definition of weak*-weak* continuous

Question

I'm reading a paper and I have couple of terms which I can't seem to find the definition for, the first one

1) what do we mean by weak*-weak* continuous map.

2) what is the definition of a left translation invariant subspace.

Thank you for your help.

Answer 1

In general,

1. A map $f:X\to Y$ is "topology 1"-"topology 2" continuous if it is continuous as a map from $(X,\text{topology 1})$ to $(Y,\text{topology 2})$. This can be described by preimages of open sets, or by nets: if $x_\alpha \to x$ in the sense of topology 1, then $f(x_\alpha)\to f(x)$ in the sense of topology 2.

2. A left translation invariant subspace is a subspace that is invariant under left translation. Which begs the question: what is left translation? I think your space is some space of functions on a group $G$, in which case left translation by $g\in G$ is the operator that transforms $x\mapsto f(x)$ into $x\mapsto f(gx)$. Example: the space consists of continuous functions on the circle $\mathbb T\subset \mathbb C$, and $g =i $. The space of functions $f$ such that $\int_{\mathbb T}f=0$ is invariant under left translation by $g$ (or by any element of $\mathbb T$ for that matter).