Ernest Ka'ai was the Ukuleles first superstar. He is most famous for his playing and teaching, (he wrote the first how to play the Ukulele book in 1906 and invented a lot of the notations and chord diagrams that are seen today), and in 1909 he started the Ernest Ka'ai Manufacturing company making Ukuleles. 

In 1917 he sold the company and bought shares in the Aloha Ukulele Manufacturing co. but the Ernest Ka'ai Mfg. Co. factory carried on without him for another three years. 

I have seen Ukuleles that don't have a Spanish heel and wooden round bodied models both of which look Mainland made but bearing the Ka'ai headstock logo? As far as I can find out they were made in the Ka'ai factory in Hawaii. I have also seen that Ka'ai distributed Chris Knutsen's Harp Ukuleles and Slide Guitars in Hawaii. 

Ka'ai left Hawaii in 1923 and though he returned in 1937 he left again in 1941 to live in Miami where he died in 1962

What I don't believe is the case though, is that the Larson Brothers, had anything to do with the Ka'ai enterprise. They spent all of their working lives in Chicago and though famous for Guitars and Mandolins don't seem to have made many Ukuleles. I also think the numbers after "Special" on the sound hole labels is a serial number and not a date.