Washington Redskins owner Jack Kent Cooke has purchased Arizona's largest hotel and office building complex, which is in Phoenix, for $52.5 million.

Cooke, who bought New York's famous Chrysler Building in 1979 and has other real estate holdings in New York and Virginia, said he will "institute improvements" in the Phoenix buildings.

He also plans to spend "the severe part of the winter in Phoenix" now that he's a property owner. The flamboyant Cooke, 69, said the purchase, "my initial investment in Sun Belt real estate, suits me just fine . . . " and gives him "roots in America's ninth largest city."

The Phoenix property, called the Rosenzweig Center, is made up of a 23-story, 400-room hotel, a 17-story office tower, a 20-story building, two smaller office buildings and parking garages, all on 15.4 acres. The buildings were constructed in 1962 by brothers Newton and Harry Rosenzweig and Del E. Webb Corp.

Cooke made a fortune in radio in his native Canada before coming to the United States, where he has built empires in cable television, sports teams and now real estate. He is the former owner of the Los Angeles Lakers basketball team and the Kings hockey team.

A recent real estate acquisition is 180 acres of land in Virginia purchased from Sen. John W. Warner (R-Va.). Cooke said his company, Jack Kent Cooke Inc., "is constantly looking for office buildings, hotels, and all sorts of industrial buildings." CAPTION: Picture, Jack Kent Cooke has purchased the Rosenzweig Center in Phoenix for $52.5 million. He called it his initial Sun Belt investment. Jack Kent Cooke Inc. Photo