This is joint work with Mariusz Wodzicki. It was conjectured in 1978 that algebraic K-theory and topological K-theory coincide on the category of stable complex C∗-algebras. This conjecture was proved by Suslin and Wodzicki about 20 years ago. In this lecture we give another proof of the conjecture which may be applied to stable real C∗-algebras or Banach algebras like the algebra of compact operators in a separable real Hilbert space. We apply our new method to compute also Hermitian K-theory of a large class of operator algebras by a comparison theorem involving the topological analog.