Let E be a Banach space. The mapping t → T (t) of ℝ (real numbers) into Lb(E), the Banach algebra of all bounded linear operators on E, is called a strongly continuous group or a C₀-group, if G = {T(t) : t ∈ ℝ} defines a group representation of (ℝ, +) into the multiplicative group of Lb(E), and if ∀f ∈ E,
[equation; see abstract in scanned thesis for details].
For example, if E = C₀(ℝ), the function space which consists of all continuous, complex functions that vanish at infinity, then (∀t ∈ ℝ) (∀f ∈ C₀(ℝ)), the function T(t)f(x) = f(x + t), x ∈ ℝ, defines a strongly continuous group, since each f ∈ E is uniformly continuous; this group is called the translation group. If we now consider E = B(ℝ), the space of bounded, continuous complex functions on ℝ, then although the translation group on E is not strongly continuous, it is strongly continuous on the subspace BUC(ℝ) of E, which consists of bounded, uniformly continuous functions. BUC(ℝ) is the largest subspace of E on which the translation group is strongly continuous.
The adjoint family of a C₀-group defined on a Banach space E, need not be strongly continuous on the Banach dual E* of E. Let E⊙ (pronounced E-sun) be the largest linear subspace of E* relative to which the adjoint family is a C₀-group:
[equation; see abstract in scanned thesis for details].
E⊙ is called the sun-dual or sun-space of E. If E = C₀(ℝ), then it follows from a well-known result of A. Plessner that E⊙ = L¹(ℝ) ([Ple]). This research paper contains a characterization of the sun-dual of BUC(ℝ) and of the subspace W AP(ℝ) of BUC(ℝ), which consists of weakly almost periodic functions on ℝ.
