 in Chapter II in [Bo72].

2. Enlargibility radii

Definition 2.1. Let 𝔤 be a contractive Banach-Lie algebra. If 𝔥 is a closed subalgebra of 𝔤, we define the enlargibility radius of 𝔤 with respect to 𝔥 as the supremum r_{𝔥}(𝔤) of the set of all real numbers R > 0 such that there exist a real Banach-Lie group G with L(G) = 𝔤 and a subgroup H of G such that

exp_{G} |_{B_{𝔤}(0,R)} is injective and exp_{G}(𝔥 ∩ B_{𝔤}(0, R)) = H ∩ exp_{G}(B_{𝔤}(0, R)),

where the supremum of an empty set is defined to be 0.

If 𝔥 = {0}, we denote simply r_{𝔥}(𝔤) = r(𝔤) and call it the enlargibility radius of 𝔤.

Definition 2.2. Let G be a connected real Banach-Lie group such that L(G) = 𝔤 is equipped with a norm making it into a contractive Banach-Lie algebra. For every R > 0 such that exp_{G} |_{B_{𝔤}(0,R)} is injective (that is, 0 < R ≤ r(𝔤) in the terminology of Definition 2.1), we denote

V_{G,R} = exp_{G}(B_{𝔤}(0, R))

and define

log_{G,R}: G → 𝔤 ∪ {∞}

by

log_{G,R} a = \{ \begin{matrix}(exp_{G} |_{B_{𝔤}(0,R)})^{−1}(a) & if a ∈ V_{G,R}, \\ ∞ & if a ∈ G \ V_{G,R}. \end{matrix}

Furthermore, we extend the norm of 𝔤 to a function ∥ · ∥: 𝔤 ∪ {∞} → [0, ∞] with ∥∞∥ = ∞.

It is clear that, if 0 < R_{1} ≤ R < r(𝔤), then log_{G,R_{1}} = log_{G,R} on V_{G,R_{1}} .

Remark 2.3. (a) A contractive Banach-Lie algebra 𝔤 is enlargible if and only if r(𝔤) > 0. Moreover, let us denote by Π(𝔤) the period group of 𝔤 and

δ := inf{∥γ∥ | 0 ≠ γ ∈ Π(𝔤)}.

We recall that Π(𝔤) is an additive subgroup of the center of 𝔤 and, if G̃ is a simply connected Banach-Lie group whose Lie algebra is 𝔤, then Π(𝔤) equals the set of all elements x in the center of 𝔤 with exp_{G} x = 1. Now, Lemma III.11 and the remark following it in [GN01] show that

min{π, r(𝔤)} = min{π, δ/2}.

In fact, that lemma implies that min(π, δ/2) ≤ r(𝔤). In particular, if δ/2 ≥ π, then r(𝔤) ≥ π. Actually, the aforementioned facts from [GN01] show that, if δ/2 ≤ π, then δ/2 is the supremum of the set of all real numbers R > 0 such that exp_{G} |_{B_{𝔤}(0,R)} is injective. Thus, if δ/2 ≤ π, then Remark 2.4 bellow shows that r(𝔤) = δ/2. Consequently, min{π, r(𝔤)} = min{π, δ/2} as claimed.

(b) In connection with Definition 2.1, we note that, if G is a Banach-Lie group with L(G) = 𝔤 (a contractive Banach-Lie algebra), 𝔥 is a closed subalgebra of 𝔤 and H is a subgroup of G such that for some real number R > 0 we have

exp_{G} |_{B_{𝔤}(0,R)} is injective and exp_{G}(𝔥 ∩ B_{𝔤}(0, R)) = H ∩ exp_{G}(B_{𝔤}(0, R)),

then H is actually a Lie subgroup of G in the sense of Definition I.4 (b) in [GN01]. In fact, H is locally closed (since 𝔥 ∩ B_{𝔤}(0, R) is closed in B_{𝔤}(0, R) and H is a subgroup), hence it is closed by Proposition 2.1 in Chapter I in [Ho65]. Now the assumption exp_{G}(𝔥 ∩ B_{𝔤}(0, R)) = H ∩ exp_{G}(B_{𝔤}(0, R)) easily implies that

𝔥 = {x ∈ 𝔤 | exp_{G}(ℝx) ⊆ H},

and then H is a Lie subgroup of G (see Remark I.5 in [GN01]).

