I had some attempts to obtain elementary proofs of (1) or (2), but I failed. Maybe these proofs should not be easy, for instance, in the case when (1) and (2) elementarily imply that the group $G$ is topological.
Since I am a specialist in paratopological groups, not semitopological, I propose a sketch of a proof that each locally compact Hausdorff paratopological group (that is, a group with jointly continuous multiplication) with (2) is topological. It is based on simple ideas and manipulations with the neighborhoods, but needs some background. So below I shall intensively cite our paper [BR], which you can look for details.
Following my teacher I say that a paratopological group $G$ is saturated if for any neighborhood $U\subset G$ of the unit the set $U^{-1}$ has nonempty interior in $G$.
Now let $U$ be an arbitrary neighborhood of the unit of the given group $(G,\tau)$. Since the set $\overline{V^{-1}}$ is compact, there is a finite subset $F$ of $\overline{V^{-1}}$ such that $\overline{V^{-1}}\subset FU$. Then $V\subset U^{-1}F^{-1}$ and therefore the set $U^{-1}$ has a nonempty interior. Thus the group $G$ is saturated.
Given a paratopological group $G$ let $\tau_\flat$ be the strongest group topology on $G$, weaker than the topology of $G$. The topological group $G^\flat=(G,\tau_\flat$), called the group reflexion of $G$, has the following characteristic property: the identity map $i:G\to G^\flat$ is continuous and for every continuous group homomorphism $h:G\to H$ from $G$ into a topological group $H$ the homomorphism $h\circ i^{-1}:G^\flat\to H$ is continuous.
The group reflexion $G^\flat$ of any abelian Hausdorff paratopological group $G$ is Hausdorff. Moreover, in this case the topology of $G^\flat$ has a very simple description: a base of neighborhoods at the unit in $G^\flat$ consists of the sets $UU^{-1}$ where $U$ runs over neighborhoods of the unit in the group $G$ (such the groups we call 2-oscillating). A bit later it was realized that the same is true for any paratopological SIN-group, that is a paratopological group $G$ possessing a neighborhood base $\mathcal B$ at the unit such that $gUg^{-1}=U$ for any $U\in\mathcal B$ and $g\in G$ (as expected, SIN is abbreviated from Small Invariant Neighborhoods). Unfortunately, Hausdorff paratopological SIN-groups do not exhaust all paratopological groups whose group reflexion is Hausdorff (for example any separated topological group has Hausdorff group reflexion but needs not be a paratopological SIN-group).
In [BR, Pr. 3] we showed that each saturated paratopological group $G$ is 2-oscillating. For this purpose fix any neighborhood $U$ of the unit $e$ in $G$. We have to find a neighborhood $W\subset G$ of $e$ such that $W^{-1}W\subset UU^{-1}$. Find an open neighborhood $V_1\subset G$ of $e$ such that $V_1^2\subset U$. Since $G$ is saturated, there are a point $x\in V_1$ and a neighborhood $W\subset G$ of $e$ such that $x^{-1}W\subset V_1^{-1}$. Then $W^{-1}x\subset V_1$ and $W^{-1}\subset V_1x^{-1}$. We can assume that $W$ is so small that $x^{-1}W\subset V_1x^{-1}$. In this case $W^{-1}W\subset V_1x^{-1}W\subset V_1V_1x^{-1}\subset V_1V_1V_1^{-1}\subset UU^{-1}$.
Now we are able to prove that the topology of $G$ coincides with the topology of its group reflexion $G^\flat$. For this purpose it suffices to show that each sufficiently small open neighborhood $U\in\tau$ is open in $G^\flat$ too. Let $W\in\tau$ be a neighborhood of the unit with compact closure in the group $(G,\tau)$. Choose a neighborhood $W_1\in\tau$ of the unit such that $W_1W_1\subset W$. Since the group $G$ is saturated, there exist a point $x\in W_1$ and a neighborhood $W_2\in\tau$ of the unit such that $W_2\subset W_1$ and $xW_2^{-1}\subset W_1$. Then the set $A=xW_2^{-1}W_2\subset W_1W_2\subset W_1W_1\subset W$ has a compact closure $\overline A$ in the group $(G,\tau)$. So the restriction $i|\overline A$ of the identity map $i:G\to G^\flat$ is a homeomorphism. Let $U\subset W_2$ be an arbitrary neighborhood of the unit it the group $(G,\tau)$. Then $xU$ is a open subset of $A\subset \overline A$. Hence $xU$ is an open subset of $\overline A$ as a subspace of $G^\flat$. Thus $xU$ is an open subset of $A$ as a subspace of $G^\flat$. Since the group $G$ is 2-oscillating, the set $A$ is open in $G^\flat$. Therefore the set $xU$ is open in $G^\flat$ too.
Update: You won’t believe me, but just now Katya Pavlyk decided to sent to me the original Ellis paper [E]. :-D The claim “locally compact Hausdorff paratopological group has (2)” is the next to the last step (Lemma 4) of the original proof of Ellis Theorem. Moreover, Katya has a question concerning the proof of this claim too. :-)
Remarks to the paper. It seems that:
– in the next to the last sequence in the proof of Lemma 4 should be “$x^{-1}\in\overline{E^{-1}_{m+1}}$” instead of “$x^{-1}\in E^{-1}_{m+1}$”;
– in the last sequence should be “$x_n^{-1}\in x^{-1}U$” instead of “$x_n^{-1}\in x^{-1}U^2$”;
– in the proof of Theorem, “$U’$” means “$X\backslash U$”;
– in the proof of Theorem, should be “$U’$” instead of “$\cal U’$”;
– in the proof of Theorem, should be “$\{e\}=$” instead of “$e=$”.
References
[BR] Taras O. Banakh, Alex V. Ravsky. Oscillator topologies on a paratopological group and related number invariants // Algebraical Structures and their Applications, Kyiv: Inst. Mat. NANU, 2002, 140--153.
[E] Robert Ellis, A note on the continuity of the inverse, Proc. Amer. Math. Soc., 8 (1957), 372-373.
