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It is shown that the Mackey Borel structures on the spectra of any two approximately finite-dimensional separable C*-algebras not of type I are isomorphic. 1. Main result. In a lecture in Bâton Rouge in 1967, J. Dixmier raised the question of comparing the Mackey Borel structures on the spectra of different separable C*-algebras. Specifically, he asked whether for a simple antiliminary separable C*-algebra the Mackey Borel structure is always the same. He suggested the study of approximately finite-dimensional C*-algebras; we have studied this case, and can report that the Mackey Borel structure on the spectrum is the same for any approximately finite-dimensional separable C*-algebra which is not postliminary. In other words, there is only one nonstandard Borel space which can arise as the spectrum of an approximately finite-dimensional separable C*-algebra. We recall that a C*-algebra is said to be approximately finite-dimensional if any finite number of elements can be approximated arbitrarily closely in norm by elements of a finite-dimensional sub-C*-algebra. 2. Theorem (cf. [6]). Let A be an approximately finite-dimensional separable C*-algebra, and let B be a separable C*-algebra which is not postliminary. Then there exist a sub-C*-algebra C of B and a surjective morphism rp: C -* A with the following properties. (i) For every state f of A, the state f ° <p ofC has a unique extension to a state ofB,sayf~. (ii) The map f \-*f~ is an affine homeomorphism from the state space of A (with the weak dual space topology) onto a locally compact face of the state space of B. (So fis pure if and only iff is.) Received by the editors August 20, 1975. AMS (MOS) subject classifications (1970). Primary 46L05, 54H05; Secondary 02K30, 04A15, 28A05.

The Mackey Borel structure on the spectrum of an approximately finite-dimensional separable 𝐶*-algebra