(c) It follows by Proposition 2.5 in [LT66] that for every finite-dimensional contractive Banach-Lie algebra 𝔤 we have r(𝔤) ≥ 2π.

Remark 2.4. If 𝔤 is a contractive Banach-Lie algebra, 0 < R < r(𝔤) and G̃ is a connected simply connected Banach-Lie group with L(G̃) = 𝔤, then exp_{G} |_{B_{𝔤}(0,R)} is injective. In fact, by the very definition of r(𝔤) (see Definition 2.1), there exists a Banach-Lie group G with L(G) = 𝔤 and exp_{G} |_{B_{𝔤}(0,R)} injective. Now let G_{0} be the connected 1-component of G. Since L(G_{0}) = 𝔤, we have a covering map p: G̃ → G_{0} such that p ∘ exp_{G} = exp_{G}. Since exp_{G} |_{B_{𝔤}(0,R)} is injective, it then follows that exp_{G} |_{B_{𝔤}(0,R)} is in turn injective.

Remark 2.5. If 𝔤_{1} and 𝔤_{2} are contractive Banach-Lie algebras and there exists an injective homomorphism of Banach-Lie algebras φ: 𝔤_{1} → 𝔤_{2} with ∥φ∥ ≤ 1, then we have r(𝔤_{2}) ≤ r(𝔤_{1}). (This is a slight improvement of assertion (∗∗∗) at page 22 in [EK64]). Using Lemma II.1 in [GN01] and the preceding Remark 2.3 (b), one can actually prove that r_{𝔥}(𝔤_{2}) ≤ r_{φ^{−1}(𝔥)}(𝔤_{1}) provided 𝔥 is a closed subalgebra of 𝔤_{2} .

The next statement is, in some respects, a slight sharpening of Theorem II.2 in [GN01], expressed in terms of enlargibility radii.

Theorem 2.6. For every real number ε > 0 there exists another real number η > 0 such that the following assertion holds: If 𝔤 is a contractive Banach-Lie algebra and 𝔥 is a closed ideal of 𝔤 such that r_{𝔥}(𝔤) ≥ ε, then r(𝔤/𝔥) ≥ η.

Proof. Let R_{0} be an arbitrary real number with

0 < R_{0} < min{ε, (1/2) log 2},

so that the Baker-Campbell-Hausdorff series H(x, y) is convergent whenever x, y ∈ B_{𝔨}(0, R_{0}) and 𝔨 is a contractive Banach-Lie algebra (see Proposition 1 in no. 2 in §7 in [Bo72]).

Then for every contractive Banach-Lie algebra 𝔤 and every closed ideal 𝔥 of 𝔤 with r_{𝔥}(𝔤) ≥ ε we have r_{𝔥}(𝔤) > R_{0} . It then follows by Remark 2.3 (b) that there exists a Banach-Lie group G with a Lie subgroup H such that L(G) = 𝔤, L(H) = 𝔥, exp_{G} |_{B_{𝔤}(0,R_{0})} is injective and exp_{G}(𝔥 ∩ B_{𝔤}(0, R_{0})) = H ∩ exp_{G}(B_{𝔤}(0, R_{0})). Clearly we may assume that H is connected. Since 𝔥 is an ideal of 𝔤, it then follows that H is a normal subgroup of G. On the other hand, since 𝔤/𝔥 is equipped with the quotient norm, it follows that for all R > 0 we have Q(B_{𝔤}(0, R)) = B_{𝔤/𝔥}(0, R), where Q: 𝔤 → 𝔤/𝔥 is the quotient map.

