Mathematics: Theory & Applications. Boston, MA: Birkhäuser. xii, 350 p. DM 170.00; öS 1241.00; sFr. 128.00 (2001).
In the late sixties and early seventies, the study of the axiomatic approach to the foundation of quantum mechanics was rife. The conventional wisdom was that the correct model for the bounded observables and states of a quantum-mechanical system consisted of self-adjoint elements of a $C^*$-algebra $A$ and normalized positive linear functionals on $A$. However, one of the axiomatical approaches held that all that could be said about the state space of the system was that it formed a compact, or possibly, linearly compact, convex subset of a locally convex Hausdorff topological vector space endowed with some properties of physical significance. This led to the question `When is a compact convex set the state space of a $C^*$-algebra?’ or, analogously `When is a linearly compact convex set the normal state space of a $W^*$-algebra?’ This excellent book was born out of the successful attempt by the authors to answer these questions. Unfortunately, readers have to wait for the publication of a second volume before the complete answers are revealed.
The present book sets the scene by giving an account of the basic theory of $C^*$-algebras and $W^*$-algebras from a point of view rather different from that found in conventional texts. At every available opportunity the authors turn to the facial structure of the state space to describe algebraic properties. The book not only gives the basic theory from this point of view but also introduces fascinating new geometric concepts in the state space, and describes their relation to the structure of the algebras. In particular, the authors recent important results on the connections between orientations of the state space of a $C^*$-algebra, and of the normal state space of a $W^*$-algebra, and the Lie algebraic structure of the $C^*$-algebra or $W^*$-algebra have not been published in a text before. These describe the connection between the authors’ notion of orientation and that of Connes, who attacked the characterization problem from a completely different point of view.
I would regard the book as essential reading for any graduate student working in $C^*$-algebras and related areas, particularly those with an interest in geometry. All those who read the book will, I am sure, wait impatiently for the second volume.