Book Review: Fundamentals in the theory of operator algebras}, III--IV. {\it An exercise approach
Gert Kjaergaard Pedersen · Bulletin of the American Mathematical Society · 1994
When Johann von Neumann in 1929 proved the double commutant theorem (the weak closure of a unital ^-invariant algebra of operators on a Hubert space is precisely the set of operators that commute with every operator commuting with the algebra), he set a train of thoughts in motion, whose destination even he could not have foreseen.The ensuing theory of operator algebras was originally motivated by the desire to create an axiomatic foundation for the emerging quantum theory, but during the long years of close collaboration between Johnny Neumann and Frank Murray things "got out of hand".The result was the monolith in twentieth century mathematics known as Rings of operators, I-IV.Respected by all, it was also daunting in its extreme technicality, and only a few dedicated mathematicians dared approach its summits.Gelfand's work, which reached the West after World War II, resulted in the theory of abstract C* -algebras and gave operator algebras a new direction, aimed at unitary representations of topological groups, mainly Lie groups.Although this multiplied the number of first-rank mathematicians working with operator algebras, the 1945-70 era may be labeled "the silent growth".Largely unnoticed by most of the mathematicial community and with only marginal interaction with other branches of mathematics, the theory gathered a number of powerful techniques and strong results.The integration in mainstream mathematics and the recognition came in the seventies and culminated in the Fields medals awarded to Alain Connes (1982) and Vaughan Jones (1990).Now we see operator algebras used in geometry (foliations), PDE (index theorems), algebraic topology (AT-theory and cyclic cohomology), and as the link between knot theory and models in statistical quantum mechanics.The theory has certainly come of age and has become fashionable.At the same time, the demand for concise textbooks that cover the subject has become noticeable, and this brings us to the books under review.Fundamentals in the theory of operator algebras, Volumes I & II were pub-