By A. Y. Helemskii

This ebook encompasses a systematic presentation of quantum useful research, a mathematical topic sometimes called operator area concept. Created within the Eighties, it these days is likely one of the so much trendy components of useful research, either as a box of energetic examine and as a resource of various vital functions. The procedure taken during this ebook differs considerably from the normal method utilized in learning operator area conception. rather than viewing "quantized coefficients" as matrices in a hard and fast foundation, during this publication they're interpreted as finite rank operators in a set Hilbert house. this enables the writer to exchange matrix computations with algebraic ideas of module idea and tensor items, hence attaining a extra invariant method of the topic. The ebook can be utilized by means of graduate scholars and examine mathematicians drawn to practical research and comparable parts of arithmetic and mathematical physics. necessities contain regular classes in summary algebra and useful research

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**Example text**

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.