nLab continuous field of C*-algebras

Idea

A continous field of $C^\ast$-algebras is a topological space $X$ and a C*-algebra $A_x$ for each point $x \in X$, such that in some sense these algebras vary continuously as $x$ varies in $X$. Hence it is a kind of topological bundle of $C^\ast$-algebras. It is also an example of a Fell Bundle over $X$ when $X$ is thought of as a topological groupoid with only identity morphisms.

Applications

In strict deformation quantization

In C* algebraic deformation quantization continuous fields of $C^\ast$-algebras over subspaces of the standard interval (tyically $\{1 , \frac{1}{2}, \frac{1}{3}, \cdots, 0\} \hookrightarrow [0,1]$) such that in the limit this becomes a Poisson algebra constitute deformation quantizations of this Poisson algebra.

References

• Marius Dadarlat, Continuous fields of $C^\ast$-algebras over finite dimensional spaces (arXiv:math/0611405)