  We introduce and solve a period-index problem for the Brauer group of a
topological space. The period-index problem is to relate the order of a class
in the Brauer group to the degrees of Azumaya algebras representing it. For any
space of dimension d, we give upper bounds on the index depending only on d and
the order of the class. By the Oka principle, this also solves the period-index
problem for the analytic Brauer group of any Stein space that has the homotopy
type of a finite CW-complex. Our methods use twisted topological K-theory,
which was first introduced by Donovan and Karoubi. We also study the cohomology
of the projective unitary groups to give cohomological obstructions to a class
being represented by an Azumaya algebra of degree n. Applying this to the
finite skeleta of the Eilenberg-MacLane space K(Z/l,2), where l is a prime, we
construct a sequence of spaces with an order l class in Br, but whose indices
tend to infinity.
