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?

  • 1
    $\begingroup$ 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 ) $\endgroup$
    – Yemon Choi
    Oct 24 '15 at 20:17
  • $\begingroup$ 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). $\endgroup$ Oct 24 '15 at 20:20
  • $\begingroup$ 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...) $\endgroup$ Oct 28 '15 at 10:50
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    $\begingroup$ 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. $\endgroup$ Oct 28 '15 at 13:52

Akemann has constructed a C*-algebra that does not contain an approximate unit of commuting elements. See Example 2.1 in

Akemann. Approximate units and maximal abelian C*-subalgebras. Pacific J. Math. 33 (1970)

As pointed out by Tristan, a C*-algebra might have no approximate unit of commuting elements but nevertheless have an almost idempotent approximate unit. (Thanks for pointing out the mistake in my original answer.) So the original question remains open.

  • $\begingroup$ 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. $\endgroup$ Feb 16 '16 at 21:45
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    $\begingroup$ Actually, Akemann and Doner have another example in a later paper "A non-separable C*-algebra with only separable abelian C*-subalgebras" (which Piotr Koszmider and I recently redid without the continuum hypothesis in "A note on the Akemann-Doner and Farah-Wofsey constructions"). The C*-algebra in question is generated by its projections which form a lattice and hence an almost idempotent approximate unit, while any commutative subnet would have to be countable and hence not large enough to be an approximate unit. $\endgroup$ Feb 16 '16 at 22:05

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