A futuristic 35-story tower where doors swoosh open with the wave of a hand houses a little-known firm long used by this emirate to deploy its oil riches around the world.

When Barclays needed to raise capital in 2008, the Abu Dhabi sovereign wealth fund, known as IPIC, invested more than $5 billion in the UK bank. Through a subsidiary, IPIC acquired holdings in German auto maker Daimler and Swiss commodities powerhouse Glencore. It helped finance the ultraluxury New York skyscraper One57, nicknamed the Billionaire Building.