For what type of separable C* algebras, there is no any integer n and any nontrivial morphism from A to the Cuntz algebra O_{n}? In the other word for what type of C* algebras, Hom(A, O_{n}) is trivial for all n?
The motivation: I was thinking to consider a NC analogy for singular homology as follows:
A singular object is a continuous map from Delta^{n}\to X. So We have a natural morphism from C(X)\to C(\Delta^{n}).
With some (or a lot of) abuse of notations and equations, the equation \sum t_{i}=1, which defines the standard simplex \Delta^{n}, could be considered as a commutative picture of universal property of Cuntz algebra O_{n}. This is a motivation to construct a complex as followos: We put C_{n}(A)= Free abelian group generated by Hom(A,, O_{n}). Now we can define a complex
....... C_{n}(A) \to C_{n+1}(A) \to....
using n+1 embeddings O_{n} to O_{n+1}
Does this idea leads to triviality?For what type of C* algebras, this construction is useful?
A separable C*-algebra embeds in the Cuntz C*-algebra \mathcal{O}_2 if and only if it is an exact C*-algebra (a result by E. Kirchberg)
Dear Etienne,
Thank you for your answer.
According to your answer, we underestand Hom(C(X), O_{n}) is nontrivial. so we obtain a sequence of (cohomological) groups(According to the above complex) Each group in this sequences is a "functor". Do you think that such functors contains some nontrivial information about topological space X?
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