I want to use the ethereum blockchain to prove ownership for example in a company (or even a hedge fund). How do I guarantee my investors (or buyers) that they bought a share of my company?
I'm assuming you already have already incorporated your company and have a shareholders agreement outlining rights, etc. Also, I am not a lawyer. I'm trying to keep this answer as general as possible, too.
Firstly, you'd have to find out if you can create a company and issue shares on a blockchain in your jurisdiction -- after all, a company is, by definition a legal entity. In fact, the whole point is to create a legal entity that replaces a person for the purposes of pursuing business (for benefits such as longevity, tax purposes, limiting personal liability, etc).
Secondly, you'd have to prove that you have the legal authority to issue shares in the company and/or that you are presently in possession of shares that you are issuing on the blockchain. In most/all jurisdictions, you will also be required to list yourself (or whoever is responsible) as the record keeper of note for shareholders in your company. You may also be legally obligated to do certain paperwork if your shareholders exceed a certain number ("going public"). In Canada, at least, you will also need things like a mailing address of shareholders so that they can receive documents that they are entitled to as shareholders.
Thirdly, you will need to prove that any securities you transfer on the blockchain can be linked to your company's list of shareholders. This may require some legal documents on your end and people checking your bricks and mortar operation or obtaining records from your incorporating jurisdiction.
Fourthly, you will need to provide a link between your on-line identity and your real-world identity. Otherwise, transactions are suspect.
There may be (and most likely are) some steps missing.
How do I guarantee my investors (or buyers) that they bought a share of my company?
If they have a "unit" of the crypto-asset, then you can trust that they only obtained it through only possible ways your contract would have defined. You can trust this because all miners on the network verify validity (this is the main innovation of the blockchain).
The crypto-asset can be implemented much like a token and all logic would live in a smart contract stored on the blockchain.
I have to create a token and a cryptocurrency? The cryptocurrency will be the shares.
This way for each 1 ether I get I will return 1 unit of my cryptocurrency (for example). If the investor does not want to have shares in my company, then I simply buy back his cryptocurrency (the one that I issued) for ethers again.
If my company has a dividend then I divide the dividend to the number of units cryptocurrency (plays the role of shares) that I gave to the investors. Next I search the ethereum addresses that invested in my company and send them the proportional dividend I would like to pay them.
I would prefer the trade of my company to be once per year and only for a period of one week. This will give me the security of using the funds invested in my company for at least a year. This can be implemented with a smart contract.
Also all profits can be stored into an account (that can not leave) and be distributed only once per year at an exact date. This way share holders will be able to see the dividend even before it is finally distributed. Of course this means that profits are not re-invested, but gives confidence to investors.
Also a smart contract can discourage selling your shares too fast back to the company. If you sell in 1 year than you get 0.95 for the ethers you invested. If it is two years you get 0.97 and so on until in 4 years you get 100%.
An example will be a company that buys real-estate. It buys houses and apartments and rents them. It will be a collective scheme where all the profit is accumulated in a single ethereum account. Once new investments are sufficient for buying a new property the company will do just that and continues to collect rent and distribute it as a dividend. Of course if all your money are invested you can not return the money to everyone at any time (just like a bank), so you would like a fixed period where you will be prepared to give money back to investors.