Approximate unit for the algebra C*(h) consisting of projectors

2010-03-22T16:26:09Z

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.

Answer by Jonas Meyer for Approximate unit for the algebra C*(h) consisting of projectors

2010-03-23T00:05:53Z

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) in Blackadar, and proved for instance as Proposition 2.2.2 (i) in Manuilov and Troitsky. (Note that in this case K(E) also has no nonzero idempotents.)

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 {u_n} 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 u_n 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 ||u_nS−Su_n|| = 1 for all n, so {u_n} 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 {u_n}, 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.

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Approximate unit for the algebra C*(h) consisting of projectors

Answer by Jonas Meyer for Approximate unit for the algebra C*(h) consisting of projectors