Journal of Lie Theory Volume 13 (2003) 263–270 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

The Lie Algebra of a Nuclear Group

Lydia Außenhofer

Communicated by K. H. Hofmann

Abstract. The Lie algebra of a nuclear group is a locally convex nuclear vector space. Mathematics Subject Classification (2000): 22B05, 22E65, 46Axx

1. The Lie Algebra of Abelian Topological Groups

Let G be an abelian topological group. The set of all one–parameter–subgroups

L(G) := {λ : ℝ → G : λ is a continuous homomorphism}

is called the Lie algebra of G. We define addition pointwise and scalar multiplication by the following formula

ℝ × L(G) → L(G), (s, λ) ↦ (s · λ : t ↦ λ(st)).

It is topologized with the compact–open topology. We use the notation P (K, U) := {f : X → Y : f(K) ⊆ U} for sets X, Y and subsets K ⊆ X and U ⊆ Y . If no confusion can arise, we will use the same notation P (·, ·) for sets of continuous functions, respectively, continuous homomorphisms. The system of neighbourhoods of the neutral element 0 of an abelian topological group G is denoted by U_{G}(0). We set G^{∗} := {χ : G → 𝕋 : χ is a continuous homomorphism} where 𝕋 is the compact group of complex numbers of modulus 1. With multiplication defined pointwise and endowed with the compact–open topology, G^{∗} is an abelian topological group, named dual group of G. We introduce the canonical homomorphism

α_{G} : G → G^{∗∗}, x ↦ (χ ↦ χ(x)).

The famous Pontryagin van–Kampen duality theorem states that if G is a locally compact abelian (LCA) group then α_{G} is a topological isomorphism. [A proof can be found in [3], p. 351 or [4], p.378 or [7], p.84.]

Lemma 1.1. α_{G} is continuous if and only if every compact subset of G^{∗} is equicontinuous.

If G is metrizable (more generally, a k–space) then α_{G} is continuous.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag [A proof of these well known facts can be found in [1] (5.10) and (5.12).]

If φ : G → H is a continuous homomorphism between abelian topological groups, we define the dual homomorphism φ^{∗} in the usual way: φ^{∗} : H^{∗} → G^{∗}, χ ↦ χ∘φ. The polar of a subset A ⊆ G (G an abelian topological group) is defined by A^{0} := {χ ∈ G^{∗}| ∀x ∈ A : Reχ(x) ≥ 0 }. The set A is called quasi–convex if for every x ∈ G \ A there exists χ ∈ A^{0} such that Reχ(x) < 0. The group G is named locally quasi–convex if the neutral element has a neighbourhood basis consisting of quasi–convex sets. This generalizes the setting of local convexity; more precisely, a Hausdorff topological vector space is locally convex if and only if it is a Hausdorff locally quasi–convex group. (See e.g. (2.4) in [2].)

It is easy to verify that α_{G} is injective for every locally quasi–convex Hausdorff group G.

Proposition 1.2. (i) For every abelian topological group G the associated Lie algebra L(G) is a topological vector space. (ii) If G is a Hausdorff space, so is L(G). (iii) If G is a locally quasi–convex Hausdorff group then L(G) is a locally convex vector space.

Proof. i) See e.g. [3] Proposition (7.36), p. 334.

ii) For λ_{1} ≠ λ_{2} ∈ L(G) there exist t ∈ ℝ such that λ_{1}(t) ≠ λ_{2}(t) and disjoint open neighbourhoods U_{i} ∈ U(λ_{i}(t)) of λ_{i}(t) (i = 1, 2). Hence the sets P ({t}, U_{i}) form disjoint neighbourhoods of the λ_{i}.

iii) According to Proposition (2.4) in [2], it is sufficient to show that L(G) is a locally quasi–convex group. Therefore, we will prove that the neighbourhood basis {P (K, U) : K compact, U ∈ U_{G}(0) quasi–convex} consists of quasi–convex sets. To this end, take λ_{0} ∉ P (K, U) and pick t_{0} ∈ K such that λ_{0}(t_{0}) ∉ U. By the quasi-convexity of U, there exists χ ∈ U^{0} such that Reχ(λ_{0}(t_{0})) < 0.

Since one–point sets are compact, the point evaluations L(G) → G, λ → λ(t) are continuous and hence X : L(G) → 𝕋, λ ↦ χ(λ(t_{0})) is a character of L(G). It is obvious that X ∈ P (K, U)^{0} and that ReX(λ_{0}) < 0.

