My brother and I sold a duplex. We received cash and a promissory note. I would like to by my brother's 50% of the promissory note. How do I do purchase it legally?

Write up a contract between you and him, and you pay him for his share. A promissory note is merely a contract to pay, normally secured with a lien when done on property. OF course you brother will want to sell his share, you can't buy it if he don't want to sell ti.