Fragmentability and Continuity of Semigroup Actions

Michael G. Megrelishvili

Communicated by Jimmie D. Lawson

Let a topologized semigroup S act continuously and linearly on a locally convex space X. We find sufficient conditions for continuity of induced actions on the spaces of linear (compact) operators and on the dual space X^{∗}, for instance. The notion of fragmentability in the sense of Jayne and Rogers and its natural uniform generalizations play a major role in this paper. Our applications show that problems concerning the continuity of induced actions have satisfactory solutions for Asplund Banach spaces X (without additional restrictions, if S is a topological group) and, moreover, for a new locally convex version of Asplund spaces introduced in the paper. The starting point of this concept was the characterization of Asplund spaces due to Namioka and Phelps in terms of fragmentability.

1. Introduction

Let π: S × X → X be a continuous linear action of a topologized semigroup S on a linear space X. For every linear space Y , denote by L(X, Y) and K(X, Y) the linear spaces of all linear and compact operators respectively, endowed with the strong, that is, the topology of bounded convergence. Consider the induced (right) action:

π^{L}: L(X, Y) × S → L(X, Y), (fs) (x) = f(sx).

(We write sx, fs instead of π(s, x) and π^{L}(f, s), respectively). The dual action, that is, the case Y := ℝ (the field of reals), is denoted by π^{∗}: X^{∗} × S → X^{∗}. The subset of all functionals f ∈ X^{∗} for which the orbit map f̃: S → X^{∗}, f̃(s) = fs, is continuous (at fixed s ∈ S) is denoted by X^{⊙} (resp.: X^{⊙}(s)).

Frequently, X^{⊙} may be a proper subset of X^{∗} even for the semigroups S := ℝ and S := [0, ∞) (see sections 1.3 and 1.5 in [36]). More generally, the space of all absolutely continuous measures on a locally compact group G is just the set of all such functionals m ∈ (C_{0}(G))^{∗} = M(G) for which the orbit mappings m̃: G → M(G) are continuous (cf. [40, 41, 11]).

We study the following general question:

1991 Mathematics Subject Classification. 54H15, 47D03, 46A99, 46B99. Key words and phrases. Asplund space, Radon-Nikodym property, fragmentable set, Namioka’s

theorem, dual space, semigroup action, compact operator, left (right) topological semigroup.

Typeset by AMS-TEX

Question 1.1. For which restrictions on the spaces X, on the operators f ∈ L(X, Y) and on the elements s ∈ S, is the pair (f, s) a point of joint (or separate) continuity for π^{L}? In particular, when do we have X^{⊙} = X^{∗}?

For Y := ℝ we refer, among others, to [5, 18, 26, 30] and to the references there. For the important case of one-parameter semigroups see, [36].

Recall some positive results.

1.2. Let S be a locally compact topological group, and let X be semireflexive. Then π^{∗} is continuous [5, 30].

1.3. If X is a Banach space such that the dual X^{∗} has the Radon-Nikodym property (by Stegall’s result [45], it is equivalent to saying that X is Asplund), and if S is the one-parameter semigroup [0, ∞), then π^{∗} is continuous at every (f, t), where f ∈ X^{∗} and t > 0. This is proved by Arendt in [3] and, later, by van Neerven [35] in a somewhat stronger form.

One of the main purposes of the present paper is to provide a unified approach to such results. We generalize 1.2 and 1.3 in several ways:

(a) We have found a more general class (NP) of spaces X, called Namioka-Phelps spaces, in which the same conclusions hold;

(b) If X is an Asplund Banach space, then π^{∗} is continuous for any topological group S. More generally, if S is a topological subsemigroup of a topological group G, then π^{∗} remains continuous at every (f, t), where f ∈ X^{∗} and t ∈ int(S) · int(S);

(c) Analogous results are valid sometimes if X^{∗} is replaced by K(X, Y) (see 6.13).

In section 3 we investigate a generalized version of fragmentability. This is an attempt to synthesize some known facts and definitions from [34, 22, 31, 20, 47, 13]. Section 4 is devoted to the class (NP) of Namioka-Phelps spaces. We say that a linear topological space X is (NP) if every equicontinuous subset F of X^{∗} is fragmented. More precisely, this means that for every element ε of the natural uniformity on X^{∗} and every non-empty subset F_{1} ⊆ F, there exists an w^{∗} -open subset O of X^{∗} such that O ∩ F_{1} is non-empty and ε-small. If X is a Banach space, then this definition gives exactly Asplund spaces. This follows directly from the well-known characterization due to Namioka and Phelps [34]. We show that (NP) contains all Asplund linear spaces in the sense of [4, 44], all semireflexive and all nuclear spaces. The class (NP) is closed under taking subspaces, arbitrary products and direct sums. Moreover, using the Diestel-Morris-Saxon result [9], it can be proved that (NP) contains the variety generated by the class of all Asplund Banach spaces.

Section 5 contains some useful “transport arguments.” Our main applications are concentrated in section 6. In particular, we generalize the above-mentioned results 1.2 and 1.3. Moreover, some “small-orbit results” are discussed there. It is a well-known observation that the smallness (separability, for instance) of the orbit of a functional f ∈ X^{∗} frequently implies that f ∈ X^{⊙}. This happens, for example, for measures on a locally compact group G. Such a result was obtained first in [25] for a second countable G and in its full generality in [46, 11]. Another example in this spirit, when f ∈ L^{∞}(G) = (L^{1}(G))^{∗}, can be found in [39]. Jan van Neerven [35] proved that for one-parameter semigroups on a Banach space X, the orbit of f ∈ X^{∗} is (weak) separable iff f ∈ X^{⊙}(t) for every t > 0. We show that some results of this kind can be unified in the framework of the general machinery developed in sections 3 and 5.

2. Preliminary results

Throughout the paper, all spaces are assumed to be Hausdorff and all linear spaces are real. The filter of all neighborhoods (nbd′s) of a point x is denoted by N_{x}. The identity of a monoid and the origin of a linear space are denoted by e and 0, respectively.

Let π: P × X → Y, π(p, x) = px be a function. Then the rules x̃(p) = px = p̃(x) define the functions: x̃: P → Y, p̃: X → Y. We use the following notation:

Con_{p}^{ℓ}(π) = {x ∈ X | x̃ is continuous at p} , Con^{ℓ}(π) = ∩ {Con_{p}^{ℓ}(π) | p ∈ P } ,

Con_{x}^{r}(π) = {p ∈ P | p̃ is continuous at x} , Con^{r}(π) = ∩ {Con_{x}^{r}(π) | x ∈ X} ,

Con_{p}(π) = {x ∈ X | π is continuous at (p, x)} ,
Con_{x}(π) = {p ∈ P | π is continuous at (p, x)} ,
Con(π) = {(p, x) ∈ P × X | π is continuous at (p, x)} .

The following useful lemma is well known (see, for example [42, p. 47]).

Lemma 2.1. Let Con^{r}(π) = P, X compact and Y a uniform space. Then for every p_{0} ∈ P, the family X̃ := {x̃: P → Y | x ∈ X} is equicontinuous at p_{0} iff X = Con_{p_{0}}(π).

The following definition generalizes the notion of uniform equicontinuity and, as we will see below, in some sense is good enough.

Definition 2.2. Let π: P × X → Y be a function, where P is a topological space, and (X, μ) and (Y, ξ) are uniform spaces. A subset A ⊆ X is called π - uniform at p_{0} ∈ P if for every entourage ε ∈ ξ, there exists (δ, U) ∈ μ × N_{p_{0}} such that for every p ∈ U and every (a, b) ∈ δ ∩ (A × A), holds (pa, pb) ∈ ε. If A is π -uniform at every p ∈ P, then A is called π -uniform. Analogously, subsets A ⊆ X which are π -uniform at p_{0} ∈ P can be defined for functions π of the form X × P → Y.

In 2.3, 2.4 and 2.5 we will keep the notation of 2.2.

Examples 2.3.

(a) Let U ∈ N_{p_{0}} and Ũ = {p̃: X → Y | p ∈ U} be uniformly equicontinuous. Then X is π -uniform at p_{0}.

(b) Let Ã = {ã: P → Y | a ∈ A} be equicontinuous at p_{0} and the restriction p̃_{0} |_{A}: A → Y be uniformly continuous. Then A is π -uniform at p_{0}.

(c) If X is compact, Con^{r}(π) = P and X = Con_{p_{0}}(π), then X is π -uniform at p_{0}.

(d) If π: G×X → X is an action of a topological group G, and μ is a uniformity on X, then X is π -uniform at e iff π is quasibounded in the sense of [27].

