Can an AW\*-algebra be recovered (up to Jordan isomorphism) from its lattice of projections? This is possible in the commutative/Boolean case.



Can an AW*-algebra be recovered from its lattice of projections?
Yes, this follows from Theorem 4.2 in Dye's Theorem and Gleason's Theorem for AW\*-algebras by Jan Hamhalter.
Link: <https://arxiv.org/abs/1408.4597>


Here the statement is: given an AW\*-algebra $A$ without type I$\_2$ summands, and an AW\*-algebra $B$, then every orthocomplement-preserving order morphism $\varphi:P(A)\to P(B)$ can be extended to a Jordan \*-homomorphism $A\to B$, i.e., the complexification of a Jordan homomorphism from the self-adjoint part of $A$ to the self-adjoint part of $B$.


You can get rid of the type I$\_2$ condition by splitting $A$ in a type I part and a type II/III part and use that the projections of a type I AW\*-algebra completely determine type I AW\*-algebras up to \*-isomorphism, which is an observation of M. Ozawa. The type II/III part follows from Hamhalter's Theorem. Details can be found in Corollary 9.2.9 of <http://repository.ubn.ru.nl/handle/2066/158429>


