Ellis's 1957 paper on Locally Compact Transformation groups proves the following:
A locally compact hausdorff topology on a group $(G, \cdot)$ for which left and right multiplication are (separately) continuous, is a group topology.
In other words, LCH semitopological groups are topological groups.
I am interested in examples demonstrating obstructions to extending this result. Specifically, I want to know if there are `nice' examples of LCH topologies on groups such that:
i) left multiplication is continuous but right is not
ii) left multiplication is continuous but the inverse map is not.
In the literature, I've found there is extensive work weakening the topological conditions. However, I've not seen examples of the above situations keeping LCH.
Any help finding such examples is greatly appreciated. Thanks!