(e) Let π: G × X → X be an action of a Baire topological group G on a metric space (X, d). Suppose that each g̃: X → X(g ∈ G) is d-uniformly continuous and every ỹ: G → X(y ∈ Y) is continuous for a certain dense subset Y ⊆ X. Then X is π -uniform.

Proof. (a) is trivial and (b) is straightforward, (c) follows from (b) and Lemma 2.1. For (e), use (d) and Theorem 1.10 from [27]. □

Lemma 2.4. Let A ⊆ X be π -uniform at p_{0} ∈ P and suppose Con^{r}(π) = P. Then

(i) The closure cℓ(A) is also π -uniform at p_{0}.

(ii) If A ⊆ cℓ(A ∩ Con_{p_{0}}^{ℓ}(π)), then the restricted map P × cℓ(A) → Y is

continuous at (p_{0}, x) for every x ∈ cℓ(A). (iii) If A = X, then the subset Con_{p_{0}}(π) is closed and coincides with Con_{p_{0}}^{ℓ}(π).

Proof. (i) For a fixed ε ∈ ξ, choose ε_{1} ∈ ξ such that ε_{1}^{3} ⊆ ε. According to Definition 2.2 there exists (δ, U) ∈ μ × N_{p_{0}} such that:

(pa, pb) ∈ ε_{1} for every (a, b) ∈ δ ∩ (A × A) and every p ∈ U.

Take δ_{1} ∈ μ with δ_{1}^{3} ⊆ δ. Then (δ_{1}, U) is the desired pair satisfying Definition 2.2 for cℓ(A). Indeed, for an arbitrary but fixed pair (x, y) from δ_{1} ∩ (cℓ(A) × cℓ(A)) and a fixed element p ∈ U, by the continuity of p̃, we can choose elements a, b ∈ A such that:

(a, x) ∈ δ_{1}, (y, b) ∈ δ_{1}, (px, pa) ∈ ε_{1}, (pb, py) ∈ ε_{1}.

Clearly, (a, b) ∈ δ_{1}^{3} ⊆ δ. Then (pa, pb) ∈ ε_{1}. Finally, observe that (px, py) ∈ ε_{1}^{3} ⊆ ε.

(ii) By (i) we may assume that cℓ(A) = A. For a fixed (a_{0}, ε) ∈ A × ξ, choose (ε_{1}, δ) ∈ ξ × μ and U_{1} ∈ N_{p_{0}} such that ε_{1}^{3} ⊆ ε and (px, py) ∈ ε_{1} for every (x, y) ∈ δ ∩ (A × A) and p ∈ U_{1}. Choose a symmetric δ_{1} ∈ μ such that δ_{1}^{2} ⊆ δ and take an element a ∈ A ∩ Con_{p_{0}}^{ℓ}(π) such that (a_{0}, a) ∈ δ_{1}. For a certain U_{2} ∈ N_{p_{0}} , (p_{0}a, pa) ∈ ε_{1} for every p ∈ U_{2}. Then if (a_{0}, x) ∈ δ_{1} , x ∈ A and p ∈ U_{1} ∩ U_{2} , we have (pa, px) ∈ ε_{1} because (a, x) ∈ δ_{1}^{2} ⊆ δ. Then (p_{0}a_{0}, px) ∈ ε_{1}^{3} ⊆ ε.

(iii) Follows easily from (ii). □

Corollary 2.5. Let every p̃: X → Y (p ∈ P) be uniformly continuous. Denote by X̂ and Ŷ the corresponding completions, and let π̂: P × X̂ → Ŷ be the induced function.

(i) If X is π -uniform at p_{0} ∈ P, then X̂ is π̂ -uniform at p_{0}. If, in addition, X = Con_{p_{0}}(π), then X̂ = Con_{p_{0}}(π̂).

(ii) If π is continuous and X is π -uniform , then π̂ is continuous and X̂ is π̂ -uniform.

Proof. For (i) apply Lemma 2.4 to π̂. The case (ii) follows from (i). □

Proposition 2.6. Let X, Y be topological groups, each endowed with its left uniformity, and let π: P ×X → Y be a function such that every p̃: X → Y (p ∈ P) is a homomorphism.

(i) X is π -uniform at p_{0} ∈ P iff (p_{0}, e) ∈ Con(π).

(ii) If (p_{0}, e) ∈ Con(π) and P = Con^{r}(π), then Con_{p_{0}}(π) = Con_{p_{0}}^{ℓ}(π).

(iii) If X = Con_{p_{0}}(π) and P = Con^{r}(π), then X̂ = Con_{p_{0}}(π̂).

(iv) If π is continuous, then π̂ is also continuous.

Proof. (i) is straightforward. For (ii), use (i) and Lemma 2.4 (iii). In order to establish (iii), combine (i) with Corollary 2.5 (i). Clearly, (iii) ⇒ (iv). □

In general, π̂ is not continuous even for group actions (see [28]).

Let X be a linear topological space, and let π: P × X → X be right linear (i.e., each p̃ is linear). We say that π is locally bounded at p_{0} ∈ P if for every bounded subset B of X there exists U ∈ N_{p_{0}} such that UB := {px | p ∈ U, x ∈ B} is bounded. The function π is called locally equicontinuous at p_{0} if there exists U ∈ N_{p_{0}} such that Ũ is equicontinuous.

Lemma 2.7.

(i) If π: P × X → X is locally equicontinuous at p_{0}, then π is locally bounded at p_{0}.

(ii) If π is locally bounded at p_{0}, then π^{∗}: X^{∗} × P → X^{∗} is continuous at

(0, p_{0}) (and therefore X^{∗} is π^{∗} -uniform at p_{0} by Proposition 2.6(i)). (iii) If π is continuous and C is a compact subset of P, then C̃ is equicontin-

uous.

Proof. (i) and (ii) are straightforward, and (iii) is a variant of Lemma 2.1. □

Proposition 2.8. Let (X, ∥ ∥) be a normed space, and let π: P × X → X be a right linear function. Then the following conditions are equivalent:

(i) π is continuous at (p_{0}, 0).

(ii) π is locally equicontinuous at p_{0}.

Proof. (i) ⇒ (ii) Since π is continuous at (p_{0}, 0), there exist U ∈ N_{p_{0}} and a number δ > 0 such that the inequality ∥a∥ ≤ δ implies that ∥pa∥ ≤ 1 for each p ∈ U. Then the norm of p̃ is not greater than \frac{1}{δ} for every p ∈ U. Indeed,

observe that ∥\frac{δ}{∥x∥} x∥ = δ for every x ≠ 0. By our choice,∥p\frac{δ}{∥x∥} x∥ ≤ 1. Therefore,

∥px∥ ≤ \frac{1}{δ} ∥x∥.

The implication (ii) ⇒ (i) is trivial. □

Let P_{1}, P_{2} be topological spaces, X_{1}, X_{2} normed spaces, and π_{k}: P_{k} × X_{k} → X_{k}, k ∈ {1, 2}, functions, where p̃_{k}: X → X is a linear bounded operator for every k ∈ {1, 2} and every p_{k} ∈ P_{k}. Consider the right linear map:

π_{1} ⊗ π_{2} : (P_{1} × P_{2}) × (X_{1} ⊗ X_{2}) → X_{1} ⊗ X_{2},

( (p_{1}, p_{2}), ∑_{j=1}^{n} x_{j} ⊗ y_{j}) ⟼ ∑_{j=1}^{n} p_{1}x_{j} ⊗ p_{2}y_{j}.

The projective and injective tensor norms ∥m∥_{p}, ∥m∥_{i} of m ∈ X_{1} ⊗X_{2} are defined by the rules:

∥m∥_{p} = inf {∑_{j=1}^{n} ∥x_{j}∥ · ∥y_{j}∥ | m = ∑_{j=1}^{n} x_{j} ⊗ y_{j}} ,

∥m∥_{i} = sup {∑_{j=1}^{n} f_{1}(x_{j})f_{2}(y_{j}) | m = ∑_{j=1}^{n} x_{j} ⊗ y_{j}, ∥f_{1}∥ ≤ 1, ∥f_{2}∥ ≤ 1, f_{1} ∈ X_{1}^{∗}, f_{2} ∈ X_{2}^{∗}} .

The projective and injective tensor products of X_{1} and X_{2} are the completions of X_{1} ⊗ X_{2} with respect to the norms ∥ ∥_{p} and ∥ ∥_{i} respectively, which we denote by X_{1} ˆ⊗_{p}X_{2} and X_{1} ˆ⊗_{i}X_{2}. For every (p_{1}, p_{2}) ∈ P_{1} × P_{2} , the corresponding π_{1} ⊗π_{2}−translation of X_{1} ⊗X_{2} is bounded with respect to both norms. Therefore, π_{1} ⊗ π_{2} can be extended to right linear functions:

π_{1} ˆ⊗_{p}π_{2} : (P_{1} × P_{2}) × (X_{1} ˆ⊗_{p}X_{2}) ⟶ X_{1} ˆ⊗_{p}X_{2},
π_{1} ˆ⊗_{i}π_{2} : (P_{1} × P_{2}) × (X_{1} ˆ⊗_{i}X_{2}) ⟶ X_{1} ˆ⊗_{i}X_{2}.

