Early Tuesday eBay said it will spin off its PayPal electronic-payments division into a separate public company — just as the corporate raider, whose firm owns 2.5 percent of eBay — had urged since January. EBay acquired PayPal, the most successful third-party company to emerge from its auction business, in 2002 for $1.5 billion in stock. [...] EBay has expanded beyond its auctioneer roots into a massive etailer rivaling Amazon, while PayPal has seen the mobile payments category undergo explosive growth — and fierce competition. [...] PayPal is on a faster growth trajectory than its parent. [...] why did eBay put up so much resistance to Icahn’s proposal to cleave the company in two?