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By Buskes G. (Ed)

This publication comprises 9 well-organized survey articles by means of top researchers in positivity, with a powerful emphasis on sensible research. It offers perception into the constitution of classical areas of continuing features, f-algebras, and necessary operators, but additionally includes contributions to fashionable issues like vector measures, operator areas, ordered tensor items, non-commutative Banach functionality areas, and frames.

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It follows that for x, y ∈ A we have Φ(xy) equals Φ((x − Φ(x))(y − Φ(y)) + Φ(Φ(x)(y − Φ(y)) + Φ(xΦ(y)) = Φ(xΦ(y)) = Φ(x)Φ(y). Hence Φ is a weak* continuous ∗-homomorphism. Thus Ker(Φ) is a weak* closed two-sided ideal in A∗∗ . 4, there exists such a projection p ∈ A∗∗ , with Ker(Φ) = pA∗∗ . If a ∈ A, then Φ((1−p)a) = Φ(a) − Φ(p)Φ(a) = a. Thus Φ restricts to a surjective weak* continuous faithful ∗-homomorphism from the von Neumann algebra (1 − p)A∗∗ onto A. Combining the last two results, it follows in a couple of easy lines that every von Neumann algebra has a unique Banach space predual.

Defines a cone A+ ⊂ A of positive elements which possess the expected properties. ” That is, first, the key point is positivity. If you like, C ∗ -algebras is a ‘positivity theory’; there is a sense in which, explicitly or implicitly, positivity underlies almost every proof in C ∗ -algebra theory (see for example the texts [18, 45, 74, 97, 114]). Note too that Connes is highlighting the miraculous relationship between the positivity and the algebra, and this miracle only gets more remarkable as one gets deeper into the theory, indeed it is a characteristic feature of the subject of operator algebras.

We remark that Sherman recently showed that there exists an equivalence relation (originally studied by Kadison and Pedersen) on B(H)+ such that the equivalence classes constitute a complete lattice. Indeed, a more general version of this holds essentially with B(H) replaced by any von Neumann algebra [105]. From time to time physicists find other ‘fixes’ to the ‘lattice failure’ problem, and some of these are quite interesting for some purposes. But essentially this ‘problem’ is just a fact of life, like the sad facts in the next paragraph.

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