Moreover, we have the following useful result.

Proposition 2.9. With the above notation, let π_{k} be continuous at (p_{k}^{0}, x_{k}) for every x_{k} ∈ X_{k} and fixed p_{k}^{0} ∈ P_{k} (k ∈ {1, 2}). Then π_{1} ˆ⊗_{p}π_{2} and π_{1} ˆ⊗_{i}π_{2} are continuous at (p_{1}^{0}, p_{2}^{0}, m) for every m ∈ X_{1} ˆ⊗_{p}X_{2} or m ∈ X_{1} ˆ⊗_{i}X_{2}, respectively.

Proof. By Proposition 2.8, each p_{k}^{0} has an nbd U_{k} such that Ũ_{k} is normbounded with respect to the norm of X_{k} (k ∈ {1, 2}). The explicit description of the norms ∥ ∥_{p}, ∥ ∥_{i} shows that for the nbd U_{1}×U_{2} of (p_{1}^{0}, p_{2}^{0}) , the family of operators

{ \atop U_{1} × U_{2}} will be norm-bounded with respect to both norms. Therefore, π_{1} ˆ⊗_{p}π_{2} , π_{1} ˆ⊗_{i}π_{2} are continuous at (p_{1}^{0}, p_{2}^{0}, 0) . By assertions (ii) and (iii) of Proposition 2.6, it suffices to prove that the orbit mappings m̃: P_{1}×P_{2} → X_{1} ˆ⊗_{p}X_{2}, m̃: P_{1}×P_{2} → X_{1} ˆ⊗_{i}X_{2} are continuous at (p_{1}^{0}, p_{2}^{0}) for every m from the dense subset X_{1} ⊗ X_{2}. Since ∥m∥_{i} ≤ ∥m∥_{p} (see Proposition 7.2.1 in [36]), we have only to examine the “projective case”. This can be done easily for elementary tensors x_{1} ⊗ x_{2} using the well known identity ∥x_{1} ⊗ x_{2}∥_{p} = ∥x_{1}∥ ∥x_{2}∥. Since such tensors span every m ∈ X_{1} ⊗ X_{2}, the proof is completed. □

3. Fragmentable functions and sets

Let (X, τ) be a topological space, and let ρ be a metric on the set X. Following Jayne and Rogers [22], we say that X is fragmented by ρ if for each non-empty subset A of X and for each ε > 0, there exists a τ -open subset O of X such that O ∩ A ≠ ϕ and ρ-diam (O ∩ A) ≤ ε.

We need the following generalization.

Definition 3.1. Let (X, τ) be a topological space, Γ a system of subsets in X, and f: X → Y a fixed (not necessarily continuous) function from X into a uniform space (Y, μ). We say that Γ is fragmented by f (with respect to the pair (τ, μ)) if for every non-empty A ∈ Γ and element ε ∈ μ, there exists a τ -open subset O of X such that O ∩ A ≠ ϕ and f(O ∩ A) is ε-small (i.e., (x, y) ∈ ε for x, y ∈ f(O ∩ A)). In the special cases Γ = N_{x} or Γ = τ, we say that X is locally fragmented at x or, respectively, locally fragmented. If 2^{A} ⊆ Γ, then we simply say that A is fragmented.

Besides the usual notion of fragmentability, Definition 3.1 generalizes one more concept. Namely, f is cliquish (at x) in the sense of Thielman [47] iff X is locally fragmented (at x) and μ is a metric. Huskable sets [13] are also a particular case of local fragmentability. It is also a remarkable fact that if a function f: X → Y is quasi-continuous in the sense of Kempisty [23, 17] (see also modified continuity in [15]), then X is locally fragmented.

Lemma 3.2. Let Γ be a system of subsets in X, and let f: (X, τ) → (Y, μ) be a fixed function from a topological space (X, τ) into a uniform space (Y, μ).

(a) If the system cℓ(Γ) := {cℓ(A) | A ∈ Γ} is fragmented by f, then Γ is fragmented by f.

(b) If h: (X_{1}, τ_{1}) → (X, τ) is a continuous function, and for a certain A ⊆ X_{1} the subset h(A) of X is fragmented by f, then A is fragmented by f ∘ h w.r.t. (τ_{1}, μ).

(c) If every non-empty closed subspace X_{1} ⊆ X is locally fragmented at some point by the restricted function f |_{X_{1}} w.r.t. (τ |_{X_{1}}, μ), then X is fragmented by f w.r.t. (τ, μ).

(d) If f is locally fragmented, (X, τ) is a Baire space and μ = ρ is a metric, then f is continuous at the points of a dense G_{δ} subset D of X.

Proof. We omit the easy proof of (a), (b), (c). In order to check (d), for a fixed ε > 0 consider the open set O_{ε} — the union of all τ -open subsets O of X such that ρ-diam (f(O)) ≤ ε. Then local fragmentability guarantees that O_{ε} is dense. It is easy to see that D := ∩ \{O_{\frac{1}{n}} | n ∈ ℕ\} is the desired dense G_{δ} subset. □

Note that the assertion (d) was actually known. Its variant has been formulated without proof in [47, Theorem IV]. The present proof, which is very close to the proof of part (i) ⇒ (ii) in Lemma 1.1 of [33] (see also [15, Theorem 1.1]), is given here for the sake of completeness.

Let a system Φ of subsets in a topological space X be directed (upwards under the inclusion), and let (Y, μ) be a uniform space. By C_{Φ}(X, Y) we will denote the set C(X, Y) of all continuous maps from X into Y endowed with the uniformity μ_{Φ} of uniform convergence on elements of Φ. Recall that a standard base of this uniformity is the system {[A, ε] | A ∈ Φ, ε ∈ μ}, where [A, ε] := {(φ_{1}, φ_{2}) | (φ_{1}(a), φ_{2}(a)) ∈ ε ∀a ∈ A}. The set C(X, Y) with the pointwise topology will be denoted by C_{p}(X, Y).

Lemma 3.3. Let Φ be a directed system of subsets in a topological space Y such that Φ contains a fundamental subsystem Φ_{1} consisting of compact subsets. Let (Z, μ) be a uniform space. Suppose that (X, τ) is a Čech-complete space and that the function

f: (X, τ) → C_{p}(Y, Z), x ↦ φ_{x}

is continuous. Then X is fragmented by f w.r.t. (τ, μ_{Φ}). Proof. Let ϕ ≠ X_{1} ⊆ X, K ∈ Φ and ε ∈ μ. We have to show that there exists a τ -open subset O of X such that O ∩ X_{1} ≠ ϕ and f(O ∩ X_{1}) is [K, ε]-small. We may assume that K is compact (by our assumption on Φ) and X_{1} is τ -closed (Lemma 3.2 (a)). There exist a metric space (M, ρ), a uniformly continuous map h of Z onto M and a positive number δ such that the inequality ρ(h(z_{1}), h(z_{2})) < δ implies that (z_{1}, z_{2}) ∈ ε. Consider the (separately continuous) evaluation map:

π_{X_{1},K}: X_{1} × K → M , (x, y) ↦ h(φ_{x}(y)).

By Namioka’s theorem [32, Theorem 1.2], there exists a dense subset D of X_{1} such that π_{X_{1},K} is jointly continuous at every (x, y), where x ∈ D and y ∈ K. Lemma 1.1 now yields that the family {h ∘ ỹ: X_{1} → M | y ∈ K} is equicontinuous at every x ∈ D. Choose arbitrarily x_{0} ∈ D. Then there exists a τ -open nbd O of x_{0} such that

ρ(h(ỹ(x_{0})), h(ỹ(x))) = ρ(h(φ_{x_{0}}(y)), h(φ_{x}(y))) < \frac{δ}{2} ∀x ∈ O ∩ X_{1} ∀y ∈ K.

This implies that f(O ∩ X_{1}) is [K, ε]-small. □

Next, we make some notational conventions. Let X be a linear topological space (ℓ.t.s.), and let μ be its natural uniformity. By μ^{∗} we denote the uniformity of the strong dual X^{∗}. The weak and weak* topologies on X and X^{∗} will be denoted by w and w^{∗} respectively. If A is a subset of X (X^{∗}), then we say that A is fragmented if A is fragmented by the inclusion map f = 1_{A}: A ↪ X (A ↪ X^{∗}) with respect to (w, μ) (resp.: (w^{∗}, μ^{∗})). The systems of all bounded subsets of X and all equicontinuous subsets of X^{∗} are denoted by Φ_{b} and Φ_{eq} respectively.

