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Maybe the following can count if you are willing to restrict to separable, simple, stable, nuclear $C^*$-algebras $A$ and $B$. Then the Kirchberg-Phillips Classification Theorem says that $A\otimes\ma …

As an alternative to Robin Chapman's solution, I would like to state Exercise 7.8 from Rørdam's, Larsen's and Laustsen's "Introduction to the K-theory of C*-algebras": For every unital AF-algebra …

This is an answer to the question: "Given a $C^\ast$-algebra $A$ that shares some properties of (higher-rank) graph algebras (such as separability, nuclearity, UCT, K-theoretic properties), can I conc …

I know the definition a spectral triple and that it is some kind of non-commutative generalisation of (the ring of functions on) a compact spin manifold. But, why is it called spectral triple?

Concerning your last question, I would say you should view your C*-algebra itself as the (coordinate ring on the) "non-commutative topological space." The spaces you suggest are commutative. Your C*-a …

Any pair of abelian groups arises, up to isomorphism, as the $K$-theory groups of a commutative $C^*$-algebra. If the groups were countable, the $C^*$-algebra can be chosen to be separable. This foll …

Expanding my comment as per suggestion above. All this is explained in more detail in the reference I gave though. A correspondence from $A$ to $B$ is a Hilbert $B$-module on which $A$ acts non-degen …

This is true. It follows essentially from the naturality of the boundary map with respect to morphisms of extensions. We need to show that the composition of the two boundary maps $$ K_*(A_{00})\to K_ …