# Almost idempotent approximate units in C*-algebras

As in Blackadar's "Operator Algebras" Definition II.4.1.1., call an approximate unit $(a_\lambda)$ in the positive unit ball of a C*-algebra almost idempotent if $a_\lambda a_\gamma=a_\lambda$ whenever $\lambda<\gamma$.

Does every C*-algebra have an almost idempotent approximate unit?

• You probably already know this, but the answer is yes for separable Cstar algebras (see Leonel Robert's comment to this question mathoverflow.net/questions/212078 ) – Yemon Choi Oct 24 '15 at 20:17
• That's right, the answer is yes for separable and, more generally, sigma-unital C*-algebras. Even more generally, I think the answer is yes for any C*-algebra A that has a "large enough" commutative C*-subalgebra B in the sense that B generates A as a hereditary C*-subalgebra (or left/right ideal). – Tristan Bice Oct 24 '15 at 20:20
• Just out of curiosity: why are you interested in this question for non separable $C^*$-algebra ? (one is generally happy when something is true for all separable $C^*$-algebras...) – Simon Henry Oct 28 '15 at 10:50
• From the point of view of, say, classification, separability might seem like a natural restriction. But from a more topological viewpoint it is not so natural. Indeed, the present question arose from another question about generalizing regularity to C*-algebras. – Tristan Bice Oct 28 '15 at 13:52

• I don't think an almost idempotent approximate unit has to consist of commuting elements. Remember in the nonseparable case that the net may not be linear. If $\alpha,\beta<\gamma$ are in your directed set defining the almost idempotent approximate unit $(a_\lambda)$ then $a_\alpha$ and $a_\beta$ certainly commute with $a_\gamma$ but not necessarily with each other. – Tristan Bice Feb 16 '16 at 21:45