The first assertion of the following lemma easily follows from the definitions.

Lemma 3.4. (a) Let Y be an ℓ.t.s. Then L(X, Y) is a uniform subspace of C_{Φ_{b}}(X, Y).

(b) [43, Ch. IV, 1.5, Corollary 4] Every locally convex space (ℓ.c.s.) X is a uniform subspace of C_{Φ_{eq}}(X^{∗}, ℝ).

The following result is well known for Banach spaces [33, Theorem 1.2].

Proposition 3.5. Every relatively weakly compact subset A of an ℓ.c.s. X is fragmented. Proof. By [43, Ch. III, 4.3], the system Φ_{eq} has a fundamental subsystem consisting of weak^{∗} -compact subsets. Therefore we can apply Lemma 3.3 to the w-continuous inclusion cℓ_{w}(A) ↪ C_{p}(X^{∗}, ℝ). Then cℓ_{w}(A) (and, hence, its subset A) is fragmented. □

Proposition 3.6. If X is semireflexive, then every relatively weak^{∗} -compact (and, hence, every equicontinuous) subset A of X^{∗} is fragmented. Proof. The semireflexivity of X means that each bounded subset of X is relatively weakly compact. Taking into consideration that every weakly compact subset of X is weakly bounded and, hence, bounded, we can apply Lemma 3.3 to the weak^{∗} -continuous inclusion map cℓ_{w^{∗}}(A) ↪ C_{p}(X, ℝ) and the system Φ_{b}. □

Definition 3.7. Let (X, μ) be a uniform space, and let ε ∈ μ. We say that X is ε-Lindelöf if the uniform cover {ε(x) | x ∈ X}, where ε(x) = {y ∈ X | (x, y) ∈ ε}, has a countable subcover. If X is ε-Lindelöf for each ε ∈ μ, then it will be called uniformly Lindelöf.

We mention that (X, μ) is uniformly Lindelöf iff it is ℵ_{0} -precompact in the sense of Isbell [20] iff X is ℵ_{0} -bounded in the sense of Guran (cf. Definition 2.4 in [48]). We prefer the name “uniformly Lindelöf” in order to avoid possible misunderstandings in linear spaces. If X, as a topological space, is either separable, Lindelöf or ccc (see [20, p.24]), then (X, μ) is uniformly Lindelöf. For a metrizable uniformity μ, (X, μ) is uniformly Lindelöf iff X is separable. Uniformly continuous maps move uniformly Lindelöf subspaces onto uniformly Lindelöf subspaces. Guran showed [16] that a topological group G endowed with its left or right uniformity is uniformly Lindelöf iff G is a topological subgroup in a product of second countable groups. As is well known, every ℓ.c.s. X is a linear topological subspace in a product of normed spaces. Combining these facts we obtain Proposition 3.8. An ℓ.c.s. X is uniformly Lindelöf iff X is a linear topological subspace in a product of separable normed spaces.

Definition 3.9. Let f: X → Y be a map from a topological space X into a uniform space (Y, μ). We say that f is locally uniformly Lindelöf at a point x ∈ X, if for every ε ∈ μ there exists U ∈ N_{x} such that f(U) is ε-Lindelöf (this holds, for example, if f(U_{0}) is uniformly Lindelöf for a certain U_{0} ∈ N_{x}).

This definition is closely related to a concept from [31] called the index of non-separability. If μ is a metric, then the condition in Definition 3.9 can be reformulated by saying that for every ε > 0 there exists U ∈ N_{x} such that the index of non-separability β(U) is less than ε. Hence, this gives a single-valued variant of the “β upper semi-continuity” [31, p. 70].

Proposition 3.10. Let Φ be a directed system of subsets in a topological space Y , and let (Z, μ) be a uniform space. Suppose that (X, τ) is Baire and the function f: X → C_{p}(Y, Z), x ↦ φ_{x} is continuous.

(i) If f: X → C_{Φ}(Y, Z) is locally uniformly Lindelöf at x_{0} ∈ X, then X is locally fragmented by f at x_{0} w.r.t. (τ, μ_{Φ}).

(ii) If X is hereditarily Baire (i.e., each closed subset is Baire) and f: X → C_{Φ}(Y, Z) is locally uniformly Lindelöf at each point, then X is fragmented by f w.r.t. (τ, μ_{Φ}).

Proof. (i). Let W be an open nbd of x_{0}, A ∈ Φ and ε ∈ μ. Our aim is to find an open subset O of W such that f(O) is [A, ε]-small. Choose symmetric δ ∈ μ such that δ^{2} ⊆ ε. We can suppose that δ is a closed subset of Z × Z. Since f is locally uniformly Lindelöf at x_{0}, there exists an open nbd U of x_{0} such that U ⊆ W and f(U) is [A, δ]-uniformly Lindelöf. Thus, there exists a sequence (x_{n}) in U such that

f(U) ⊆ ∪ {[A, δ] (f_{x_{n}}) | n ∈ ℕ} ,

where

[A, δ] (f_{x_{n}}) = {φ ∈ C(Y, Z) | (f_{x_{n}}(y), φ (y)) ∈ δ ∀y ∈ A} .

Then

U = ∪{M_{n} | n ∈ ℕ} for M_{n} := {x ∈ U | (f_{x_{n}}(y), f_{x}(y)) ∈ δ ∀y ∈ A}.

Since δ is closed and f is (τ, p)-continuous, one can easily see that each M_{n} is closed in U. By the Baire category theorem, a certain M_{n_{0}} contains a non-empty τ -open subset O. Then, (f_{x_{n_{0}}} , f_{x}) ∈ [A, δ] for every x ∈ O. This implies that f(O) is [A, ε]-small, because δ^{2} ⊆ ε.

For (ii), use (i) and Lemma 3.2 (c). □

Corollary 3.11. Let Y be an ℓ.t.s. (or an ℓ.c.s.), and let X be a uniformly Lindelöf subset of Y^{∗} (resp.: Y ). If X is hereditarily Baire in the weak^{∗} (resp.: weak) topology, then X is fragmented.

Proof. Apply Proposition 3.10(ii) to the inclusion (X, w^{∗}) ↪ C_{p}(Y, ℝ) and the system Φ = Φ_{b} (resp.: (X, w) ↪ C_{p}((Y^{∗}, ℝ) and Φ = Φ_{eq} ). □

Corollary 3.12. Let X be a Baire topological space.

(i) Suppose that (Y, τ) is an ℓ.c.s and that f: X → Y is weakly continuous. If f(X) is second countable w.r.t. τ, then f is τ -continuous at each point of a dense G_{δ} subset of X.

(ii) Suppose that (Y, τ) is an ℓ.t.s., (Y^{∗}, τ^{∗}) denotes its dual, and f: X → Y^{∗} is weak^{∗} -continuous. If f(X) is second countable w.r.t. τ^{∗}, then f is τ^{∗} -continuous at each point of a dense G_{δ} subset of X.

Proof. Use Proposition 3.10 (i) and Lemmas 3.4, 3.2. □

Remark 3.13. If, in the assertion (i) of Corollary 3.12, the space Y is assumed to be normed, then we obtain a result of Alexiewicz and Orlicz [1].

Recall that a real-valued function f defined on an open convex subset U of a linear topological space X is called Fréchet differentiable at x_{0} ∈ U whenever there exists u ∈ X^{∗} such that for every bounded B ⊆ X and every ε > 0, there exists δ > 0 such that for all x ∈ B and for all t with the property 0 < |t| < δ, the following inequality is satisfied:

|\frac{f(x_{0} + tx) − f(x_{0})}{t} − u(x)| < ε.

The function u is denoted by f′(x_{0}). In the definition, δ can be chosen so small that for every t ∈ (0, δ) and every x ∈ B, holds

(D)  f(x_{0} + tx) + f(x_{0} − tx) − 2f(x_{0}) < t\frac{ε}{3}.

A weak^{∗} slice of a nonempty subset A ⊆ X^{∗} is a subset of A of the form:

S(x_{0}, A, α) = {f ∈ A | f(x_{0}) > σ_{A}(x_{0}) − α} ,

where x_{0} ∈ X, α > 0 and σ_{A}(x) := sup {f(x) | f ∈ A}.

