By H. Araki, V. Bach, J. Yngvason (Editors)

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Axioms 2 and 4 are direct analogues of Axioms AHS2 and AHS4, respectively (also cf. the end of Sec. 2). The ‘bootstrap’ Axiom 5 restricts the possible uniformities on P; it is somewhat analogous to Axiom AHS3. In the previous section we have seen that the pure state space of a unital ∗ C -algebra satisfies Axioms 1–5. The remainder of this paper is devoted to the proof of the following Theorem. If a set P satisfies Axioms 1–5 (with P as a transition probability space containing no sector of dimension 3), then there exists a unital C ∗ -algebra AC , whose self-adjoint part is A (defined through Axiom 1).

23, 24] in the present context), or abandon the use of Hilbert lattices and develop a spectral theory of well-behaved transition probability spaces, analogous to the spectral theory of compact convex sets of Alfsen and Shultz [6, 7]. Despite considerable efforts in both directions the author has failed to remove the restriction. The theorem lays out a possible mathematical structure of quantum mechanics with superselection rules. Like all other attempts to do so (cf. [43, 44, 49, 37, 14]), the axioms appear to be contingent.

This is particularly true of Axiom AHS2 and of our Axiom 2, which lie at the heart of quantum mechanics. One advantage of the axiom schemes in [5] and the present paper is that they identify the incidental nature of quantum mechanics so clearly. , no uniformity is present), then the above still holds, with the obvious modifications. In that way, however, only perfect C ∗ -algebras [50, 2] can be reconstructed (cf. Sec. 4). 3. From Transition Probabilities to C ∗ -algebras The proof of the theorem above essentially consists of the construction of a C -algebra AC from the given set P.

### Reviews in Mathematical Physics - Volume 9 by H. Araki, V. Bach, J. Yngvason (Editors)

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