In the book K-Theory and $C^*$-Algebras: A Friendly Approach by Niels Wegge-Olsen I came across the following notion: in Lemma 16.4 authors assume that the $C^*$-algebra $A$ possess "(commuting) approximate unit $(a_n)_n$ so that $a_{n+1}a_n=a_n$". In the next lemma the only assumption about $A$ is that $A$ is $\sigma$-unital, which means that there is some countable approximate unit. However the authors claim that in this case we may assume that this approximate unit is as above.

Why we can assume that this approximate unit have this additional property (having assumed that $A$ is just $\sigma$-unital?

  • 3
    $\begingroup$ This is a functional calculus argument, IIRC. See the comments on this older MO question mathoverflow.net/questions/212078/… $\endgroup$
    – Yemon Choi
    Jun 26 '16 at 0:32
  • 1
    $\begingroup$ N.E. Wegge-Olsen is one, indivisible person. $\endgroup$ Jun 26 '16 at 15:02
  • $\begingroup$ Ok, so the construction would go as follows: we fix an increasing sequence $(f_n)_n \subset C_0(0,1]$ of nonnegative functions, with the properties that $\|f_n\| \leq 1$ and $f_{n+1}(x)=1$ whenever $f_n(x)>0$. Then if we fix a nonnegative element $a$ in our algebra $A$ (with norm $\leq 1$) we can form $u_n=f_n(a)$ via functional calculus. It is clear for me that we therefore get an increaing sequence of nonnegative elements in the unit ball of $A$. However I don't quite understand why we get approximate unit, in other words why $\|u_nx-x\| \to 0$ for each $x \in A$? $\endgroup$
    – truebaran
    Jun 26 '16 at 18:51
  • $\begingroup$ I also wonder where exactly do we use the assumption that our algebra is $\sigma$-unital. In the comment to the question for which you gave a link one speaks about strictly positive element: I have problems in understanding what does it mean in the nonunital case (in the unital my guess would be positive and invertible) and how to obtain such an element (my guess is that here the assumption of being $\sigma$-unital is used and one has to take something like $\sum_{n=1}^{\infty}\frac{1}{2^n}e_n$ where $(e_n)_n$ is some countable approximate unit). $\endgroup$
    – truebaran
    Jun 26 '16 at 18:56
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    $\begingroup$ Strictly positive elements are defined here en.wikipedia.org/wiki/C*-algebra#Quotients_and_approximate_identities and it is not quite right to talk about invertibility in algebras which do not have identity elements. If your algebra is not sigma-unital then it won't have any strictly positive elements, see Section 1.4 of math.ucr.edu/~monnot/c%20algebras.pdf $\endgroup$
    – Yemon Choi
    Jun 27 '16 at 17:57

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