Proposition 1.3. Let φ : H → G be a continuous homomorphism between abelian topological groups. Then the induced mapping φ_{∗} : L(H) → L(G), λ ↦ φ ∘ λ is a continuous linear mapping. Moreover, if φ is an embedding, so is φ_{∗}.

Proof. A straightforward calculation shows that φ_{∗} is a linear mapping. The assertion follows from the fact that φ_{∗}(P (K, φ^{−1}(U)) = P (K, U) ∩ imφ_{∗} (for K ⊆ ℝ compact and U ∈ U_{G}(0)).

Proposition 1.4. Let H be an abelian locally quasi–convex Hausdorff group such that α_{H} is continuous. Then

Φ : L(H) → Hom_{co}(H^{∗}, ℝ^{∗}), λ ↦ λ^{∗}

is an embedding.

Proof. It is easily verified that Φ is a group homomorphism. Since H is a locally quasi–convex Hausdorff group, α_{H} is injective, and from this fact it is straightforward that Φ is also injective. For every compact subset K ⊆ ℝ and every quasi–convex neighbourhood U ∈ U_{H}(0) we have

Φ(P (K, U)) = im(Φ) ∩ P (U^{0}, K^{0}).  (∗)

[Proof of ⊆: Let λ ∈ P (K, U), χ ∈ U^{0}, and x ∈ K. Then we obtain Φ(λ)(χ)(x) = λ^{∗}(χ)(x) = χ(λ(x)). Since Reχ(λ(x)) ≥ 0 for every χ ∈ U^{0}, we get λ(x) ∈ U. Proof of ⊇: Let λ^{∗} ∈ P (U^{0}, K^{0}), x ∈ K, and χ ∈ U^{0}. It follows from λ^{∗}(χ) ∈ K^{0} that Reχ(λ(x)) = Reλ^{∗}(χ)(x) ≥ 0. Since U is quasi–convex, this yields λ(x) ∈ U.]

By assumption, α_{H} is continuous, hence the sets {U^{0} : U ∈ U_{H}(e)} form a cobasis for the compact subsets of H^{∗} (1.1). So the assertion follows from (∗).

2. The Lie algebra of a nuclear group

The class of nuclear groups was introduced by Banaszczyk in [2]. It is a Hausdorff variety consisting of abelian groups which contains all LCA groups and all locally convex nuclear vector spaces and which is closed with respect to forming arbitrary products, subgroups and Hausdorff quotient groups. (See e.g. section 7 in [2].)

Here we use a technical description making use of the Kolmogoroff–diameters whose original setting is in the context of locally convex spaces.

Let V be a vector space. For symmetric and convex subsets X, Y ⊆ V we define

d_{k}(X, Y) := inf{c > 0| ∃L_{c} ≤ V : dimL_{c} < k and X ⊆ c · Y + L_{c}}.

As usual, inf Ø = ∞. The number d_{k}(X, Y) is called the k–th Kolmogoroff– diameter of X with respect to Y .

A locally convex vector space V is called a nuclear vector space if for every symmetric and convex neighbourhood U of 0 there exists W, another symmetric and convex neighbourhood of 0, such that d_{k}(W, U) ≤ k^{−1} for all k ∈ ℕ. Roughly speaking, W is considerably smaller than U.

Next, we wish to define the Kolmogoroff–diameters for groups.

Let W, U be subsets of a group G. We set (d_{k}(W, U)) ≤ (c_{k}) ((c_{k}) a sequence of real numbers) if there exist: a vector space V , symmetric and convex subsets X, Y ⊆ V which satisfy d_{k}(X, Y) ≤ c_{k} for all k ∈ ℕ, a subgroup H of V , and a homomorphism φ : H → G which fulfills W ⊆ φ(X∩H) and φ(H∩Y) ⊆ U.

An abelian Hausdorff topological group G is called a nuclear group if for every neighbourhood U of the neutral element 0, for every c > 0, and every m ∈ ℕ, there exists a neighbourhood W of 0 which satisfies (d_{k}(W, U)) ≤ (c·k^{−m}).

Lemma 2.1. Let G be a nuclear group such that α_{G} is continuous. Then for every compact subset K ⊆ G^{∗}, every c > 0, and every m ∈ ℕ there exists a compact subset S ⊆ G^{∗} which satisfies (d_{k}(K, S))_{k∈ℕ} ≤ (c · k^{−m})_{k∈ℕ}.

Proof. Fix a compact subset K ⊆ G^{∗}, c > 0, and m ∈ ℕ.