Proposition 3.14. Let X be a linear topological space, and let F be an equicontinuous subset of X^{∗}. Suppose that for every non-empty relatively w^{∗} - closed subspace A of F, the sublinear functional σ_{A}: X → ℝ, defined by the rule σ_{A}(x) = sup {f(x) | f ∈ A}, is Fréchet differentiable at some point (depending on A) x of X. Then F is fragmented. Proof. Fix ε > 0 and a bounded set B ⊆ X. We have to show that for every non-empty subset A of F, there exists a relatively w^{∗} -open non-empty subset O which is [B, ε]-small. It suffices to find a [B, ε]-small weak^{∗} slice of A. By Lemma 3.2 (a), we may suppose that A is relatively w^{∗} -closed in F. Consider the function σ_{A}. It is continuous because A is equicontinuous. Suppose that every weak^{∗} slice of A is not [B, ε]-small. As in the proof of Lemma 2.18 in [38], we will show that σ_{A} is nowhere Fréchet differentiable. Let x ∈ X. By our assumption, for each n ≥ 1 the weak^{∗} slice S (x, A, \frac{ε}{3n}) is not [B, ε]-small. Therefore, there exist f_{n}, h_{n} ∈ S (x, A, \frac{ε}{3n}) and x_{n} ∈ B which satisfy

|f_{n}(x_{n}) − h_{n} (x_{n})| > ε.

On the other hand, by the definition of slice we have:

f_{n}(x) > σ_{A}(x) − \frac{ε}{3n} , h_{n}(x) > σ_{A}(x) − \frac{ε}{3n}.

Then

σ_{A} (x + \frac{x_{n}}{n} ) + σ_{A} (x − \frac{x_{n}}{n} ) − 2σ_{A}(x) ≥

≥ f_{n} (x + \frac{x_{n}}{n} ) + h_{n} (x − \frac{x_{n}}{n} ) − (f_{n} + h_{n}) (x) − \frac{2ε}{3n} =

= \frac{1}{n}(f_{n} − h_{n}) (x_{n}) − \frac{2ε}{3n} > \frac{ε}{n} − \frac{2ε}{3n} = \frac{1}{n} · \frac{ε}{3}.

This contradicts the inequality (D). □

4. Namioka-Phelps spaces

Recall [4, 44] that a linear topological space X is said to be Asplund if every continuous convex real function defined on an open convex subset of X is Fréchet differentiable on a dense G_{δ} subset of its domain. If, in the definition, “dense G_{δ} ” is replaced by “dense”, X is called a Fréchet differentiable space.

For Banach spaces, these definitions give the same classes. General Asplund spaces are studied systematically in [44, 14]. For information on Asplund Banach spaces, see for example [34, 45, 6, 38]. Among various characterizations of Asplund Banach spaces, we mention here only two. The first one states that a Banach space X is Asplund iff X^{∗} has the Radon-Nikodym property [45]. In 1975 Namioka and Phelps [34] proved that a Banach space X is Asplund iff every bounded subset of X^{∗} is fragmented. The last criteria justifies the following main definition.

Definition 4.1. We say that an ℓ.t.s. X is Namioka-Phelps (abbr.: (NP)) if every equicontinuous subset of X^{∗} is fragmented.

Recall that if X is a normed space then a subset F of X^{∗} is equicontinuous if and only if F is bounded.

Proposition 4.2. Every Fréchet differentiable (and, hence, every Asplund) space is (NP). Proof. Directly follows from Proposition 3.14. □

Proposition 4.3. Every semireflexive space is (NP). Proof. Directly follows from Proposition 3.6. □

An ℓ.c.s. X is called semi-Montel [37] if every bounded subset of X is relatively compact. We say that an ℓ.c.s. X is quasi-Montel if every bounded subset of X is precompact. By [43, ch. III, §7, Corollary 2], every nuclear space is quasi-Montel.

Proposition 4.4. Let X be a quasi-Montel space, and let Y be a linear topological space. Then for every equicontinuous subset F of L(X, Y), the topology of pointwise convergence and the strong topology coincide on F. Proof. The strong topology is the topology of bounded convergence. By our assumption, every bounded subset of X is precompact. Now our assertion follows from the fact that the topologies of precompact and simple convergence coincide on F (see [43, III, 4.5]). □

Corollary 4.5. Every quasi-Montel space is (NP).

Proposition 4.6. Let X be a linear topological space. If X^{∗} (or, at least, each equicontinuous subset of X^{∗}) is uniformly Lindelöf, then X is (NP).

Proof. By [43, III, 4.3], every equicontinuous subset F of X^{∗} is contained in a weak^{∗} -compact equicontinuous subset cℓ_{w^{∗}}(F), which is fragmented by Corollary 3.11. □

Remark 4.7. (a) The last result generalizes the well-known fact in the theory of Asplund spaces, which states that each Banach space X with the separable dual X^{∗} is Asplund.

(b) Corollary 4.5 can also be derived from Proposition 4.6 because, as follows easily from [43, III, 4.3], each equicontinuous subset of X^{∗} is relatively compact in the strong topology whenever X is quasi-Montel.

The following result is a minor modification of Lemma 2.1 from [33].

Lemma 4.8. Let (X, τ_{X}) and (Y, τ_{Y}) be compact (Hausdorff) spaces, and let μ_{X} and μ_{Y} be uniformities on the sets X and Y respectively. Suppose that there is a continuous surjection f: X → Y which is also uniformly continuous w.r.t. μ_{X} and μ_{Y} . If X is fragmented by the identity map 1_{X}: X → X w.r.t. (τ_{X}, μ_{X}), then Y is fragmented by the identity map 1_{Y} : Y → Y w.r.t. (τ_{Y} , μ_{Y}).

Proof. Let A be a non-empty τ_{Y} -compact subset of Y, and let ε ∈ μ_{Y} . Choose δ ∈ μ_{X} such that (f × f) (δ) ⊆ ε. By Zorn’s Lemma, there exists a minimal τ_{X} - compact subset M of X such that f(M) = A. Since X is fragmented, there exists V ∈ τ_{X} such that V ∩ M ≠ ϕ and V ∩ M is δ -small. Then the set f(V ∩ M) is ε-small. Consider the set W = A \ f(M \ (V ∩ M)). Then

(a) W is ε-small, being a subset of f(V ∩ M);

(b) W is relatively τ_{Y} -open in A;

(c) W is non-empty (otherwise M \ (V ∩ M) is a proper τ_{X} -compact subset of M such that f(M \ (V ∩ M)) = A).

Therefore, by Lemma 3.2 (a), the proof is complete. □

Proposition 4.9. If E is (NP) and M is a linear subspace of E, then M is (NP).

Proof. Let Y be an equicontinuous subset of M^{∗}. By [37, 9.11.4 (a)] there exists an equicontinuous (and, hence, fragmented) subset X of E^{∗} such that for the canonical mapping q^{∗}: E^{∗} → M^{∗} (where q: M ↪ E is the inclusion), holds q^{∗}(X) = Y. By the Alaoglu-Bourbaki theorem and Lemma 3.2 (a), we may assume that X and Y are weak^{∗} -compact. The fragmentability of Y follows from Lemma 4.8, applied to the map q^{∗} |_{X}: X → Y , which is weak^{∗} -weak^{∗} continuous and also uniformly continuous when X and Y carry the uniformities μ_{X} and μ_{Y} inherited from the strong uniformities. □

A linear map q: X → Y is said to be bound covering if for every bounded subset A of Y , there exists a bounded subset B of X such that q(B) = A. A linear open map of a normed space onto a normed space is a bound covering. For more information see [8].

Proposition 4.10. Let q: X → Y be a continuous bound covering linear map. If X is (NP), then Y is (NP). Proof. Let F ⊆ Y^{∗} be equicontinuous. In order to establish the fragmentability of F, fix: a non-empty subset F_{1} of F, a bounded subset A of Y, and a number ε > 0. Since q is bound covering, there exists a bounded subset B of X such that q(B) = A. The set

F_{1}q = {f ∘ q ∈ X^{∗} | f ∈ F_{1}}

is an equicontinuous subset of X^{∗} and, hence, it is fragmented because X is (NP). Therefore, for [B, ε] there exist a finite sequence {x_{1}, x_{2}, …, x_{n}} in X, a number δ > 0 and a functional h_{0} ∈ F_{1}q such that for every h ∈ F_{1}q satisfying the condition

|h_{0}(x_{i}) − h(x_{i})| < δ ∀ i ∈ {1, 2, …, n},

we have

|h_{0}(x) − h(x)| < ε for every x ∈ B.

Clearly, h_{0} = f_{0} ∘ q for a certain f_{0} ∈ F_{1}. Then

|f_{0}(y) − f(y)| < ε for every y ∈ A,

whenever

|f_{0}(y_{i}) − f(y_{i})| < δ ∀ i ∈ {1, 2, …, n},

where f ∈ F_{1} and y_{i} denotes q(x_{i}). This proves our assertion. □

Example 4.11. The class (NP) is not closed under quotients. In particular, Proposition 4.10 may be false if q is not bound covering. Indeed, there is [43, IV, Ex. 20] a Fréchet Montel (and, hence, (NP)) space E in which there exists a closed subspace M such that E/M is isomorphic with the Banach space ℓ^{1} which is not Asplund (or, equivalently, is not (NP)) by [34, Corollary 10]. This example also shows that the class of all ℓ.c. (NP)-spaces is not a variety in the sense of [9].

