Is mining the unfair part of bitcoin in the same way banks (or rather, fractionary reserve systems) are to current monetary system? I've been reading a lot about bitcoin lately and I'm very interested.

Especially, when it comes to making a currency that is way more fair, because it avoids a centralized control of it.

However, in the same way today's banks/centralbanks/FED/regulators benefit from how the current monetary system works, wouldn't an economy solely based on bitcoin make the big IT corporations take advantage of everyone due to their computing (mining) power?

Cannot we build a currency that is decentralized like bitcoin, but that doesn't have the "mining" part of it? (I know that mining concept is very tied to cryptography, but maybe a less cryptographic-centric system can be built, like one based on trust/untrust between nodes?)

Thanks very much, and sorry if this has already been asked, but it's difficult to look for a question like this.
Question: Cannot we build a currency that is decentralized like bitcoin, but that doesn't have the "mining" part of it?

No need to, one was already created years ago and is still up and running today. Bitcoin depends on currency creation to record transactions, you can't get around this because that is the design of it and how it works. The only one that approaches this in a different way is Timekoin. Transactions and Currency Creation are separate. Transactions can be processed like clockwork every 5 minutes with no currency being created, provided some currency already exist to spend. Currency Creation is also produced like clockwork in very small amounts. So anyone running the server software can create currency, but in a very gradual, controlled amounts as long as the strict rules are being followed.