K-theory for banach *-algebras

There are two ways to define the K-0-group of a Banach *-algebra, one based on idempotents and algebraic equivalence and another based on projections and Murray-von Neumann equivalence. We show that these two K-0-groups are not in general isomorphic. There are also two ways to define the K-1-group of a Banach *-algebra, one based on homotopy of invertibles and another based on homotopy of unitaries. Again, we show that these two K-1-groups need not be isomorphic.