Rieffel deformation of group coactions

Let G be a locally compact group, H an abelian subgroup and let f be a continuous 2-cocycle on the dual group of H. Let B be a C*-algebra equipped with a continuous right coaction of G. Using Rieffel deformation, we can construct a quantum group G(f) and the deformed C*-algebra B(f). The aim of this paper is to show that B(f) is equipped with a continuous coaction of G(f). The transition from the original coaction to its deformed counterpart is nontrivial in the sense that the deformed one contains complete information about the undeformed one. In order to illustrate our construction we apply it to the action of the Lorentz group on the Minkowski space obtaining a C*-algebraic quantum Minkowski space

Rieffel deformation of homogeneous spaces

AbstractLet G1⊂G be a closed subgroup of a locally compact group G and let X=G/G1 be the quotient space of left cosets. Let X=(C0(X),ΔX) be the corresponding G-C∗-algebra where G=(C0(G),Δ). Suppose that Γ is a closed abelian subgroup of G1 and let Ψ be a 2-cocycle on the dual group Γˆ. Let GΨ be the Rieffel deformation of G. Using the results of the previous paper of the author we may construct GΨ-C∗-algebra XΨ – the Rieffel deformation of X. On the other hand we may perform the Rieffel deformation of the subgroup G1 obtaining the closed quantum subgroup G1Ψ⊂GΨ, which in turn, by the results of S. Vaes, leads to the GΨ-C∗-algebra GΨ/G1Ψ. In this paper we show that GΨ/G1Ψ≅XΨ. We also consider the case where Γ⊂G is not a subgroup of G1, for which we cannot construct the subgroup G1Ψ. Then generically XΨ cannot be identified with a quantum quotient. What may be shown is that it is a GΨ-simple object in the category of GΨ-C∗-algebras

CNC machine tool error compensation system implementation strategies and their constraints

This paper deals with the constraints imposed on error compensation systems by their implementation strategies. Constraints stemming from the structure of the control system with an NCK architecture of the ADCBI type are presented. Practical realizations of compensation and potential solutions offered by contemporary commercial CNC controllers are discussed