This is probably well-known to the experts but I could not find any answer neither in my head nor in the literature: Is there a (unital) C\*-algebra such that its projections do not form a lattice (under the usual ordering)? Certainly, this cannot be a von Neumann algebra.



C*-algebras with bizzarre structure of projections
You can find examples of AF algebras without the lattice property in Section 2 of *AF Algebras with a Lattice of Projections* by Aldo J. Lazar [here](http://www.mscand.dk/article.php?id=2618).


