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?
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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.
She is definitely the Shadow Broker in Mass Effect 3, regardless of if you played the DLC or not. I didn't play the DLC, but there were signs that she had become the Broker regardless. In one conversation with EDI, she lets slip that Liara has become the new shadow broker and also when you look at any war assets involving the shadow broker, there is usually a picture of Liara.