State spaces of operator algebras: Basic theory, orientations, and C*-products, by

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‖, and von Neumann algebras are C*-algebras that have a Banach space predual. We write A,B, . . . for elements of concrete operator algebras and x, y, . . . for elements of abstract operator algebras. What are they good for? The central motivation in the subject has always been physics — more about this later — but in recent decades attention has been shifting toward connections with other areas of mathematics. Indeed, operator algebras arise naturally in a wide range of settings. If Ω is a compact Hausdorff topological space, then C(Ω), the set of continuous functions from Ω into C, is a C*-algebra. If X is a σ-finite measure space, then L∞(X) is a von Neumann algebra. If G is a locally compact group, then its left representation on the Hilbert space L(G) generates both a C*-algebra C∗(G) and a von Neumann algebra W ∗(G). There are operator algebras naturally associated to foliated manifolds [5], directed graphs [6], Euclidean Bruhat-Tits buildings [13], and Poisson manifolds [15]. It sometimes seems that almost every mathematical object has a naturally associated operator algebra! Moreover, operator algebra techniques have paid off handsomely with, for example, major applications to group representations [7], the Novikov conjecture [12], Connes’ index theorem for foliations [5], and Jones’ work in knot theory [10]. In order to appreciate Alfsen and Shultz’s contribution, one needs to know a little about states and order. Every concrete operator algebra admits a partial order defined by setting A ≤ B if B − A is positive semidefinite, i.e., 〈(B − A)v, v〉 ≥ 0 for all v ∈ H . This is equivalent to the abstract definition x ≤ y if y − x = z∗z for some z, so it does not depend on the representation.