On a generalization of the topological Brauer group

Abstract

In the present paper we propose some generalization of the topological Brauer group that includes higher homotopical information and contains the classical one as a direct summand. Our approach is based on some kind of bundle-like objects called ``lax algebra bundles'' that occupy an intermediate position between ``Morita bundle gerbes'' and matrix algebra bundles. The main results of the paper include the descripion of the homotopy type of their classifying space. The obtained results can be applied to the twisted $K$-theory because the lax algebra bundles are geometric representatives of the ``higher'' twists of topological $K$-theory that have finite order. v.2: major changes, especially in the second half of the paper v.3: to clarify the presentation the significant part of the text has been rewritten v.4: major changes, completely different methods comparing with previous versions v.5: major changes and corrections v.6: section 3 added v.7: the definition of equivalence of LABs fixed v.8: section 3 has been rewritten v.9: remark 3.2 and some explanation in subsection 3.3 have been added v.10 in this version we omit the UHF algebra approach; otherwise, correction and clarifications have been made, in subsection 4.3 an outline of the theory of modules over LABs has been added v.11: some corrections and clarifications v.12: theorem 6.2 added, minor corrections v.13: some additions (the most important are section 7 and remark 3.5) v.14: subsections 6.2-6.4 added v.15: subsection 6.5 added v.16: The appendix has been completely rewritten v.17: Some issues with simplicial sets and simplicial spaces in the Appendix were corrected