I'm building an application which uses last.fm API. I want my server to communicate with last.fm and the users of my application would communicate with the server. So the user is indirectly communicating with last.fm. I'm doing this to speed up the whole communication by caching some data on my server.
Is this OK?
Yes, this is OK. Many other services do this.
If you think about it, any action any app takes is always indirect. There is no requirement that the application acting on behalf of the user runs on the same computer as the user. Quite often the application runs on a web server.
Some examples of apps which do this include http://tweekly.fm/ and http://hypem.com/ .
In order for the service to act on the user's behalf (for certain methods such as scrobbling), you have to authorise your application as the user, and this is achieved using the web auth flow described at http://www.last.fm/api/webauth .
(This is one of those flows where the user is directed to a page on Last.fm to confirm that they authorise your app. Your app receives a session key in return, which allows your app to act on behalf of the user).