Since α_{G} is continuous, there exists U ∈ U_{G}(0) such that K ⊆ U^{0} (1.1). By assumption, G is a nuclear group, hence we can find a neighbourhood W ∈ U_{G}(e) such that (d_{k}(W, U)) ≤ c̃k^{−m̃} for c̃ = \frac{c}{γm̃} and m̃ = m + 5. (As usual, γm̃ denotes the universal constant defined in (2.14) in [2].) According to (16.4) in [2], we get (d_{k}(U^{0}, W^{0})) ≤ (c̃γm̃ · k^{−m̃+5}) = (c · k^{−m}).

Since S := W^{0} is compact ((3.5) in [1]) and K ⊆ U^{0} we get (d_{k}(K, S)) ≤ (c · k^{−m}) and hence the assertion follows.

Theorem 2.2. Let G be a nuclear group such that α_{G} is continuous. Then Hom_{co}(G^{∗}, ℝ) is a nuclear group, too.

Proof. For K ⊆ G^{∗} compact, c > 0, and m ∈ ℕ we put:

m̃ := m + 4 und c̃ < min(\frac{c}{2γ_{m+4}γ_{m+2}} , \frac{\sqrt{6}}{2πγ_{m+4}} ).

According to the Lemma 2.1, there exists a compact subset S ⊆ G^{∗} which satisfies (d_{k}(K, S)) ≤ (c̃k^{−m̃}). Thus there exist a vector space V , a subgroup H ≤ V , two symmetric and convex subsets X, Y ⊆ V which satisfy d_{k}(X, Y) ≤ c̃k^{−m̃} for all k ∈ ℕ, and a homomorphism φ : H → G^{∗} such that K ⊆ φ(X ∩ H) and φ(Y ∩ H) ⊆ S.

According to (2.14) in [2], there exist pHsns (pre-Hilbert seminorms) p, q defined on ⟨X⟩_{ℝ} such that their closed unit balls B_{p}, respectively B_{q}, satisfy X ⊂ B_{p}, B_{q} ⊆ Y and d_{k}(B_{p}, B_{q}) ≤ γm̃c̃k^{−m̃+2}. Without loss of generality ((2.13) in [2]) we may assume that V = ⟨X⟩_{ℝ}. In particular, we obtain:

K ⊆ φ(H ∩ B_{p}) and φ(H ∩ B_{q}) ⊆ S.  (1)

By L(H), respectively A(H), we denote the free vector space, respectively, the free abelian group generated by η(H) where η : H ↦ η(H) is a bijection; further, we introduce the following mappings:

π : L(H) → V, ∑ λ_{x}η(x) ↦ ∑ λ_{x}x

and

π′ : A(H) → H, ∑ k_{x}η(x) ↦ ∑ k_{x}x.

The compositions p̃:= p ∘ π and q̃:= q ∘ π are pHsns on L(H) whose closed unit balls satisfy Bp̃ = π^{−1}(B_{p}) and Bq̃ = π^{−1}(B_{q}). Since π is surjective, we obtain ((2.8) b) in [2]):

d_{k}(Bp̃, Bq̃) = d_{k}(B_{p}, B_{q}) ≤ γm̃c̃k^{−m̃+2}  (2)

and hence ∑_{k∈ℕ}d_{k}(Bp̃, Bq̃)^{2} ≤ (γm̃c̃)^{2} ∑_{k∈ℕ}k^{−2(m̃−2)} < \frac{1}{4} . Let

X̃ := conv(A(H) ∩ Bp̃) and Ỹ := conv(A(H) ∩ Bq̃).  (3)

Hence (3.20) in [2] implies d_{k}(X̃, Ỹ) ≤ 2d_{k}(Bp̃, Bq̃) ≤ 2γm̃c̃k^{−m̃+2}. Again by (2.14) in [2], there exist pHsns p′, q′ on ⟨X̃⟩_{ℝ} ≤ L(H) which satisfy

X̃ ⊆ B_{p′}, B_{q′} ⊆ Ỹ, and d_{k}(B_{p′}, B_{q′}) ≤ 2γm̃γ_{m̃−2}c̃k^{−m̃+4} ≤ c · k^{−m}.  (4)

[figure: L(H); A(H); H; π′; G^{∗}; φ; ℝ; λ]

We define

Ψ : aHom(G^{∗}, ℝ) → aHom(L(H), ℝ), λ ↦ { \atop λ ∘ φ},  (5)

where { \atop λ ∘ φ} denotes the unique linear extension on L(H) of the mapping λ ∘ φ : H → ℝ and aHom all (not necessarily continuous) homomorphisms. Observe that the spaces aHom(·) are vector spaces and that Ψ is a linear mapping. As a consequence of π′(A(H) ∩ Bp̃) = H ∩ B_{p}, π′(A(H) ∩ Bq̃) = H ∩ B_{q}, and

