The meaning of the notation $\mathbb C_{\text{Re}z\ge \alpha}$?

Question

I have come across the following notation in some papers I have been reading. It is a notation pertaining to a certain subset of the complex plane. What does the notation $$\mathbb C_{\text{Re}z\ge \alpha}$$ mean, where $\alpha\in\mathbb R$? Is this a common notation for any particular subset of $\mathbb C$?

I thought that this was meant to represent the portion of $\mathbb C$ obtained on cutting the plane down the vertical line at $\text{Re}z=\alpha$, and considering only those points to the right of it. But, is it in fact to do with a ball of some sort?

Answer 1

My guess: $$ \{z \in \mathbb C : \text{Re }z \ge \alpha\} $$ Thus it is a half-plane, not a ball.

If (as you note) it is not clear, then it should be explained (either here, or earlier in the text).