Mach's principle and infinite speed of distant stars If rotation was relative, we could say that the Universe is revolving around the earth. But how could this be true, since at some distant point the speed of a star would be greater than the speed of light ? How does Mach answer to this question ?
 A: One answer is: you won't know until you try.  
Another answer could be that the aggregate influences of all matter are what create the rotational rest frame.  If there were no other matter than our tiny planet in the Universe, would there be any way to detect rotation? 
I'm not wording that well, but it might add to  your insight.
"It is justified to consider Mach as the precursor of the General Theory of Relativity." Albert Einstein
The Ehrenfest paradox may interest you:
https://en.wikipedia.org/wiki/Ehrenfest_paradox
It was one of the thought experiments that help Einstein may the leap the General Relativity.
There is a modified Mach's principle in General Relativity.  I haven't read this paper, but it is a topic of great interest to me.
https://arxiv.org/abs/gr-qc/9607046
"We define a new parameter `cumulative drag index' for a particle in circular orbit in a stationary, axisymmetric gravitational field and study its behaviour in the two well known solutions of general relativity {\it viz.}, the Kerr spacetime and the G\"odel spacetime, wherein the inertial frame dragging has an important role. As it shows similar behaviour for both co and counter rotating particles, it may indeed be an indication of the influence of the faraway universe on local physics and thus Machian. "
A: In your rotating frame the star has a Lorentz Boost $\lambda=\frac {\omega r} {c}$ perpendicular to the radius vector to the star.  If $\lambda << 1$, then $\frac{v}{\not{c}} = \frac {\omega r} {\not{c}} $ which is the familiar relation you used to suggest superluminal $v$. However, there is no limit to how large $\lambda$ can be as $\omega r$ increases, but always $v \lt c $ because $\frac{v}{c} =\tanh(\lambda) \lt 1$.
I don't know if Mach answered your question this way. Before Einstein published Special Relativity in 1905, Lorentz transformations were known from the invariance of Maxwell's Equations. Mach and Einstein talked to each other. Mach died in 1916 so he certainly knew about Special Relativity and how velocities add.  He knew that successive boosts could not add up to greater than c, so he could have had a similar explanation perhaps without using the words "Lorentz Boost Parameter".