Now denote V = B_{𝔤/𝔥}(0, R_{0}) and W = B_{𝔤/𝔥}(0, η), where η stands for the value of δ given by Lemma 1.2 for ε = R_{0} . Also denote U = B_{𝔤}(0, R_{0}) and A = B_{𝔤}(0, η), so that H(A × A) ⊆ U according to the choice of η. Also, Q(A) = W. It then follows as in the proof of Theorem II.2 in [GN01] that the mapping

E: B_{𝔤/𝔥}(0, η) → G/H, E(Q(X)) := q(exp_{G} X),

is correctly defined and an analytic homomorphism of local analytic groups. Here q: G → G/H stands for the quotient map, and we think of B_{𝔤/𝔥}(0, η) as a local analytic group with the Baker-Campbell-Hausdorff multiplication. Thus, in view of the theorem on extension of analytic structure (see [Sw65], page 213), it follows that there exists a Banach-Lie group K with L(K) = 𝔤/𝔥 and exp_{K} |_{B_{𝔤/𝔥}(0,η)} injective, hence η ≤ r(𝔤/𝔥) according to Definition 2.1.

3. Enlargibility of asymptotic products

Here is the main result of the present section.

Theorem 3.1. For every real number ε > 0 there exists another real number η > 0 such that the following assertion holds: If J is a directed set and {𝔤_{j}}_{j∈J} is a family of contractive Banach-Lie algebras with the asymptotic product 𝔤 and such that inf_{j∈J} r(𝔤_{j}) ≥ ε, then r(𝔤) ≥ η.

We now establish some notation needed in the proof of Theorem 3.1.

Notation 3.2. In the present section, until the proof of Theorem 3.1, we keep the notation in its statement, assuming that inf_{j∈J} r(𝔤_{j}) ≥ ε. We fix an arbitrary

real number R_{0} with 0 < R_{0} < ε.

For each j ∈ J, it then follows by Remark 2.3 (a) that the Banach-Lie algebra 𝔤_{j} is enlargible, and we denote by G_{j} a connected simply connected Banach-Lie group with L(G_{j}) = 𝔤_{j}.

Remark 3.3. It follows by Proposition III.12 (ii)–(iii) in [GN01] and the preceding Remark 2.3 (a) that there exist a connected Banach-Lie group G and a group homomorphism ψ: G → ∏_{j∈J} G_{j} such that, if

π_{k}: ℓ^{∞}({𝔤_{j}}_{j∈J}) → 𝔤_{k} and p_{k}: ∏_{j∈J} G_{j} → G_{k}

are the canonical projection maps whenever k ∈ J, then the following assertions hold.

(i) We have L(G) = ℓ^{∞}({𝔤_{j}}_{j∈J}) := 𝔤̃.

(ii) The homomorphism ψ is injective and continuous.

(iii) For every k ∈ J we have L(p_{k} ∘ ψ) = π_{k}.

(iv) For every x = (x_{j})_{j∈J} ∈ 𝔤̃ we have ψ(exp_{G} x) = (exp_{G_{j}} x_{j})_{j∈J}.

Notation 3.4. Until the proof of Theorem 3.1 we shall keep the notation of Remark 3.3. Using Definition 2.2, we introduce the following subset of G:

H := {h ∈ G | lim_{j∈J} ∥(log_{G_{j},R} ∘p_{j} ∘ ψ)(h)∥ = 0},

where 0 < R ≤ R_{0} . Note that the definition of H does not depend on the choice of R (see the remark concluding Definition 2.2).

We also denote

𝔥 := c_{0}({𝔤_{j}}_{j∈J}).

Remark 3.5. We recall from the Introduction that 𝔥 = c_{0}({𝔤_{j}}_{j∈J}) is a closed ideal of the contractive Banach-Lie algebra 𝔤̃ = ℓ^{∞}({𝔤_{j}}_{j∈J}. In this connection, we shall eventually see from Lemmas 3.6–8 that H is a normal Lie subgroup of G corresponding to the ideal 𝔥 of 𝔤̃ = L(G).

Lemma 3.6. The set H is a subgroup of G.

Proof. Let h_{1}, h_{2} ∈ H and fix δ > 0 given by Lemma 1.2 for ε = R_{0} . We may suppose that 0 < δ < R_{0} . It then follows by Notation 3.4 that, for i ∈ {1, 2}, we have lim_{j∈J} ∥(log_{G_{j},δ} ∘p_{j} ∘ ψ)(h_{i})∥ = 0, hence there exists j_{0} ∈ J such that

(∀j ∈ J, j ≥ j_{0}) ∥(log_{G_{j},δ} ∘p_{j} ∘ ψ)(h_{i})∥ < δ for i ∈ {1, 2}.

Then for all j ∈ J with j ≥ j_{0} we have

(log_{G_{j},R_{0}} ∘p_{j} ∘ ψ)(h_{1}h_{2}) = log_{G_{j},R_{0}}((p_{j} ∘ ψ)(h_{1}) · (p_{j} ∘ ψ)(h_{2}))

= H((log_{G_{j},δ} ∘p_{j} ∘ ψ)(h_{1}), (log_{G_{j},δ} ∘p_{j} ∘ ψ)(h_{2})), where the latter equality follows in view of the choice of δ. Now, since

lim_{j∈J} ∥(log_{G_{j},δ} ∘p_{j} ∘ ψ)(h_{i})∥ = 0

for i ∈ {1, 2}, it easily follows by Lemma 1.2 that lim_{j∈J} ∥(log_{G_{j},R_{0}} ∘p_{j}∘ψ)(h_{1}h_{2})∥ =

0. Thus h_{1}h_{2} ∈ H.

Finally, note that for every h ∈ G and j ∈ J we have

(log_{G_{j},R_{0}} ∘p_{j} ∘ ψ)(h^{−1}) = \{ \begin{matrix}−(log_{G_{j},R_{0}} ∘p_{j} ∘ ψ)(h) & if h ∈ V_{G,R_{0}}, \\ ∞ & if h ∈ G \ V_{G,R_{0}}, \end{matrix}

which implies at once that h ∈ H is equivalent to h^{−1} ∈ H, and the proof ends.

Lemma 3.7. The subgroup H of G is normal.

Proof. Since the group G is connected, it clearly suffices to show that gHg^{−1} ⊆ H whenever g ∈ V_{G,R_{0}} . Let us fix such an element g and denote x := log_{G,R_{0}} g, so that x ∈ B_{𝔤}(0, R_{0}) and exp_{G} x = g. Also denote g_{j} := (p_{j} ∘ ψ)(g) ∈ G_{j} whenever j ∈ J. By Remark 3.3 (iii)–(iv) we then have g_{j} ∈ V_{G_{j},R_{0}} , hence denoting x_{j} := log_{G_{j},R_{0}} g_{j} we get x_{j} ∈ B_{𝔤_{j}}(0, R_{0}) and exp_{G_{j}} x_{j} = g_{j}.

Now let h ∈ H arbitrary, so that lim_{j∈J} ∥(log_{G_{j},R_{0}} ∘p_{j} ∘ ψ)(h)∥ = 0. For

each j ∈ J we have

(log_{G_{j},R_{0}/e^{R_{0}}} ∘p_{j} ∘ ψ)(ghg^{−1})
= log_{G_{j},R_{0}/e^{R_{0}}} ((p_{j} ∘ ψ)(g) · (p_{j} ∘ ψ)(h) · (p_{j} ∘ ψ)(g)^{−1})
= log_{G_{j},R_{0}/e^{R_{0}}} (g_{j} · (p_{j} ∘ ψ)(h) · g_{j}^{−1})
= log_{G_{j},R_{0}/e^{R_{0}}} ((exp_{G_{j}} x_{j}) · (p_{j} ∘ ψ)(h) · (exp_{G_{j}} x_{j})^{−1})
= e^{ad_{𝔤_{j}} x_{j}} log_{G_{j},R_{0}}((p_{j} ∘ ψ)(h)).

