By A.M. Fink
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Extra resources for Almost Periodic Differential Equations
The ampliﬁcation of ϕ is, by deﬁnition, the operator 1F ⊗ ϕ : FE → FF ; thus it is well deﬁned by ax → aϕ(x); x ∈ E, a ∈ F. We denote it, for brevity, by ϕ∞ . In the particular case of a functional, say f : E → C, its ampliﬁcation f∞ takes values in F = FC, and it is well deﬁned by sending ax to f (x)a. 1) (ϕψ)∞ = ϕ∞ ψ∞ , which holds whenever the composition ϕψ makes sense. The following algebraic observation will be used frequently. 1. For linear spaces E and F , the map Φ : FE → FF is an ampliﬁcation of some operator between E and F ⇐⇒ it is a morphism of bimodules.
Take an arbitrary u and then P such that u ∈ FP A. The restriction of · to FP A is a C ∗ -norm on the latter algebra, and therefore it must coincide with · P . Thus u = u . Return, for a moment, to the C ∗ -algebra B(K) ⊗ A, where A is a C ∗ -algebra and K is a ﬁnite-dimensional Hilbert space. It is well known that The C ∗ -norm in B(K) ⊗ A is a cross-norm; that is, for every a ∈ B(K) and x ∈ A we have a ⊗ x = a x . 3 This norm, of course, is not complete. 28 1. 5. The norm in the ampliﬁcation of a C ∗ -algebra is also a cross-norm.
A vector functional on an operator space B(H, K) is, by deﬁnition, the one acting as a → aξ, η for some ﬁxed ξ ∈ H, η ∈ K. We shall always use the symbol ⊗ . for the Hilbert tensor product of Hilbert spaces as well as for the Hilbert tensor product of operators acting in these spaces; 12 1. , [93, Ch. 6] or [83, Ch. 2, §8]. ) Recall that, for a Hilbert space H, we cc cc have H cc ⊗ . H = (H ⊗ . H) up to the isometric isomorphism, leaving elementary tensors unmoved. The symbol ⊕ denotes the Hilbert sum of Hilbert spaces.