Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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:

Bohrification of operator algebras and quantum logic

share|improve this question
    
Based on the remarks in your paper, this may deserve the "open-problem" tag. –  Jonas Meyer Jan 14 '10 at 8:57
add comment

Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook.

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.