A C*-algebra is *Rickart* if for each $x\in A$ there is a projection $p\in A$ so that
$R(x)=pA$.
Here the right-annihilator $R(S)$ of $S\subset A$ is defined
as $$R(S)=\{a\in A\mid xa=0\, \forall x\in S\}$$ and $R(x)\equiv R(\{x\})$.

In:

Kazuyuki Saito and J. D. Maitland Wright. $C^∗$-algebras which are Grothendieck spaces.

Rend. Circ. Mat. Palermo(2), 52(1):141–144, 2003.

an alternative definition is studied:
define a C*-algebra to be *Rickart* if each maximal Abelian *-subalgebra of $A$ is Rickart (or, equivalently, monotone $\sigma$-complete).
Equivalently, one may require that every Abelian *-subalgebra is contained in an
Abelian Rickart C*-algebra.

This definition is more general and seems to be sufficient for many applications.

Is this definition in fact equivalent to the original one?

This question recently came up in our investigations in the foundations of quantum theory: