In the book "Noncommutative Geometry" by Alain Connes (link to the book) on page 162 the author defines new homotopy groups $\pi_{n,k}(X,*)$ for a locally compact pointed space $(X,*)$ as the group of homotopy classes of morphisms $C_0(X,*)\to M_k(C_0(\mathbb{R}^n))$. The group law comes from the natural morphism $C_0(\mathbb{R}^n)\bigoplus C_0(\mathbb{R}^n)\to C_0(\mathbb{R}^n)$. The motivation for this definition is that for $k=1$ this coincides with the usual homotopy groups due to Gelfand's theorem which states that the category of commutative $C*$-algebras is antiequivalent to the category of locally compact spaces; for $k>1$ we are replacing the algebra $C_0(\mathbb{R}^n)$ with a Morita equivalent one $M_k(C_0(\mathbb{R}^n))$, in the book there is an example which shows that $\pi_{3,2}(S^1)$ is nontrivial, but there are no links to further reading - Where can I read more about this construction?
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