{ \atop λ ∘ φ}|_{A(H)} = λ ∘ φ ∘ π′, we obtain

Ψ^{−1}(P (B_{p′}, [−1, 1])) = \{λ ∈ aHom(G^{∗}, ℝ) : { \atop λ ∘ φ}(B_{p′}) ⊆ [−1, 1]\}

⊆^{(4)} \{λ ∈ aHom(G^{∗}, ℝ) : { \atop λ ∘ φ}(X̃) ⊆ [−1, 1]\}

=^{(3)} \{λ ∈ aHom(G^{∗}, ℝ) : { \atop λ ∘ φ}(A(H) ∩ Bp̃) ⊆ [−1, 1]\}

= {λ ∈ aHom(G^{∗}, ℝ) : λ ∘ φ ∘ π′(A(H) ∩ Bp̃) ⊆ [−1, 1]}

= {λ ∈ aHom(G^{∗}, ℝ) : λ ∘ φ(H ∩ B_{p}) ⊆ [−1, 1]}

⊆^{(1)} {λ ∈ aHom(G^{∗}, ℝ) : λ(K) ⊆ [−1, 1]}, and

Ψ^{−1}(P (B_{q′}, [−1, 1])) = \{λ ∈ aHom(G^{∗}, ℝ) : { \atop λ ∘ φ}(B_{q′}) ⊆ [−1, 1]\}

⊇^{(4)} \{λ ∈ aHom(G^{∗}, ℝ) : { \atop λ ∘ φ}(Ỹ) ⊆ [−1, 1]\}

=^{(3)} \{λ ∈ aHom(G^{∗}, ℝ) : { \atop λ ∘ φ}(A(H) ∩ Bq̃) ⊆ [−1, 1]\}

= {λ ∈ aHom(G^{∗}, ℝ) : λ ∘ φ ∘ π′(A(H) ∩ Bq̃) ⊆ [−1, 1]}

= {λ ∈ aHom(G^{∗}, ℝ) : λ ∘ φ(H ∩ B_{q}) ⊆ [−1, 1]}

⊇^{(1)} {λ ∈ aHom(G^{∗}, ℝ) : λ(S) ⊆ [−1, 1]}.

Trivially,

{λ ∈ Hom(G^{∗}, ℝ) : λ(S) ⊆ [−1, 1]} ⊆ Ψ^{−1}(P (B_{q′}, [−1, 1])) ∩ Hom(G^{∗}, ℝ) and

{λ ∈ Hom(G^{∗}, ℝ) : λ(K) ⊆ [−1, 1]} ⊇ Ψ^{−1}(P (B_{p′}, [−1, 1])) ∩ Hom(G^{∗}, ℝ).

Since these sets are symmetric and convex, we obtain:

d_{k}({λ ∈ Hom(G^{∗}, ℝ) : λ(S) ⊆ [−1, 1]}, {λ ∈ Hom(G^{∗}, ℝ) : λ(K) ⊆ [−1, 1]})

≤ d_{k}(Ψ^{−1}(P (B_{q′}, [−1, 1]))∩Hom(G^{∗}, ℝ), Ψ^{−1}(P (B_{p′}, [−1, 1]))∩Hom(G^{∗}, ℝ))

≤^{⋄} d_{k}(Ψ^{−1}(P (B_{q′}, [−1, 1])), Ψ^{−1}(P (B_{p′}, [−1, 1])))

≤^{∗} d_{k}(P (B_{q′}, [−1, 1]) ∩ im(Ψ), P (B_{p′}, [−1, 1]) ∩ im(Ψ))

≤^{•} d_{k}(P (B_{q′}, [−1, 1]), P (B_{p′}, [−1, 1]))

=^{∘} d_{k}(B_{p′}, B_{q′}) ≤^{(4)} c · k^{−m}.

⋄: Observe that polars of and inverse images under linear mappings of closed unit balls of pHsns are closed unit balls of pHsns; hence we may apply (2.13) in [2].

∗ is a consequence of (18.5) in [1],

• follows from (2.13) in [2], ∘ is (2.16) in [2].

Corollary 2.3. Let G be a nuclear group such that α_{G} is continuous. Then Hom_{co}(G^{∗}, ℝ) is a locally convex nuclear vector space.

