Spectrum of A C*-algebra - Mackey Borel Structure

Mackey Borel Structure

 is a topological space and thus can also be regarded as a Borel space. A famous conjecture of G. Mackey proposed that a separable locally compact group is of type I if and only if the Borel space is standard, i.e. is isomorphic (in the category of Borel spaces) to the underlying Borel space of a complete separable metric space. Mackey called Borel spaces with this property smooth. This conjecture was proved by James Glimm for separable C*-algebras in the 1961 paper listed in the references below.

Definition. A non-degenerate *-representation π of a separable C*-algebra A is a factor representation if and only if the center of the von Neumann algebra generated by π(A) is one-dimensional. A C*-algebra A is of type I if and only if any separable factor representation of A is a finite or countable multiple of an irreducible one.

Examples of separable locally compact groups G such that C*(G) is of type I are connected (real) nilpotent Lie groups and connected real semi-simple Lie groups. Thus the Heisenberg groups are all of type I. Compact and abelian groups are also of type I.

Theorem. If A is separable, Â is smooth if and only if A is of type I.

The result implies a far-reaching generalization of the structure of representations of separable type I C*-algebras and correspondingly of separable locally compact groups of type I.