Algebraic and Hermitian K-theory of K-rings

The main purpose of the present article is to establish the real case of "Karoubi's conjecture" in algebraic K-theory. The complex case was proved in 1990-91 by the second author and Andrei Suslin. Compared to the case of complex algebras, the real case poses additional difficulties. This is due to the fact that topological K-theory of real Banach algebras has period 8 instead of 2. The method we employ to overcome these difficulties can be used for complex algebras, and provides some simplifications to the original proofs. We also establish a natural analog of "Karoubi's conjecture" in Hermitian K-theory.