Given today's focus on profit from short-term fluctuations in share prices (even down to milliseconds), share prices are highly volatile and has very little to do with the profitability of the company. Since trading in share prices is a zero-sum game, the main effect is that the clever (and fast) take money from the not-so clever (and not-so fast). The society as a whole gains zip from this. Quite the opposite, in fact, as we have to bail out banks that made the wrong bets, while the banks that make the right bets channel their profit to huge bonuses to its CEO, CFO etc, who immediately send it to the Cayman Islands.
So, IMO, shares should only be traded at par, so the only potential gain from shares would be payment of dividends. This is, by its nature, not a zero-sum game, and millisecond trading will not profit from it.
I don't see this happening, though. Not only will many financial institutions do what they can to prevent it, but they will find ways around a law that requires shares to be sold at a fixed price: Rather than trading the shares themselves at varying prices, they will trade papers that promise to buy or sell shares (at the fixed price) at a specified later date. These papers will be subject to price fluctuations, unless the laws forbid these too. And if they do, the banks will just find some other form of derivative to trade.