For the latter equality, see Notation 3.4 and note that, since the Banach-Lie algebra 𝔤_{j} is contractive, we have ∥ad_{𝔤_{j}}x_{j}∥ ≤ ∥x_{j}∥ < R_{0} , hence ∥e^{−ad_{𝔤_{j}} x_{j}}∥ ≤ e^{∥ad_{𝔤_{j}} x_{j}∥} ≤ e^{R_{0}} , which in turn implies e^{−ad_{𝔤_{j}} x_{j}}(B_{𝔤_{j}}(0, R_{0}/e^{R_{0}})) ⊆ B_{𝔤_{j}}(0, R_{0}).

Using the inequality ∥e^{ad_{𝔤_{j}} x_{j}}∥ ≤ e^{R_{0}} , it follows from the above equalities that

(∀j ∈ J) ∥(log_{G_{j},R_{0}/e^{R_{0}}} ∘p_{j} ∘ ψ)(ghg^{−1})∥ ≤ e^{R_{0}} · ∥(log_{G_{j},R_{0}} ∘p_{j} ∘ ψ)(h)∥.

Now, since h ∈ H, we get ghg^{−1} ∈ H (see Notation 3.4) as desired.

Lemma 3.8. For every R ∈ (0, R_{0}] we have

exp_{G} |_{B_{𝔤}(0,R)} injective and exp_{G}(𝔥 ∩ B_{𝔤}(0, R)) = H ∩ V_{G,R}.

Proof. The fact that exp_{G} |_{B_{𝔤}(0,R)} is injective follows by Remark 3.3 (i), (ii),

(iv), along with the fact that exp_{G_{j}} |_{B_{𝔤_{j}} (0,R)} is injective for all j ∈ J (according to the choice of R_{0} ).

The proof of the other assertion has two steps.

1^{∘} We first prove that exp_{G}(𝔥) ⊆ H. To this end, let x = (x_{j})_{j∈J} ∈ 𝔥 = c_{0}({𝔤_{j}}_{j∈J}). Then

(1)  lim_{j∈J} ∥x_{j}∥ = 0,

hence there exists j_{0} ∈ J such that

(2)  (∀j ∈ J, j ≥ j_{0})  ∥x_{j}∥ < R.

On the other hand, for every j ∈ J, we have (p_{j} ∘ ψ)(exp_{G} x) = exp_{G_{j}} x_{j} by Remark 3.3 (iv). Then (2) implies that

(3)  (∀j ∈ J, j ≥ j_{0})  (log_{G_{j},R} ∘p_{j} ∘ ψ)(exp_{G} x) = x_{j}.

It then follows by (1), (3) and Notation 3.4 that exp_{G} x ∈ H.

2^{∘} It follows by Step 1^{∘} that exp_{G}(𝔥 ∩ B_{𝔤}(0, R)) ⊆ H ∩ V_{G,R}.

To prove the converse inclusion, take h ∈ H ∩ V_{G,R} arbitrary. Since h ∈ V_{G,R}, there exists a unique x = (x_{j})_{j∈J} ∈ B_{𝔤}(0, R) such that h = exp_{G} x (see the beginning of the present proof, and also Remark 3.3 (i)). The fact that x ∈ B_{𝔤}(0, R) shows that for every j ∈ J we have ∥x_{j}∥ < R.

Now note that, in Step 1^{∘}, we actually proved that (2) ⇒ (3), more precisely that (log_{G_{j},R} ∘p_{j} ∘ ψ)(exp_{G} x) = x_{j} provided ∥x_{j}∥ < R. This fact shows that, in the present situation, we have

(∀j ∈ J) (log_{G_{j},R} ∘p_{j} ∘ ψ)(h) = x_{j}.

