Albert coached at the Lawn Tennis Club of
Marseille in 1920 but soon after, following the death of his father, he returned to the Carlton Club's seven court-complex at Cannes. In 1922 he won the international professional invitation tournament at Lloyd Park,
Walthamstow (sponsored by the
News of the World), defeating
Joseph Negro in straight sets in the semi-final and overcoming
Charles Read of Queen's Club in the final "after a magnificent contest that lasted two and a half hours". When organising tournaments at the Carlton Courts he was able to attract the best players of the day, both professional and amateur, and in the course of a 1923 competition four hundred and twenty matches were played there in the course of a single week. These included a “memorable” contest between
Suzanne Lenglen and
Elizabeth Ryan and a match between Burke and the Frenchman Henri d'Arsonville, then regarded as the world professional champion. The latter encounter attracted heavy betting on its outcome, "thrilled a large audience and ended in a win for the French player". In 1925 Burke, partnered by Lenglen, played in an exhibition match against
René Lacoste and
Yvonne Bourgeois at Nice. Albert Burke won the Bristol Cup in France in 1924 and 1925, beating
Roman Najuch in both finals. He was also losing finalist in the Bristol Cup in 1926, 1929 and 1931, defeated in all three finals by
Karel Koželuh. In the absence of Koželuh he won the
Deauville tournament in 1927, meeting Joseph Negro,
Howard Kinsey and his own brother Edmund in the final pool, and beating them all without losing a set. On account of this victory he was saluted as "champion of the world", and he was the acknowledged Champion of France when, in the following year, he played in the 1928 Bristol Cup competition and appeared at the World Professional Singles Championship at
Queen's Club, defeating the title holder,
Dan Maskell, in the third round. Playing against his brother Edmund and their assistant coach at Cannes,
Robert Ramillon, Albert won the doubles final at Queen's with Roman Najuch as his partner, but it was Ramillon who defeated Edmund to take the singles title. Burke finished second in the 1930
French Pro Championship round robin, while
Karel Koželuh was the winner. At the
US Pro Tennis Championships Burke was a quarter finalist in 1931 (losing to
Howard Kinsey) and 1932 (losing to
Bill Tilden). In 1931, at an exhibition match at
Crawford Notch, he had defeated Tilden in what was described as "the best feat of his career". At the 1935
Wembley Championships, Burke lost in the quarter finals to
Ellsworth Vines. The following year he partnered
Henri Cochet to the doubles title at the French Professional Championships. ==Death==