A Baer \*-ring is an \*-algebra whose lattice of projections is complete. I know two well-handed kinds of these structures: 


1- W\*-algebras (abstract case of von Neumann algebras). 


2- The inverse limit of W\*-algebras (called locally W\*-algebra). 


Q. Does there exist any other type(s) of Baer \*-rings? 



Some non-trivial Baer *-rings
*More examples*. Take any field $F$ with a positive definite involution (that is, $\sum a\_ia\_i\* = 0$ implies all $a\_i = 0$). Form the ring of $n \times n$ matrices, $M\_n F$, equipped with $\*$-transpose; it is a Baer\* ring. Moreover, we can take the bounded subring of $F$, $F\_b$, consisting of those elements of $F$, $a$, for which we can solve $aa^\* +\sum x\_i x\_i^\* = m$ for some integer $m$). Then $M\_n F\_b$ contains all the projections of $M\_n F$, hence is itself a Baer\* ring.


For example, let $F = K(x\_{\alpha})$, equipped with the identity involution, where $K$ is any subfield of the reals, and $\{x\_{\alpha}\}$ is an arbitrary family of indeterminates (I wanted to give a field which could not be embedded in the complexes; choose the cardinality of the set of indeterminates to exceed that of the complexes). 


*And still more*. Let $R$ be a self-injective von Neumann regular ring with positive definite involution. Then $R$ and all its matrix rings (and matrix rings over the bounded subrings as defined above) are Baer\*. This includes the regular ring of AW\* algebras of finite type, and also weird examples constructed from the field of fractions of the Weyl algebra. 