But h ∈ H, so by Notation 3.4 we have lim_{j∈J} ∥x_{j}∥ = 0, that is, x ∈ 𝔥. Thus

h = exp_{G} x with x ∈ 𝔥 ∩ B_{𝔤}(0, R), which concludes the proof of the desired equality. Proof of Theorem 3.1. For R_{0} as in in Notation 3.2, we get by Remark 3.3, Notation 3.4, and Lemmas 3.6–8 that there exist a Banach-Lie group G and a subgroup H of G such that L(G) = 𝔤̃ := ℓ^{∞}({𝔤_{j}}_{j∈J}), the function exp_{G} |_{B_{𝔤}(0,R_{0})} is injective and exp_{G}(𝔥∩B_{𝔤}(0, R_{0})) = H ∩exp_{G}(B_{𝔤}(0, R_{0})),

where 𝔥 = c_{0}({𝔤_{j}}_{j∈J}) as in Notation 3.4.

Thus R_{0} ≤ r_{𝔥}(𝔤̃) by Definition 2.1. It then follows by Theorem 2.6 that there exists η > 0 depending only on R_{0} such that η ≤ r(𝔤̃/𝔥). But 𝔤̃/𝔥 = 𝔤 (see the definition of asymptotic products in the Introduction), hence η ≤ r(𝔤), as desired.

Corollary 3.9. If J is a directed set and {𝔤_{j}}_{j∈J} is a family of contractive Banach-Lie algebras such that inf_{j∈J} r(𝔤_{j}) > 0, then the asymptotic product of the

family {𝔤_{j}}_{j∈J} is an enlargible Banach-Lie algebra. Proof. Use Theorem 3.1 and Remark 2.3 (a).

4. The local theorem on enlargibility

In the present section, we make use of the previous results on asymptotic products to prove the Local Theorem on Enlargibility of V. Pestov (see [Pe88] and [Pe92]). The next two lemmas claim their origins from a typical ultraproduct method of proof. (For this method, see the comments following the proof of Proposition 6.2 in [He80].)

Lemma 4.1. Let 𝔤 be a contractive Banach-Lie algebra, J a directed set and {𝔤_{j}}_{j∈J} a family of closed subalgebras of 𝔤 such that the following conditions hold.

(j) If j_{1}, j_{2} ∈ J and j_{1} ≤ j_{2} , then 𝔤_{j_{1}} ⊆ 𝔤_{j_{2}} .
(jj) We have 𝔤 = ⋃_{j∈J} 𝔤_{j}.

If 𝔤̂ stands for the asymptotic product of the family {𝔤_{j}}_{j∈J}, then there exists an isometric homomorphism of Banach-Lie algebras ψ: 𝔤 → 𝔤̂.

Proof. Let a ∈ 𝔤. According to the hypothesis (jj), there exists j_{0} ∈ J with a ∈ 𝔤_{j_{0}} . Define

a_{j} = \{ \begin{matrix}a & if j ≥ j_{0}, \\ 0 & otherwise. \end{matrix}

Then (a_{j})_{j∈J} ∈ ℓ^{∞}({𝔤_{j}}_{j∈J}), and it is easy to see that

ψ(a) := (a_{j})_{j∈J} + c_{0}({𝔤_{j}}_{j∈J}) ∈ 𝔤̂

does not depend on the choice of j_{0} and moreover ∥ψ(a)∥ = ∥a∥ (see Lemma 1.1). Furthermore, it is clear that ψ: 𝔤 → 𝔤̂ is a Lie algebra homomorphism.

Lemma 4.2. Let 𝔤 be a contractive Banach-Lie algebra, I a directed set and {𝔥_{i}}_{i∈I} a family of closed subalgebras of 𝔤 such that the following hypotheses hold.

(i) If i_{1}, i_{2} ∈ I and i_{1} ≤ i_{2} then 𝔥_{i_{1}} ⊆ 𝔥_{i_{2}} .
(ii) We have 𝔤 = \overline{⋃_{i∈I} 𝔥_{i}}.

Then there exist a directed set J and a family {𝔤_{j}}_{j∈J} of closed subalgebras of 𝔤 such that the following conditions hold.

