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Major additions:

  • Added subsection 3.4.4 discussing hereditarily atomic von Neumann algebras, and a subsection 4.3.4 arguing they form a model of the quantum lambda calculus.

  • Changed the formulation of Tomiyama's theorem (128II), added a remark about this (128IIa), and added a proof (changing 128III).

    As a result, 128VI and 128VII had to be changed slightly.

  • Added a proof of Russo--Dye's theorem (34a), and adapted 128IV,V,VII accordingly.

  • Added an example in 84a showing that not all equalisers in CStarPU exist, and changed 10IX accordingly.

  • Changed point 119V (showing that several categories of von Neumann algebras are symmetric monoidal endowed with the tensor product) from an exercise to a theorem. To this end I changed 119IV accordingly, and added 119IVa-c.

Major corrections:

  • Corrected an error in 119IV(1) and 119V: when S and T are ultraweakly dense in A and B, resp., then S(x)T need not be ultraweakly dense in A(x)B. The conclusion should have been the linear span S(x)T is ultraweakly dense in A(x)B.

  • Removed the superfluous condition that t>=1/2||a|| from the definition of a positive element of a Cstar-algebra (9IV), changed 9II and 9VIII accordingly, and added a remark 9Va on the matter.

  • Replaced the dubious exercise 20V by a lemma with proof.

  • Since more information on the positive elements of a C-star algebra is needed to solve them then is available there, moved the exercises on the categorical products and equalisers in CStarMIU and CStarPU from 10VII, 10VIII, and 18I to 20a.

  • Corrected the definition of the ultrastrong topology. (It's not the least topology that makes all seminorms induced by np-functionals continuous.)

  • Corrected the definition of completely positive bilinear map in 112(4)d.

Small changes:

  • Replaced "c.q.".

  • Added the exception (121II) to the claim (in 1III) that all proofs are given.

  • Added definitions for := and \equiv in 1Va.

  • Simplifies 5X by assuming that f(x')=1 instead of just f(x')=0.

  • Added a definition of [a,b] for C-star algebras in 8IVa.

  • Added "normal" to the list of properties of linear maps in 10II.

  • Expands 11VIII a bit to prevent the 2^n from looking like a typo.

  • Added a line at the end of 11III explaining that the series converges absolutely.

  • Added an introduction to 19, moving 19I to 19Ia.

  • Added a remark in 10VIa on the boundedness of p-maps.

  • Added a remark, 22IIa, on other interpretations of the word "order ideal".

  • Added explicit explanation of "n" as abbreviation of "normal" in 38Ia, and added a reference back to it from 42IIa.

  • Explicitly attributed Definition 42I to Kadison.

  • Added a description of ultraweak and ultrastong convergence of nets in 42III, and a description of the ultraweakly and ultrastrongly open subsets.

  • Added a definition of W*(S) in 42V4a.

  • Added an exercise 43Ia on filtered infima in a von Neumann algebra.

  • Added a reminder of the meaning of finite (measure space) in 43V(7).

  • Added a parenthetical remark in 44XV.

  • Added the first line of 45III.

  • Added some words on the relation between meagreness and negligibility in 52IIIa and 52IIIb.

  • Improved formulation of 56II,VII.

  • Added a line explaining the "Nevertheless" in 65III(4).

  • Improved formulation of 69IV.

  • Added a 'factorisation theorem' in 69IVa for nmiu-maps (into a nmiu-surjection followed by a nmiu-injection.)

  • Added the observation in 69IVb that the range of an nmiu-map is a von Neumann subalgebra of its codomain.

  • Added a hint to 70II.

  • Replaced the formula in 73III(3) by a drawing.

  • Added a remark, 96Ia, on the choice of 'filter'.

  • Added a remark, 101Ia, on the use of the symbol \diamond.

  • Added a warning on 'normal bilinear maps' in 102IIa.

  • Added more examples of central similarity in 104III2a.

  • Added an introductory remark, 116IIIa, to 116IV.

  • Added the interpretation of lambda terms in 120.

  • Added a short remark on recursion in the quantum lambda calculus in 120II.

  • Attributed Lemma 125IV to Kornell.

  • Added an exercise on preimages and tensors in 125VIIa,b.

  • Added some explanation after the definition of S in 125IX.

  • Added a reference to 128X.

  • Added a remark on not yet using the normality of duplications in 128Xa.

  • Fixed various typos and other small errors.