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