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 wellknown references.

1$\begingroup$ For a notion of a noncommutative Grassmannian, see the MO question mathoverflow.net/questions/168993/… $\endgroup$ – Pedro Lauridsen Ribeiro Mar 17 '16 at 22:53

$\begingroup$ These discussions are all valid approaches to quantum grassmannians. But more similar to the quantum projective space is the notion of quantum flag varietiesI.e. Those coming from quantum groupswhich I worked out the gluing for the affine patches in my thesis... $\endgroup$ – B. Bischof Mar 18 '16 at 0:20
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 Lietheoretic origins, give a rich structure of their quantized coordinate rings, the quantum Grassmannians.

$\begingroup$ Thanks for the links Carlo! However, the second is not a noncommutative space, but something coming from quantum cohomology/string theory. The others are quantized coordinate algebras, not quantized homogeneous coordinate rings, as in the linked question. $\endgroup$ – Lars Pettersen Mar 17 '16 at 22:11

$\begingroup$ There are also various papers by Stokman and Letzter $\endgroup$ – AHusain Apr 16 '16 at 23:06
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:qalg/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).