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The tangent groupoid can be used in constructing the index map and proving the Atiyah-Singer index theorem. This may illustrate its importance. Higson and Roe will write a book on it.

Let $\theta$ be a real number. We define $A_{\theta}$, the algebra of continuous functions on a noncommutative $2$-torus, to be the universal $C^*$-algebra generated by two generators $U$ and $V$ whic …

We could talk about the formal smoothness of an algebra. See for example Ginzburg's lecture notes For an associative algebra $A$ over a field $k$ we define $$ D(A)=T(A+\bar{A})/(\bar{ab}=a\bar{b}+\bar …

In the equivariant Atiyah-Singer index theorem, when $G$ is a compact group acting on a manifold $M$ and $R(G)$ is the representation ring of $G$. We have the analytic index $$ \text{a-ind}: K^*_G(TM) …

In Alain Connes' $C^*$-algèbres et géométrie différentielle (an English translation is here,), for a $C^*$-algebra $A$, we consider a $C^*$-dynamic system $(A,G,\alpha)$, where $G$ is a Lie group and …

In Kasparov's 1988 paper Equivariant KK-theory and the Novikov conjecture section 4 he defined the Dirac element for a (non-spin) $G$- Riemanian manifold $X$ as an element in the $K$-homology $K^0_G(C …

Fix a real number $0<q<1$. We consider the standard Podles sphere $A_q$ as the universal unit $C^*$-algebra generated by $a$ and $b$ with relations \begin{equation*} \begin{split} &a=a^*,~ ab=q^2ba, ~ …

Let $M$ be a smooth manifold with a Poisson bracket $\{-,-\}$. Kontsevich proved that there exists a deformation quantization of $M$, i.e. let $C^{\infty}(M)[[\hbar]]=C^{\infty}(M)\otimes_{\mathbb{R}} …