I heard a shiur about it last year.
Basically, nobody gives a hechsher on the store itself. Regarding the kashrus of the utensils used, there is a machlokes between the OU and the CRC regarding it. Basically, the OU holds that the cleaning process doesn’t make the keilim treif, while the CRC holds that it does.
So, I guess in New York, you can go to Starbucks, and in Chicago you can’t.
Just to note, cleaning the utensils can be an issue because some of the items aren’t kosher, and all the utensils get washed together.