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The first statement is not quite right - $C([0,1])$ is a commutative C*-algebra but not an AW*-algebra, even though its only projections are the constant 0 and 1 functions, which certainly constitutes a complete Boolean algebra. However, it is true that AW*-algebras are precisely the real rank zero C*-algebras whose projections form a complete lattice. In ...


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Consider the given representation as sitting inside the universal representation. Then there exists a projection $E$ in the commutant of the enveloping von Neumann algebra which projects onto the representation subspace. Now consider the family $F$ of central projections that contain $E$. This is clearly non-empty as the identity is always one such ...



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