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Let $A$ be a commutative unital $C^*$ algebra. Then $A=C(X)$ for some compact Hausdorff space $X$. Topological $K$-theory group (namely $K_0$) is defined in terms of vector bundles as a Grothendieck g …

Let $A$ be a $C^*$-algebra: then one defines topological $K_1$ group as $GL_{\infty}(A^+)/\Big(GL_{\infty}(A^+)\Big)_0$ where $A^+$ denotes $A$ with the unit adjointed (even if $A$ already had a unit: …

There are several ways in which one can define $K$-theory for $C^*$-algebras: for $K_0(A)$ group two aproaches: algebraic (using idempotents) and topological (using projections, i.e. self-adjoint idem …

In the K-theory formulation of the index theorem one defines the topological index in terms of the so called wrong way maps. Those maps are defined for embeddings of compact manifolds $i:X \to Y$: see …

I posted this question on StackExchange however no one has answered it therefore I though that it may be appriopraiet for Mathoverflow. So here is the question: Let $M$ be a manifold (closed). I kno …

I would like to understand the proof of Proposition 15 (see page 70 in this link ). More precisely: I would like to understand a particular step in the proof namely: Why $\frac{d}{dt}(\varphi \# …

It is in some sense folklore that given two arbitrary abelian groups $G,H$ one can find a $C^*$ algebra $A$ such that $K_0(A)=G$ and $K_1(A)=H$. My question is the following: what is known in the case …

There are (at least) two approaches to K-homoology: one is via the so called dual algebra which is due to Paschke. The second is via the Fredholm modules and is due to Kasparov. In Nigel Higson's book …

Recently I asked the following question, about the separability of the underlying $C^*$-algebra in the definition of $K$-homology: mathoverflow.net/questions/181361 As far as I understood, i …

In noncommutative geometry when one want to go to the differentiable level, one is forced to work with algebras which are no longer $C^*$. It is nice if we don't loose much information by the replacem …

Let me recall the definition of the Bott periodicity isomorphism in the context of $C^*$-algebras. We take a (class of) projection $p \in M_n(A^+)$ and map it to the class of $M_n(A)$ valued loop $f_p …

In this paper Aityah, Bott and Shapiro give an alternative definition of (relative) $K$-theory groups $K(X,Y)$ using sequences of bundles (this group is denoted by $L_n(X,Y)$ where $n$ is the length o …

In the usual setting of the Atiyah-Singer index theorem the situation is as follows: we have a closed smooth manifold $M$ without boundary and $D$ is some elliptic differential operator acting on sect …

There is a general (abstract) index theorem in noncommutative geometry: you take a K-theory class and K-homology class (which is represented by a triple $(A,H,F)$) and you pair them together. This p …

I would like to understand the proof of Lemma 1, page 339 in this book. Very briefly, the context is as follows: we have even dimensional oriented conformal manifold with the Hodge star operator chose …