Carl Icahn has taken a stake in Ebay Inc and is proposing a spin-off of the companys fast-growing PayPal division, but the e-commerce giant rebuffed the overture, setting the stage for a potential battle with the activist investor. EBay, which bought PayPal for $1.5 billion in 2002 and has considered hiving off the multibillion-dollar payments service, argued on Wednesday that the business would lose synergies with the overall e-commerce business as an independent unit. Shares of Ebay, which also reported earnings per share a penny above Wall Street expectations, jumped as much as 12 percent. Ebay Chief Executive John Donahoe said he had heard Icahn out but rejected his proposal.