(j) If j_{1}, j_{2} ∈ J and j_{1} ≤ j_{2} , then 𝔤_{j_{1}} ⊆ 𝔤_{j_{2}} .

(jj) We have 𝔤 = ⋃_{j∈J} 𝔤_{j}.

(jjj) For every j ∈ J there exists a sequence i_{0}(j) ≤ i_{1}(j) ≤ ⋯ in I such that, if 𝔥̂_{j} stands for the asymptotic product of the family {𝔥_{i_{n}(j)}}_{n∈ℕ}, then there exists an isometric homomorphism of Banach-Lie algebras φ_{j}: 𝔤_{j} → 𝔥̂_{j}.

Proof. The proof has several stages.

1^{∘} We define J := {j: ℕ → I | j(0) ≤ j(1) ≤ ⋯ } endowed with the partial ordering

j_{1} ≤ j_{2} ⟺ (∀n ∈ ℕ) j_{1}(n) ≤ j_{2}(n).

It is easy to check that J is a directed set, since I is directed.

For each j ∈ J we define

𝔤_{j} := {a ∈ 𝔤 | (∃ (a_{n})_{n∈ℕ} ∈ ∏_{n∈ℕ} 𝔥_{j(n)}) lim_{n→∞} ∥a_{n} − a∥ = 0}.

Since 𝔥_{j(0)} ⊆ 𝔥_{j(1)} ⊆ ⋯, it is easy to check that

𝔤_{j} = \overline{⋃_{n∈ℕ} 𝔥_{j(n)}}

which implies at once that 𝔤_{j} is a closed subalgebra of 𝔤.

2^{∘} We now show that the family {𝔤_{j}}_{j∈J} satisfies conditions (j) and (jj). Condition (j) is straightforward. To check condition (jj), let a ∈ 𝔤 arbitrary.

By hypothesis (ii), there exist i_{0}, i_{1}, i_{2}, … ∈ I and a_{0} ∈ 𝔥_{i_{0}}, a_{1} ∈ 𝔥_{i_{1}}, … such that lim_{n→∞} ∥a_{n} − a∥ = 0. To construct j ∈ J with a ∈ 𝔤_{j}, first define j(0) = i_{0} . Then pick j(1) ∈ I such that j(1) ≥ j(0) and j(1) ≥ i_{1} , so that a_{1} ∈ 𝔥_{i_{1}} ⊆ 𝔥_{j(1)} . If j(0) ≤ ⋯ ≤ j(k) have been constructed, let j(k + 1) ∈ I with j(k + 1) ≥ j(k) and j(k + 1) ≥ i_{k+1} , so that a_{k+1} ∈ 𝔥_{i_{k+1}} ⊆ 𝔥_{j(k+1)} , and so on. Since lim_{n→∞} ∥a_{n} − a∥ = 0 and a_{n} ∈ 𝔥_{j(n)} for all n ∈ ℕ, we get a ∈ 𝔤_{j} according to the definition of 𝔤_{j} at stage 1^{∘}.

Consequently, 𝔤 = ⋃_{j∈J} 𝔤_{j}, which is just condition (jj).

3^{∘} To check condition (jjj), let j ∈ J be arbitrary, and define i_{n}(j) = j(n) for all n ∈ ℕ. If 𝔥̂_{j} denotes the asymptotic product of the family {𝔥_{j(n)}}_{n∈ℕ}, we define

φ: 𝔤_{j} → 𝔥̂_{j} = ℓ^{∞}({𝔥_{j(n)}}_{n∈ℕ})/c_{0}({𝔥_{j(n)}}_{n∈ℕ})

in the following way: for arbitrary a ∈ 𝔤_{j}, there exists (a_{n})_{n∈ℕ} ∈ ∏_{n∈ℕ} 𝔥_{j(n)} with

lim_{n→∞} ∥a_{n} − a∥ = 0. Then (a_{n})_{n∈ℕ} ∈ ℓ^{∞}({𝔥_{j(n)}}_{n∈ℕ}), so that we may define

φ_{j}(a) := (a_{n})_{n∈ℕ} + c_{0}({𝔥_{j(n)}}_{n∈ℕ}) ∈ 𝔥̂_{j}.

Then it is clear that φ_{j}(a) does not depend on the choice of the sequence (a_{n})_{n∈ℕ}, and Lemma 1.1 shows that

∥φ_{j}(a)∥ = lim sup_{n→∞} ∥a_{n}∥ = lim_{n→∞} ∥a_{n}∥ = ∥a∥.

Also, it is clear that φ_{j}: 𝔤_{j} → 𝔥̂_{j} is a Lie algebra homomorphism.

Concerning the equality 𝔤_{j} = \overline{⋃_{n∈ℕ} 𝔥_{j(n)}} in stage 1^{∘} of the proof of

Lemma 4.2, it is noteworthy that a related fact is noted in Remark 1.2 in [Me71]. We now come to the main result of the present paper.

Theorem 4.3. For every real number ε > 0 there exists another real number η > 0 such that r(𝔤) ≥ η provided 𝔤 is a contractive Banach-Lie algebra, I is a directed set and {𝔥_{i}}_{i∈I} is a family of closed subalgebras of 𝔤 such that inf_{i∈I} r(𝔥_{i}) ≥ ε and the following conditions hold:

(i) If i_{1}, i_{2} ∈ I and i_{1} ≤ i_{2} then 𝔥_{i_{1}} ⊆ 𝔥_{i_{2}} .
(ii) We have 𝔤 = \overline{⋃_{i∈I} 𝔥_{i}}.

Proof. Let J and {𝔤_{j}}_{j∈J} given by Lemma 4.2. By Theorem 3.1, condition (jjj) in Lemma 4.2 and Remark 2.5 we get ε_{1} ≤ inf_{j∈J} r(𝔤_{j}), where the real

number ε_{1} > 0 depends only on ε.

On the other hand, by Theorem 3.1, Lemma 4.1 and Remark 2.5 again, we have η ≤ r(𝔤), for some real number η > 0 depending only on ε_{1} , that is, on ε.

The following result is just the Local Theorem on Enlargibility proved in [Pe92].

Corollary 4.4. A contractive Banach-Lie algebra 𝔤 is enlargible if and only if there exist a directed set I and a family {𝔥_{i}}_{i∈I} of closed subalgebras of 𝔤 satisfying the following conditions:

(i) If i_{1}, i_{2} ∈ I and i_{1} ≤ i_{2} then 𝔥_{i_{1}} ⊆ 𝔥_{i_{2}} .

(ii) We have 𝔤 = \overline{⋃_{i∈I} 𝔥_{i}}.

(iii) We have inf_{i∈I} r(𝔥_{i}) > 0.

Proof. If 𝔤 is enlargible, we may let the family {𝔥_{i}}_{i∈I} consist in 𝔤 alone. For the converse assertion, use Theorem 4.3 and Remark 2.3 (a).

For the sake of completeness, we conclude by mentioning the result contained in Corollary 3.5 in [Pe92] (or Corrolaire 1 in [Pe88]).

Remark 4.5. It follows by Remark 2.3 (c) that condition (iii) in Corollary 4.4 is satisfied if each 𝔥_{i} is finite dimensional. Consequently a Banach-Lie algebra is enlargible provided it has a dense locally finite subalgebra.

Acknowledgment: I thank Professor K.-H. Neeb for kindly drawing my attention to the paper [GN01], and to Professor V.G. Pestov for sending me, upon my request, a lot of offprints of his papers on enlargibility. I also thank the referee for a number of useful suggestions and interesting remarks.

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Daniel Beltiţă Institute of Mathematics “Simion Stoilow” of the Romanian Academy P.O. Box 1-764 RO-70700 Bucharest Romania dbeltita@imar.ro

Received December 02, 2002 and in final form April 7, 2003