nLab continuous field of C*-algebras

Idea

A continous field of $C^\ast$-algebras is a kind of topological bundle of $C^\ast$-algebras: A topological space $X$ and a C*-algebra $A_x$ for each point $x \in X$, such that these algebras vary continuously, in some sense, as $x$ varies in $X$. When $X$ is thought of as a topological groupoid with only identity morphisms, this may be understaood as a special case of Fell bundles over $X$.

Applications

In strict deformation quantization

In C* algebraic deformation quantization continuous fields of $C^\ast$-algebras over subspaces of the standard interval (tyically $\{1 , \frac{1}{2}, \frac{1}{3}, \cdots, 0\} \hookrightarrow [0,1]$) which at $\hbar = 0$ are Poisson algebras constitute non-perturbative deformation quantizations of this Poisson algebra (hence of the phase space of some physical system that it represents).

References

The notion originates with:

• Jacques Dixmier, §10 in: $C^\ast$-algebras, North Holland (1977)

An efficient reformulation is due to:

• Eberhard Kirchberg, Simon Wassermann, Def. 1.1 in: Operations on continuous bundles of $C^\ast$-algebras, Mathematische Annalen 303 4 (1995) 677-698 [eudml:165387, pdf]

See also:

• Marius Dadarlat, Continuous fields of $C^\ast$-algebras over finite dimensional spaces, Advances in Mathematics 222 5 (2009) 1850-1881 [arXiv:math/0611405, doi:10.1016/j.aim.2009.06.019]