Quantum Grassmannians? In noncommutative algebraic geometry a commonly studied family of objects are  quantum projective spaces. Theses are certain deformations of the homogeneous coordinate ring of $\mathbb{CP}^n$. For example, see this mathoverflow post. The obvious question I would like to ask is whether or not people consider a Grassmannian generalisation of such objects, and if so, what are some well-known references.
 A: apparently, quantum grassmannians come in many variations --- this may be what you are looking for:
Graded quantum cluster algebras and an application to quantum Grassmannians, Grabowksi & Launois, 2010.

Among those who study quantized coordinate rings, it is widely
  acknowledged that Grassmannians have a special place. The intricate
  geometric structures associated to Grassmannians, due in part to their
  Lie-theoretic origins, give a rich structure of their quantized
  coordinate rings, the quantum Grassmannians.

A: There is a deformation quantization approach to the quantization of the Grassmannians taking their Kähler symplectic form as the starting point. You can find this in the preprint by Schirmer arXiv:q-alg/9709021 from the nineties. He wrote a PhD on that. There are many earlier works on the quantization of $\mathbb{CP}^n$ in the same spirit, starting perhaps with works of Moreno and collaborators in the early eighties. You find many references in Schirmer's preprint.
However, this is not directly a $C^*$-algebraic approach, if you are interested in things like that. Nevertheless, the (a priori formal) star product is algebraic on many nice functions (the representative functions).
