Approximate unit for the algebra C*(h) consisting of projectors - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:22:57Z http://mathoverflow.net/feeds/question/19032 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19032/approximate-unit-for-the-algebra-ch-consisting-of-projectors Approximate unit for the algebra C*(h) consisting of projectors Kolya Ivankov 2010-03-22T16:26:09Z 2010-03-24T00:08:44Z <p>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.&nbsp;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 <code>$C^*(h)$</code>, 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.</p> http://mathoverflow.net/questions/19032/approximate-unit-for-the-algebra-ch-consisting-of-projectors/19075#19075 Answer by Jonas Meyer for Approximate unit for the algebra C*(h) consisting of projectors Jonas Meyer 2010-03-23T00:05:53Z 2010-03-24T00:08:44Z <p>No. In fact, K(E) need not even contain any nonzero projections. Take a (nontrivial) C*-algebra B with no nonzero projections<sup>1</sup> and take E=B as a right module over B with inner product &#9001;a,b&#9002;=a*b. Then K(E)&#8773;B, as mentioned in <a href="http://books.google.com/books?id=_YQvFHnD6bQC&amp;lpg=PA109&amp;ots=w9jB2qq8x4&amp;dq=blackadar%252013.2.4&amp;pg=PA109#v=onepage&amp;q=&amp;f=true" rel="nofollow">Example 13.2.4 (a)</a> in Blackadar, and proved for instance as <a href="http://books.google.com/books?id=s_pNcoqXwFoC&amp;lpg=PA203&amp;dq=manuilov%2520troitsky&amp;lr=&amp;pg=PA19#v=onepage&amp;q=&amp;f=true" rel="nofollow">Proposition 2.2.2 (i)</a> in Manuilov and Troitsky. (Note that in this case K(E) <a href="http://mathoverflow.net/questions/16943/reference-needed-for-every-idempotent-in-a-c-algebra-is-similar-to-a-hermitian" rel="nofollow">also has no nonzero idempotents</a>.)</p> <p><strong>Edit:</strong> 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.</p> <p>In the case when E is a Hilbert space over B=&#8450;, 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<sub>n</sub>} consisting of increasing projections. Because h is <em>strictly</em> 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&#8838;B(E). E.g., suppose you have u<sub>n</sub> 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<sub>n</sub>S&#8722;Su<sub>n</sub>|| = 1 for all n, so {u<sub>n</sub>} 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<sub>n</sub>}, but you typically will not be able to find one consisting of projections, even in the Hilbert space case.</p> <p><sup>1</sup> E.g., take B to be C<sub>0</sub>(X) for some noncompact, locally compact, connected space X, or if you prefer simple algebras see <a href="http://www.jstor.org/stable/2042420" rel="nofollow">Blackadar's 1980 paper</a>.</p>