However, we have:

Proposition 4.12. The class (NP) is closed under products and l.c. direct sums. Proof. Let X = ∏_{i∈I} X_{i} be a topological product of (NP) spaces. Suppose F

is an equicontinuous subset of X^{∗}. Fix: a nonempty subset F_{1} ⊆ F, a bounded subset B of X, and a number ε > 0. Since F_{1} is equicontinuous, the polar

F_{1}^{0} = {x ∈ X | |f(x)| ≤ 1 ∀ f ∈ F_{1}}

is a neighborhood of 0 in X. Therefore, for a certain finite J ⊆ I, the projection pr_{i} (F_{1}^{0}) is X_{i} for i ∉ J. This implies that for every functional f = ∑_{i∈I} f_{i} from F_{1},

the functional f_{i} ∈ X_{i}^{∗} is trivial for each i ∉ J. In fact, f can be represented as

f = ∑_{j∈J} f_{j}.

Every projection pr_{j}(B) is a bounded subset of X_{j}(j ∈ J). Since pr_{j}(F) is an equicontinuous subset of X_{j}^{∗} and X_{j} is (NP), for every j ∈ J there exist:

(a) a functional h^{j} ∈ X_{j}^{∗},

(b) A finite set {x_{1}^{j}, x_{2}^{j}, …, x_{n_{j}}^{j}} ⊆ X_{j},

(c) a number δ_{j} > 0 such that if

|f_{j} (x_{k}^{j}) − h^{j} (x_{k}^{j})| < δ_{j} ∀ k ∈ {1, 2, …, n_{j}},

then

|f_{j}(y) − h^{j}(y)| < \frac{ε}{|J|} ∀ y ∈ pr_{j}(B).

Now, consider:

(1) the finite set {x̃_{k}^{j} | j ∈ J, k ∈ {1, 2, …, n_{j}}} ⊆ X, where x̃_{k}^{j} denotes the

element of ∏_{i∈I} X_{i} having x_{k}^{j} in the j -th coordinate and all other coordinates

are zero;

(2) the number δ := min{δ_{j} | j ∈ J};

(3) the functional h_{0} := ∑_{j∈J} h^{j} ∈ X^{∗}.

We claim that for every f = ∑_{j∈J} f_{j} ∈ F_{1} , the finite system of inequalities:

|f (x̃_{k}^{j}) − h_{0} (x̃_{k}^{j})| ≤ δ ∀ j ∈ J, ∀ k ∈ {1, 2, …, n_{j}}

imply that

|f(y) − h_{0}(y)| ≤ ε ∀ y ∈ B.

Indeed,

|f (x̃_{k}^{j}) − h_{0} (x̃_{k}^{j})| = |f_{j} (x_{k}^{j}) − h^{j} (x_{k}^{j})| ≤ δ ≤ δ_{j}.

Then for every y = (y_{i})_{i∈I} ∈ B, holds:

|f(y) − h_{0}(y)| = |∑_{j∈J} (f_{j}(y_{j}) − h^{j}(y_{j}))| ≤ ∑ |f_{j}(y_{j}) − h^{j}(y_{j})| ≤ |J| · \frac{ε}{|J|} = ε.

This proves our assertion for products.

For direct sums, the proof is quite similar. The following standard fact from [43, II, 6.3] plays the major role in the proof.

Fact. For every bounded subset B of a locally convex direct sum ⊕_{i∈I} X_{i}, there

exists a finite set J ⊆ I such that pr_{i}(B) is zero for every i ∉ J.

□

Proposition 4.13. The class (NP) contains the variety generated by the Asplund Banach spaces. In particular, every quotient of a subspace M of ∏_{i∈I} X_{i},

where each X_{i} is an Asplund Banach space, belongs to (NP).

Proof. An easy consequence of [9, Theorem 1.4] and the results of this section. □ Remark 4.14. (a) By Proposition 4.13, the (NP)-space E from Example 4.11 is not contained in a product of Asplund Banach spaces.

(b) The class (ASP) of Asplund spaces is a proper subclass of (NP). Indeed, (ASP) is not closed under locally convex direct sums [43, Example 6.1], in contrast to (NP).

(c) Recall that if the dual E^{∗} of a Banach space is weakly compactly generated (in the sense of [2]), then E is Asplund (cf. [34]). Is it true that an l.c.s. E is (NP) if E^{∗} is weakly compactly generated (cf. [19]) ?

5. A “transport” argument

Definition 5.1. Let S be a topologized semigroup, and let P, Q be subsets of S. We say that an element t ∈ S is left (right) P -reachable from Q if for every nonempty open subset O ⊆ Q there exists p ∈ P such that pO ∈ N_{t} (resp.: Op ∈ N_{t}). We write s ◁^{ℓ} t or s ◁^{r} t if there exists a non-empty compact subset C_{s,t} of S such that for every nbd P of C_{s,t} there exists an nbd Q of s such that t is left or, respectively, right P -reachable from Q. One gets the weaker relations writing: s ◁_{w}^{ℓ} t or s ◁_{w}^{r} t, whenever there exists Q ∈ N_{s} such that t is left or, respectively, right S -reachable from Q. Denote by S^{◁^{ℓ}} (S^{◁^{r}}, S^{◁_{w}^{ℓ}}, S^{◁_{w}^{r}}) the set of all t ∈ S

such that for a certain s ∈ S, holds s ◁^{ℓ} t (resp.: s ◁^{r} t, s ◁_{w}^{ℓ} t, s ◁_{w}^{r} t).

Lemma 5.2. Let S be a topological subsemigroup of a topological group G. Then int(S) · int(S) ⊆ S^{◁^{ℓ}} ∩ S^{◁^{r}}. More precisely, if t = s_{1}s_{2}, where s_{1}, s_{2} ∈ int(S), then s_{2} ◁^{ℓ} t with C_{s_{2},t} = {s_{1}} and s_{1} ◁^{r} t with C_{s_{1},t} = {s_{2}}. Proof. For every nbd P of s_{1} in S, choose an nbd U of e in G so small that s_{1}U ⊆ P and U^{−1}s_{2} ⊆ int(S). Consider nbd Q := U^{−1}s_{2} of s_{2}. Then t is left P -reachable from Q because for every non-empty open subset O of Q, we can take u ∈ U such that u^{−1}s_{2} ∈ O. Then for p := s_{1}u, holds t = (s_{1}u) (u^{−1}s_{2}) ∈ pO and pO is open in S. This proves s_{2} ◁^{ℓ} t with C_{s_{2},t} = {s_{1}}. Easy modifications prove the second case. □

Corollary 5.3. (a) If G is a topological group, then G^{◁^{ℓ}} = G^{◁^{r}} = G. (b) [0, ∞)^{◁^{ℓ}} = [0, ∞)^{◁^{r}} = (0, ∞). A topologized semigroup S is called left (right) topological if, for the multipli-

cation π: S × S → S, we have Con^{ℓ}(π) = S (resp.: Con^{r}(π) = S).

Lemma 5.4. Let S be a left (right) topological monoid, and let H(e) denote the group of all units in S. If sH(e) (resp.: H(e)s) is dense in S, then s ∈ S^{◁_{w}^{r}} (resp.: s ∈ S^{◁_{w}^{ℓ}}). Proof. It is trivial to show that e ◁_{w}^{r} s (resp.: e ◁_{w}^{ℓ} s) . □

Lemma 5.5. (a) Let π: S × X → X be a semigroup action on a uniform space (X, μ). Assume that x_{0} ∈ X, s, t ∈ S and the following conditions are satisfied:

(1) s ◁^{ℓ} t ;

(2) X is π -uniform at every c ∈ C_{s,t};

(3) S is locally fragmented at s by the orbit map x̃_{0} : S → (X, μ). Then x̃_{0} is continuous at t.

(b) The same is true for a right action π: X ×S → X provided that “s ◁^{r} t” takes the place of (1).

(c) If S̃ is μ-uniformly equicontinuous, then in (a) and (b), condition (2) can be dropped, and (1) can be replaced by the weaker assumptions: s ◁_{w}^{ℓ} t and s ◁_{w}^{r} t respectively. Proof. We prove only (a). Case (b) is similar, and (c) can be obtained by a minor modification of (a).

