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](http://link.springer.com/article/10.1007%2FBF02871928). *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:


[Bohrification of operator algebras and quantum logic](http://arxiv.org/abs/0905.2275)



Saito-Wright definition of Rickart C*-algebras
They are equivalent. See [*On Defining AW*\*-*algebras and Rickart C*\*-*algebras* by K. Saitô, and J.D.M. Wright](http://arxiv.org/abs/1501.02434) (arXiv:1501.02434)