Proof. According to the above theorem, Hom_{co}(G^{∗}, ℝ) is a nuclear group and, obviously, a topological vector space. Now (8.9) in [2] implies that it is a nuclear locally convex vector space.

Remark 2.4. A similar result has been shown in (20.25) in [1], namely: If G is a nuclear group and α_{G} is continuous then Hom_{co}(G^{∗}, 𝕋) is a nuclear group.

This is a rather trivial consequence of the estimate proved in (16.4) in [2]. The problem which prevents us to copy that proof is the following difference

between characters (i.e. continuous homomorphism into 𝕋) and real characters (i.e. continuous homomorphisms into ℝ):

Let H be a subgroup of a nuclear group G. Then every character χ : H → 𝕋 can be extended to a character χ_{G} : G → 𝕋 ((8.3) in [2]).

The analogous assertion for real characters is not true as the following example will show:

It is well known that the space of rapidly decreasing sequences s := {(x_{n})_{∈ℕ} : (∀m ∈ ℕ) p_{m}((x_{n})_{n∈ℕ}) < ∞} endowed with the topology induced by the norms p_{m}((x_{n})_{n∈ℕ}) := sup{n^{m}|x_{n}| : n ∈ ℕ} is a locally convex nuclear vector space (see e.g.(6.1.3) in [8]). The subgroup H := ⟨e_{k} : k ∈ ℕ⟩ where e_{k} = (δ_{n,k})_{n∈ℕ} is discrete and free. Hence every real sequence (x_{n}) gives rise to a real character H → ℝ, ∑ k_{n}e_{n} ↦ ∑ k_{n}x_{n}.

For a continuous linear form φ of s, there exists there exists m ∈ ℕ such that φ(B_{p_{m}}) is bounded; this implies that (φ(e_{n})) = O(n^{m}).

We wish to use the result (2.2) in order to prove that the Lie algebra of a nuclear group is a nuclear vector space.

Proposition 2.5. Let G = ∏_{i∈I} G_{i}, then L(G) is topologically isomorphic to ∏_{i∈I} L(G_{i}).

Proof. See (7.38) in [3].

Theorem 2.6. If G is a nuclear group then L(G) is a locally convex nuclear vector space.

Proof. Since we already know that L(G) is a Hausdorff topological vector space ((1.2) i)), because of (8.9) in [2], it is sufficient to show that L(G) is a nuclear group.

According to (21.3) in [1], G can be embedded into a product of metrizable nuclear groups G_{i}. Hence L(G) is topologically isomorphic to a subspace of ∏_{i∈I} L(G_{i}) (1.3 and 2.5). Since the class of nuclear groups forms a variety, we may assume that G is metrizable.

By (1.1), the mapping α_{G} is continuous so (1.4) implies that L(G) can be identified with a subgroup of Hom_{co}(G^{∗}, ℝ^{∗}). According to (23.27) (e) in [4], ℝ^{∗} is topologically isomorphic to ℝ, hence the groups Hom_{co}(G^{∗}, ℝ^{∗}) and Hom_{co}(G^{∗}, ℝ) are topologically isomorphic. Now 2.2 implies that this group is nuclear and hence the assertion follows.

Note: The question underlying this paper was posed by Helge Glöckner at the conference “Nuclear Groups and Lie Groups” in September 1999 in Madrid.

References

[1] Außenhofer, L., Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, Dissertationes Mathematicae 384 (1999).

[2] Banaszczyk, W., “Additive Subgroups of Topological Vector Spaces, ” Springer–Verlag, 1991.

[3] Hofmann, K. H., and S. A. Morris, “The Structure of Compact Groups,” de Gruyter Studies in Mathematics, 1998.

[4] Hewitt, E., and K. A. Ross, “Abstract Harmonic Analysis, ” Springer– Verlag, 1979.

[5] Kōmura, T., and Y. Kōmura, Über die Einbettung der nuklearen Räume in (s)^{A}, Math. Ann. 162 (1966), 284–288.

[6] Martín Peinador, E., and J. Núñez García (Eds.), “Nuclear Groups and Lie Groups,” Heldermann Verlag, 2001.

[7] Morris, S. A., “Pontryagin Duality and the Structure of Locally Compact Abelian Groups,” Cambridge University Press, 1977.

[8]  Pietsch, A., “Nuclear Locally Convex Spaces,” Springer–Verlag, 1972.

Lydia Außenhofer Mathematisch–Geographische Fakultät Katholische Universität Eichstätt Ostenstr. 26 D–85071 Eichstätt lydia.aussenhofer@ku-eichstaett.de

Received March 5, 2002 and in final form August 21, 2002