  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
