Who Spread the Rumors About Alice?
The simple answer is… The Knights. The more complicated question is why did the Knights bother to spread a rumor about Alice?
As explained in "What happened to Mika?", the Knights are forever plotting to keep Lain form embracing her humanity, thus having more power in the Wired. They realize that in order for Lain to be compelled to give up her flesh, she will need to be rejected as a human.
The Knights use "Lain in the Wired" to spread the rumor about
Alice's forbidden relationship with her teacher. They want to make it
clear that Lain is the one spreading the rumor and that Alice finds this
out. Once this happens, it is their hope that Alice will reject Lain
completely. This, in turn, would cause Lain to see the futility and
fragility of human relationships and push Lain to retreat into the Wired.
The Knights, however, underestimate Lain's bond with Alice. Instead of
giving up, Lain fights to clear her name and salvage their friendship.
Lain discovers she has the power to access people's memories and uses this
power to erase the rumor from the collective unconsciousness.