How did America establish an informal empire in China from 1899 to 1912?

The US established an informal empire in China by sending troops to quell the Boxer Rebellion in 1900. The US, along with other nations, sent troops to China in order to ensure that the country remained open to trade.
China lost the Sino-Japanese War (1894–1895) to Japan, and that defeat highlighted China’s weakness to the rest of the world. Many European nations occupied parts of China and created “spheres of influence” in order to maintain control of trade between China and their home country. For example, British troops occupied Hong Kong in order to guarantee that trade between China and Great Britain remained intact. The US Secretary of State, John Hay, issued the “Open Door Note,” in which he called for all nations that had a “sphere of influence” in China to respect the rights of other nations and the rights of Chinese nationals. However, few nations adhered to the Open Door Policy, which created resentment and hostility towards Europeans from many Chinese citizens. As a result, Chinese nationals “rebelled” against the Europeans in China in what became known as the Boxer Rebellion. The rebellion was crushed within a matter of weeks, and the European nations (along with the US and Japan) continued to control “spheres of influence” in China.
The US established its “empire” in China by controlling a specific region’s trade and using military force to guarantee that it remained in control of that area. By controlling parts of China during this time, the US was merely adding to its overseas claims, which included Cuba, the Philippines, Guam, Samoa, and Puerto Rico.