Algebraic and Hermitian \(K\)-theory of \(\mathcal K\)-rings (Q2851206)

The first author developed \(K\)-theory for the category of Banach algebras denoted \(K^{\mathrm{top}}_*\) [Lect. Notes Math. 725, 254--290 (1979; Zbl 0444.46054)]. This theory is different from algebraic \(K\)-theory \(K_*\) of unital rings defined by Quillen. In 1978, Karoubi observed that both theories possess \(\mathbb{Z}\)-graded multiplicative structures that are associative, graded-commutative and compatible with the natural comparison map from \(K_*\) to \(K^{\mathrm{top}}_*\). He conjectured that the natural comparison map is an isomorphism. In the paper under review, the authors establish the real case of this conjecture. The complex case was proved by \textit{A. A. Suslin} and the second author [Ann. Math. (2) 136, No. 1, 51--122 (1992; Zbl 0756.18008)]. The real case poses more difficulties than the complex case due to the fact that topological \(K\)-theory of real Banach algebras is periodic of period eight instead of two. The authors also establish a similar result in the case of Hermitian \(K\)-theory.NEWLINENEWLINEIn an appendix to this paper [ibid. 64, No. 3, 941--945 (2013; Zbl 1283.46037)], \textit{T. Fack} provides a self-contained proof of the exactness of the maximal tensor product functor for real \(C^*\)-algebras.