I am sorry if this question is elementary:
What is a precise definition of a contractible Banach algebra?
What is my mistake to think that there is a possible (conflict) contradictory situation in the following paper and "Wolfram" Note?:
1)Corollary 4.9 page 368 of the following paper:
http://www.ams.org/journals/tran/1962-102-02/S0002-9947-1962-0170219-7/S0002-9947-1962-0170219-7.pdf
2)Finite dimensionality of contractible $C^*$ algebras as described here: