\(K\)-theory for real \(C^*\)-algebras via unitary elements with symmetries (Q323978)

\(K\)-theory is a fundamental tool in the study of complex \(C^*\)-algebras. The real variant of \(K\)-theory is much less studied, but has recently received new interest because of applications in the theory of topological insulators and in string theory. By Bott periodicity, complex \(K\)-theory consists of two different groups and real \(K\)-theory of eight different groups. The complex \(K\)-groups may be defined through projections and unitaries, respectively. The same is true for the zeroth and first real \(K\)-groups. This article develops similar pictures for the higher real \(K\)-groups: they are all described as groups of unitaries in stabilisations that satisfy certain equations that involve the involution, the real structure, and certain real structures on matrix algebras. In addition, the boundary maps for long exact sequences are described explicitly in this picture. The formulas are similar to the usual ones for complex \(K\)-theory. Some low-dimensional examples are computed explicitly, including formulas for the generators of the real \(K\)-groups in question.NEWLINENEWLINEThe first important step is to describe the \(i\)th real \(K\)-group \(KO_i(B)\) of a real \(C^*\)-algebra \(B\) as the inductive limit of \([A_i, M_n(B)]\) for \(n\to\infty\), for certain real \(C^*\)-algebras \(A_i\). Here, \([,]\) denotes the set of homotopy classes of \(*\)-homomorphisms. For this purpose, the notions of homotopy symmetric \(C^*\)-algebras and semiprojective \(C^*\)-algebras are carried over to the real context, and some results about these notions are extended to real \(C^*\)-algebras. The \(C^*\)-algebras \(A_i\) are shown to be homotopy symmetric and semiprojective. This implies that the inductive limit of \([A_i,M_n(B)]\) gives the \(E\)-theory \(E(A_i,B)\). This is identified with \(KO_i(B)\) by the UCT for real \(C^*\)-algebras.