A non-commutative model for higher twisted \(K\)-theory (Q2797309)

The possible twists of \(K\)-theory of a space are classified by a generalized cohomology theory denoted by \(gl_1(KU)\). All these twisted \(K\)-theory and corresponding twisted \(K\)-homology groups have been defined using tools from homotopy theory and the theory of \(\infty\)-categories. For those twisted \(K\)-theory groups coming from a twist in \(H^3(X,\mathbb Z)\), an operator-algebraic description is known: it is the \(K\)-theory of a locally trivial bundle of \(C^*\)-algebras over \(X\) with the fibre \(\mathbb K(\ell^2\mathbb N)\), the compact operators on a separable Hilbert space. The twists coming from \(H^3(X,\mathbb Z)\) are only a rather small subset of the possible twists.NEWLINENEWLINEThis article describes general twists of \(K\)-theory and of \(K\)-theory with coefficients through \(C^*\)-algebras and their bivariant \(K\)-theory. The main point is to replace the \(C^*\)-algebra of compact operators by a general strongly self-absorbing \(C^*\)-algebra \(D\). Roughly speaking, the \(K\)-theory of such a bundle gives a general twisted \(K\)-theory of the underlying space. Bivariant \(K\)-theory allows to describe twisted \(K\)-homology in this fashion as well. The operator-algebraic definitions of twisted \(K\)-theory and \(K\)-homology are shown to be equivalent to previous definitions from homotopy theory.NEWLINENEWLINEThe starting point of the construction is a symmetric ring spectrum \(KU^D\) and a module spectrum over it, which carries a natural action of the automorphism group of \(D\otimes \mathbb K(\ell^2\mathbb N)\) and which has the same homotopy type as the ring spectrum \(KU^D\). Both are constructed by putting the \(C^*\)-algebra \(D\) in appropriate places into a previous construction of a symmetric \(K\)-theory spectrum, which is based on \textit{J. Trout}'s definition of \(K\)-theory for \(\mathbb Z/2\)-graded \(C^*\)-algebras [Ill. J. Math. 44, No.~2, 294--309 (2000; Zbl 0953.19002)].