State spaces of operator algebras : basic theory, orientations, and C*-products
Erik M. Alfsen, Frederic W. Shultz · 2001
One of the most important auxiliary objects associated with an operator algebra is its state space. The two books under review describe the authors ’ solutions, obtained together with H. Hanche-Olsen and B. Iochum [1], [2], [9], to the problems: What data must be added to a state space so that the operator algebra can be recovered? and Which convex sets can arise as state spaces? As that work is now around twenty years old, they are able to present it here in a very finished form. Operator algebras come in two varieties, C*-algebras and von Neumann algebras. (My friends who work in the non self-adjoint theory will forgive me for using the term in this way for the purposes of this review.) Concretely, a C*-algebra is a linear subspace of B(H) (the space of bounded operators on a complex Hilbert space H) which is algebraically closed under operator products and adjoints and is topologically closed in norm. Concrete von Neumann algebras are defined similarly, now requiring closure in the weak * topology. There are abstract characterizations as well: C*-algebras are complex Banach algebras equipped with an involution satisfying ‖x ∗ x ‖ = ‖x ‖ 2, and von Neumann algebras are C*-algebras that have a