Let E be a Hilbert C\*-module over some C\*-algebra and let $h \in K(E)$. Due to B. Blackadar's, "K-Theory for Operator algebras" Thm. 17.11.4 for a separable C\*-algebra $A$, represented by elements of $B(E)$, it is possible to construct a countable approximate unit $\{u\_n\}$ contained in $C^\*(h)$, such that $\{u\_n\}$ is quasicentral for $A$ and $u\_{n+1} u\_n=u\_n$. The question is: is it always possible to make $u\_n$ be projectors (or, at least, idempotents). The question seems to be obvious if $E$ is just a Hilbert space, but I'm not sure for Hilbert modules.



Approximate unit for the algebra C*(h) consisting of projectors
No. In fact, K(E) need not even contain any nonzero projections. Take a (nontrivial) C\*-algebra B with no nonzero projections1 and take E=B as a right module over B with inner product 〈a,b〉=a\*b. Then K(E)≅B, as mentioned in [Example 13.2.4 (a)](http://books.google.com/books?id=_YQvFHnD6bQC&lpg=PA109&ots=w9jB2qq8x4&dq=blackadar%252013.2.4&pg=PA109#v=onepage&q=&f=true) in Blackadar, and proved for instance as [Proposition 2.2.2 (i)](http://books.google.com/books?id=s_pNcoqXwFoC&lpg=PA203&dq=manuilov%2520troitsky&lr=&pg=PA19#v=onepage&q=&f=true) in Manuilov and Troitsky. (Note that in this case K(E) [also has no nonzero idempotents](https://mathoverflow.net/questions/16943/reference-needed-for-every-idempotent-in-a-c-algebra-is-similar-to-a-hermitian).)


**Edit:** I removed an overly complicated comment on the Hilbert space case, forgetting to take into account that h is strictly positive, and in particular positive. I added a comment on trouble that may arise even in this case.


In the case when E is a Hilbert space over B=ℂ, taking into account the fact that h is self-adjoint, C\*(h) is the C\*-algebra generated by a self-adjoint compact operator, and therefore the spectral projections of h provide an approximate identity {un} consisting of increasing projections. Because h is *strictly* positive, its range is dense, so this will be a sequence of projections converging weakly to the identity operator. However, this approximate identity need not be quasicentral for A⊆B(E). E.g., suppose you have un equal to the projection onto the span of the first n elements of an orthonormal basis. If S is the unilateral shift with respect to that basis, then ||unS−Sun|| = 1 for all n, so {un} is not quasicentral for C\*(S). Pedersen uses the Hahn-Banach theorem and the axiom of choice to show the existence of a quasi-central approximate identity in the closed convex hull of {un}, but you typically will not be able to find one consisting of projections, even in the Hilbert space case.


1 E.g., take B to be C0(X) for some noncompact, locally compact, connected space X, or if you prefer simple algebras see [Blackadar's 1980 paper](http://www.jstor.org/stable/2042420).


