The first development on the CENTURY SQUARE site was the Gateway West tower, known in later years as 1801 Avenue of the Stars. The building was completed in 1963, the year before business began at the adjacent shopping center. This historic Mid-Century Modern (and Welton Becket-designed) structure was knocked down in April 2015, in the name of "progress." It has been replaced with an enlargement of the mall.

A poster drawn by famed fashion illustrator Betty Brader. It announced Joseph Magnin, which opened at CENTURY SQUARE in August 1965.
Advert from http://americanhistory.si.edu

An exterior view of the mall's Mayfair Market grocery store, which was accessed from the upper floor of a subterranean parking deck. A ramp up to the main retail level of the mall is seen on the right.
Photo from http://www.groceteria.com

An interior view of the Mayfair Market. It was rebranded as a Gelson's Market in 1971. By this time, the mall was often promoted as the CENTURY CITY SHOPPING CENTER, although ads mentioning CENTURY SQUARE were still appearing in the late '70s. Most Los Angelino's just call the complex the CENTURY CITY MALL.
Photo from http://www.groceteria.com