algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
(geometry $\leftarrow$ Isbell duality $\to$ algebra)
If the $A$ is a C*-algebra of functions on some topological space $X$ and the action of $G$ on $A$ is induced from an action on $X$, then in good cases the crossed product algebra is equivalent to the convolution algebra of the action groupoid.
e.g. (Land 13, prop. 4.2, Soltan, section 5.1, Bos 07, example 12.2.8) also (Connes 06, around p. 13)
A standard textbook account (with an eye towards KK-theory) is section 10 of
Original articles include
M. A. Rieffel, Proper actions of groups on C∗-algebras Mappings of operator algebras (Philadelphia, PA, 1988), 141–182, Progr. Math. 84, Birkhäuser Boston, Boston, MA, 1990. (pdf)
Jonathan Henry Brown, Proper actions of groupoids on $C^*$-algebras (arXiv:0907.5570)
Detailed lecture notes are in
A survey of noncommutative geometry with a bunch of examples of crossed product $C^\ast$-algebras is
Further discussion in the context of the Baum-Connes conjecture is in
Applications to geometric quantization and specifically geometric quantization by push-forward are indicated in
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