Let ε ∈ μ. Since C_{s,t} is compact, by elementary compactness arguments, making use (2), we can pick nbd P of C_{s,t} such that: (*) There exists δ ∈ μ such that (px, py) ∈ ε for every (x, y) ∈ δ and p ∈ P.

According to Definition 5.1, choose for P nbd Q of s such that t is P - reachable from Q. By (3) there exists an open non-empty subset O of Q such that the set x̃_{0}(O) = Ox_{0} is δ -small. By our choice, for a certain p ∈ P, holds pO ∈ N_{t}. Then the set x̃_{0}(pO) = pOx_{0} is ε-small by (∗). This proves the continuity of x̃_{0} at t. □

Several transport arguments and their applications can be found in [26, 42].

6. Applications

Lemma 6.1. (i) Let X be an ℓ.t.s., P be a topological space and π: P ×X → X a right linear map satisfying Con^{r}(π) = P. For a normed space Z, consider the induced map π^{L}: L(X, Z) × P → L(X, Z). If (p_{0}, 0) ∈ Con(π), then for every equicontinuous subset F of L(X, Z), there exists U ∈ N_{p_{0}} such that F U is equicontinuous in L(X, Z).

(ii) If we replace the assumption (p_{0}, 0) ∈ Con(π) with the assumption that Q̃ is equicontinuous for a certain Q ∈ N_{p_{0}}, then Z may be an arbitrary ℓ.t.s.

Proof. (i) Since Z is normed, the system \{\frac{1}{n} B | n ∈ ℕ\} , where B is the unit ball in Z, is a local base at 0. As in the case Z := ℝ, it is easy to show that a subset E of L(X, Z) is equicontinuous iff E ⊆ W^{0}, where W is a certain nbd of 0 in X, and W^{0} denotes the “polar” of W, i.e., the set:

W^{0} = {f ∈ L(X, Z) | ∥f(x)∥ ≤ 1 ∀x ∈ W }.

Since F is equicontinuous in L(X, Z), there exists an nbd V of 0 such that F ⊆ V^{0}. The continuity of π at (p_{0}, 0) implies that UW ⊆ V for certain nbd’s U ∈ N_{p_{0}}, W ∈ N_{0}. Then, eventually, F U ⊆ W^{0}.

The proof of (ii) is analogous and even easier and, hence, is omitted. □ Now we are ready to prove the following main lemma.

Lemma 6.2. Let X be (NP), and let π: S × X → X be a continuous linear action. Denote by π^{∗} the dual action X^{∗} × S → X^{∗}. If s ◁^{r} t (s ◁_{w}^{r} t) and π is locally bounded at every q ∈ C_{s,t} ∪ {t} (resp.: S̃ is equicontinuous), then X^{∗} = Con_{t}(π^{∗}). Proof. We prove only the “locally bounded” case. The second case is an easy modification.

Fix f ∈ X^{∗}. In order to establish that π^{∗} is continuous at (f, t), by Lemma 2.7(ii) and the ‘right’ version of Proposition 2.6(ii), it suffices to show that f̃: S → X^{∗}, s ↦ fs is continuous at t. By our hypothesis, condition (1) in Lemma 5.5(b) is satisfied. Let c ∈ C_{s,t}. Since π is locally bounded at c, π^{∗} is continuous at (0, c) (Lemma 2.7(ii)). By the ‘right’ version of Proposition 2.6(i), X^{∗} is π^{∗} -uniform at c. Therefore, condition (2) also holds. In order to check the validity of (3), take, due to Lemma 6.1, nbd U ∈ N_{s} such that fU is an equicontinuous subset of X^{∗}. Since X is (NP), fU is fragmented. On the other hand, the continuity of π guarantees that f̃: S → X^{∗} is weak^{∗} -continuous. Therefore, Lemma 3.2(b) establishes that U is fragmented by f̃: S → (X^{∗}, μ^{∗}). Hence, S is locally fragmented at s by f̃. Now we can use Lemma 5.5(b) which yields that f̃ is continuous at t. □

The following applications are divided into several subsections.

A. Locally compact (semi) group actions.

Theorem 6.3. Let X be (NP), let S be a topological subsemigroup of a locally compact topological group G, and let π: S × X → X be a continuous linear action. Then π^{∗}: X^{∗} × S → X^{∗} is continuous at every (f, t), where f ∈ X^{∗} and t ∈ int(S) · int(S). In particular, if S = G, then π^{∗} is continuous. Proof. Let f ∈ X^{∗} and t = s_{1}s_{2}, where s_{1}, s_{2} ∈ int(S). By Lemma 5.2, s_{1} ◁^{r} t with C_{s_{1},t} = {s_{2}}. The points s_{2} and t, being in int(S), have locally compact nbd’s in S. Therefore, by Lemma 2.7, π is locally equicontinuous (and, hence, locally bounded) at each q ∈ {s_{2}, t}. Now, we can apply Lemma 6.2 . □

Corollary 6.4. (Moore [30, Ch. 5, Theorem 5], Bourbaki [5, Ch. 8, §2, Ex. 3(c)]) For every locally compact topological group G and every continuous linear action on a semireflexive space X, the dual action is continuous. Proof. By Proposition 4.3, X is (NP). □

Corollary 6.5. (Generalized Arendt Theorem.) Let X be (NP). For every continuous linear one-parameter semigroup action π: [0, ∞) × X → X, the dual action π^{∗}: X^{∗} × [0, ∞) → X^{∗} is continuous at every (f, t), where f ∈ X^{∗} and t > 0.

Remark 6.6.

(a) Helmer has shown [18, Corollary 4.3], that if S is a locally compact topological monoid, then for every continuous linear action of S on a semireflexive space X, the dual action is continuous at every (f, t), where f ∈ X^{∗} and t is a unit.

(b) Arendt [3] proved Corollary 6.5 for Banach spaces X whose duals have the Radon-Nikodym Property. By [45], such X are Asplund. Therefore, Corollary 6.5 is stronger. A “non-adjoint” generalization of Arendt’s result was obtained by van Neerven [35, Lemma 3.1].

(c) It is actually well known that some restrictions on points t ∈ S are really needed. For example, if S = [0, ∞) and X is a (non-reflexive) Asplund Banach space, then “t > 0” is essential even for X = c_{0}, or for X = Y^{∗} where Y is the (quasi-reflexive) James space (cf. [36, Examples 2.3.5, 1.5.3]).

(d) Generally, the local boundedness in Lemma 6.2, as well as the local compactness in Theorem 6.3 and Corollary 6.4, cannot be dropped even for a Baire topological group G and a reflexive ℓ.c.s. X. Indeed, if G = ℝ_{+}^{ℵ_{0}} is the ℵ_{0} -power of the multiplicative group of all positive real numbers, X = ℝ^{ℵ_{0}} and π: G × X → X is defined coordinatewise, then it is easy to see that π^{∗} is not continuous at any (0, g), where g ∈ G.

Theorem 6.7. Let X be (NP) and let S be a compact left topological monoid such that the group of all units H(e) is dense in S. For every linear continuous action π: S × X → X, the dual action π^{∗} is continuous at every (f, t), where f ∈ X^{∗} and t ∈ H(e). Proof. Apply Lemma 6.2, taking into account Lemmas 2.7(iii) and 5.4. □

B. Actions on normed spaces.

Theorem 6.8. Let X be a normed space whose dual X^{∗} has the Radon-Nikodym property, and let S be an arbitrary topologized semigroup. For every continuous linear action π: S × X → X, the dual action π^{∗} is continuous at every (f, t), where f ∈ X^{∗} and t ∈ S^{◁^{r}} (or even t ∈ S^{◁_{w}^{r}}, if the action is contracting). Proof. By Proposition 2.6(iv), we may assume that X is Banach. Abovementioned characterization from [45] implies that X is Asplund. By Proposition 4.2, X is (NP). By Proposition 2.8, π is locally equicontinuous and, hence, locally bounded by Lemma 2.7(i). Finally, use Lemma 6.2. □

Corollary 6.9. For an arbitrary topological group G and any continuous linear action of G on an Asplund Banach space, the dual action is also continuous.

Remark 6.10. (a) For G = ℝ, see van Neerven [36, Corollary 6.2.6].

(b) Christensen and Kenderov proved in [7] that every weak^{∗} -continuous mapping F from a Baire space S into X^{∗}, where X is Banach and X^{∗} has the Radon-Nikodym property, is norm-continuous at any point of a certain dense G_{δ} subset of S. This result, together with Propositions 2.8, 2.6, Lemma 2.7 and easy transport arguments, provide an alternative proof of Corollary 6.9 in the case of a Baire topological group G.

