When an AW*-algebra is a W*-algebra

Question

In a very old book of Kaplansky "Rings of operators", on p. 123 one can find the following sentence:

It is a standing conjecture that an AW${}^\ast$-algebra is W${}^\ast$ if its center is W${}^\ast$.

I am wondering what is the current status of this conjecture.

Thank you, J.

Answer 1

I don't know who solved this problem originally, but see J.D. Maitland Wright, Wild AW∗-factors and Kaplansky-Rickart algebras, J. London Math. Soc. 13 (1976), 83–89 for a construction of a family of AW* factors. These have trivial, hence commutative, center.