From: baez@math.ucr.edu (John Baez) Subject: Re: W*-algebras (aka von Neumann algebras) Date: Thu, 9 Mar 2000 04:21:26 GMT Newsgroups: sci.physics.research Summary: [missing] In article <8a0ul9$4q6$1@nnrp1.deja.com>, wrote: >[...] what are W*-algebras >(and how are they used in physics)? I know what C*-algebras are, but >not W*-algebras. If you consider this "a long story", at least provide >me with references. Various equivalent definitions of W*-algebra were discussed earlier in this thread so I won't repeat all that stuff. The basic idea is that they are C*-algebras that have "enough projections", allowing you to do quantum mechanics to your heart's content. In quantum mechanics, projections correspond to yes-or-no questions. A typical commutative C*-algebra is the algebra of all bounded continuous functions on some topological space; a typical commutative W*-algebra is the algebra of bounded measurable functions on some measure space. The latter sort of algebra has "enough projections", namely the characteristic functions of measurable sets. From this you can begin to see another important point: to combine the math of probability theory and quantum mechanics in truly royal style, you need W*-algebras. They let you do "noncommutative integration theory". A lot of the discussion on this thread has concerned the trick for turning any old C*-algebra into a W*-algebra. Thanks to this trick, the theory of C*-algebras and W*-algebras get so intimately intertwined that it becomes unthinkable to study one without studying the other. Alain Connes and Vaughan Jones recently won Fields medals for their work on W*-algebras - work deeply inspired by physics. Clearly anyone who wants to win a Fields medal should learn this stuff, preferably before the age of 40. Here are my favorite books that talk about the physical applications of C*-algebras and W*-algebras. I list them in rough order of increasing difficulty: Gerard G. Emch, Algebraic methods in statistical mechanics and quantum field theory, Wiley-Interscience, New York, 1972. Rudolf Haag, Local quantum physics: fields, particles, algebras, Springer-Verlag, Berlin, 1992. Ola Bratteli and Derek W. Robinson, Operator algebras and quantum statistical mechanics, 2 volumes, Springer-Verlag, Berlin, 1987-1997. For books that talk about the math but not the physics, try these (again in rough order of increasing difficulty): William Arveson, An invitation to C*-algebras, Springer-Verlag, New York, 1976. Masamichi Takesaki, Theory of operator algebras I, Springer-Verlag, Berlin, 1979. Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras, 4 volumes, Academic Press, New York, 1983-1992. Shoichiro Sakai, C*-algebras and W*-algebras, Springer-Verlag, Berlin, 1971.