Let Is(X) be the group of all linear isometries of a normed space X. Denote by p (and p^{∗}) the group topology on Is(X) generated by the system of all orbit mappings

{x̃: Is(X) → X, x̃(g) = gx | x ∈ X}
(respectively: {f̃: Is(X) → X^{∗}, f̃(g) = fg | f ∈ X^{∗}}) .

Theorem 6.8 implies that if X^{∗} has the Radon-Nikodym property, then the dual action of (Is(X), p) on X^{∗} is continuous. Therefore, p^{∗} ⊆ p. This yields:

Corollary 6.11. If X is a reflexive Banach space, then the topologies p and p^{∗} on Is(X) coincide.

C. Construction of minimal topological groups.

Recall that a topological group G is said to be minimal [10] if it does not admit a strictly coarser Hausdorff group topology. If X × G is minimal for every minimal group X, then G is called perfectly minimal [10]. The following theorem provides additional information to the results of [29].

Theorem 6.12. Let X be an Asplund Banach space. Then every topological subgroup of (Is(X), p) is a topological group retract of a perfectly minimal topological group. Proof. By Corollary 6.9, the dual action of (Is(X), p) on X^{∗} is continuous. Therefore, G is an HBR-group in the sense of [29, Definitions 4.2, 4.7]. Now our assertion follows directly from Theorem 4.8 of [29]. □

D. Actions on compact operators.

Let X and Y be Banach spaces. If X^{∗} or Y has the approximation property, then the injective tensor product X^{∗} ˆ⊗_{i}Y is naturally isomorphic to K(X, Y) [24, p. 268].

Theorem 6.13. Let X and Y be Banach spaces. Suppose that X is Asplund and that either X^{∗} or Y has the approximation property. Then for every continuous linear action π: S × X → X, the induced action π^{L}: L(X, Y) × S → L(X, Y) is continuous at every (f, t), where f is a compact operator and t ∈ S^{◁^{r}}. In particular, the induced action π^{K}: K(X, Y) × G → K(X, Y) is jointly continuous for arbitrary topological group S = G. Proof. By Proposition 2.8, π is locally equicontinuous. Then the same is true for the right action π^{L}. Since “equicontinuous” implies “uniformly equicontinuous” in linear spaces, Example 2.3(a) shows that L(X, Y) is π^{L} -uniform. Fix (f, t) ∈ K(X, Y) × S^{◁^{r}}. By Lemma 2.4(iii), it suffices to show that f̃: S → L(X, Y) is continuous at t. Since f is compact, f̃(S) ⊆ K(X, Y). Therefore, we may assume that f̃ has the form S → K(X, Y). By the above-mentioned fact, we have K(X, Y) = X^{∗} ˆ⊗_{i}Y. Moreover, the action of S on K(X, Y) naturally coincides with the action π^{∗} ˆ⊗_{i}π_{0}, where π_{0} is the trivial action of the one-point monoid. By Proposition 4.2 and Lemma 6.2, π^{∗} is continuous at (f, t). Now, Proposition 2.9 completes the proof. □

The compactness of f_{0} is essential. For an appropriate counterexample, take f_{0} = 1_{X} and a non-uniform continuous semigroup representation.

E. “Small” orbits.

Theorem 6.14. Let π: S × X → X be a continuous linear action of a Baire topologized semigroup on a normed space X, let Y be an ℓ.t.s., and let f_{0} : X → Y be a continuous linear operator. Suppose that for every s ∈ S there exists U ∈ N_{s} such that f_{0}U is uniformly Lindelöf (e.g., precompact, separable, Lindelöf, ccc). Then the action π^{L}: L(X, Y) × S → L(X, Y) is continuous at (f_{0}, t) for every t ∈ S^{◁^{r}}. Proof. As in the proof of Theorem 6.13, it suffices to show that f̃_{0} : S → L(X, Y) is continuous at every t ∈ S^{◁^{r}}. This can be done by combining Proposition 3.10(i) and Lemma 5.5(b). □

We list here some known results which can easily be obtained by adapting 6.14.

(a) [35, Theorem 3.4] Let π: [0, ∞) × X → X be a continuous linear action of [0, ∞) on a Banach space X, and let f_{0} ∈ X^{∗}. Then f̃_{0} : [0, ∞) → X^{∗} is continuous at every t > 0 iff the orbit of f_{0} is (weakly) separable.

(b) (See [46, 11], [25] (for second countable G)) Let G be a locally compact group. If the G-orbit of a measure m ∈ M(G) = (C_{0}(G))^{∗} is separable, then the orbit map m̃ is continuous (and, hence, m is absolutely continuous).

(c) [39] If G is a locally compact group and f_{0} ∈ L^{∞}(G) = (L^{1}(G))^{∗} has a separable orbit, then the orbit map f̃_{0} is continuous.

Recall that a Banach space X is said to have the point of continuity property (abbr.: (PC)) if each bounded closed set C admits a point of continuity of the identity map (C, weak) → (C, norm). This property can be characterized in terms of fragmentability. By [13, 3.13] X has the property (PC) iff each weakly closed bounded subset of X is huskable (in our terminology: locally fragmented). Another way to say this is: every bounded subset of X is fragmented [21, p. 665]. Banach spaces with the Radon-Nikodym property or with (weakly) Čech-complete ball have the property (PC) [13, 3.14].

Sometimes the weak continuity of an orbit mapping implies its continuity. Several results of this kind can be found in [12, 18, 26, 30].

Theorem 6.15. Let π: S ×X → X be a linear action on an ℓ.c.s. (X, τ) such that π is continuous at (p, 0) for every p ∈ S. Suppose that s ◁^{ℓ} t, x_{0} ∈ X and x̃_{0} : S → X is weakly continuous. Then x̃_{0} is τ -continuous at t under each of the following conditions:

(i) X is a Banach space with the property (PC); (ii) There exists a separable Baire nbd U of s. If the action is equicontinuous, then in both cases we may assume that s ◁_{w}^{ℓ} t.

Proof. Proposition 2.6(i) shows that X is π -uniform.

(i) By proposition 2.8, Ux_{0} is norm-bounded for a certain nbd U of s. Abovementioned characterization of (PC) implies that Ux_{0} is fragmented. Then by Lemma 3.2(b) the map x̃_{0} is locally fragmented at s. For the rest use Lemma 5.5(a).

(ii) Since x̃_{0} is weakly continuous, Ux_{0} is weakly separable, and, hence, even τ -subseparable (a subset of a separable set) because τ is locally convex. Therefore, Ux_{0} is uniformly Lindelöf. By Lemma 3.4(b) and Proposition 3.10(i), S is locally fragmented by x̃_{0} at s. Now Lemma 5.5(a) completes the proof.

If the action is equicontinuous then use Lemma 5.5(c). □

Corollary 6.16. Let S be a Baire locally separable semitopological group. Then RUC(S) = W RUC(S) (for definitions see, for example, [12]).

F. Actions on quasi-Montel spaces.

Theorem 6.17. Let X be a quasi-Montel space, P a topological space and π: P × X → X a continuous right linear function.

(i) For every normed space Y , the induced map π^{L}: L(X, Y) × P → L(X, Y) is separately continuous.

(ii) If P is locally compact, then π^{L} is jointly continuous for every ℓ.t.s. Y.

Proof. (i) For each f_{0} ∈ L(X, Y), use Lemma 6.1(i) (with F = {f_{0}}) and Proposition 4.4. (ii) Instead of Lemma 6.1(i), use Lemmas 6.1(ii), 2.7 and 2.4(iii). □

G. Sublinear functionals defined over orbits.

Theorem 6.18. Let π: S×X → X be a continuous linear action of a topologized semigroup S on a normed space X. Suppose that f_{0} ∈ X^{∗}, s_{0} ∈ S, U ∈ N_{s_{0}}, and for every non-empty relatively w^{∗} -closed subset A of f_{0}U, the sublinear functional σ_{A}, defined by the rule σ_{A}(x) = sup {f_{0}(sx) | f_{0}s ∈ A}, has a point of Fréchet differentiability. Then π^{∗} is continuous at (f_{0}, t) for every t satisfying s_{0} ◁^{r} t. Proof. Let s_{0} ◁^{r} t. Our aim is to show that all conditions of Lemma 5.5(b) are satisfied. First, choose, by Lemma 6.1, nbd V of s_{0} such that V ⊆ U and f_{0}V is equicontinuous (equivalently, bounded). Proposition 3.14 states that f_{0}V is fragmented. By Lemma 3.2(b) and the weak^{∗} -continuity of f̃_{0}, we obtain that V is fragmented by f̃_{0}. In particular, S is locally fragmented at s_{0} by f̃_{0}. By Proposition 2.8 and Lemma 2.7(ii), X^{∗} is π^{∗} -uniform. Therefore, we can use Lemma 5.5(b), which completes the proof. □

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