The Mass Effect 2 Lair of the Shadow Broker DLC sees Liara assume the role of the Shadow Broker.
Will Mass Effect 3 assume that she took over that role even if you don’t import a character in which you completed that DLC?
If you have not played the Lair of the Shadow Broker, Liara is initially referred to as an information broker, like she was on Illium.
…until you get onto the privacy of the Normandy, where she has set up shop, and admits to Shepard that she has tracked down and usurped “the galaxy’s most notorious information broker”.
The dialog is a bit different if you actually helped her (via the DLC), but the outcome is the same regardless.