  The usual T-duality that relates the type IIA and IIB theories compactified
on circles of inversely-related radii does not operate if the dimensional
reduction is performed on the time direction rather than a spatial one. This
observation led to the recent proposal that there might exist two further
ten-dimensional theories, namely type IIA^* and type IIB^*, related to type IIB
and type IIA respectively by a timelike dimensional reduction. In this paper we
explore such dimensional reductions in cases where time is the coordinate of a
non-trivial U(1) fibre bundle. We focus in particular on situations where there
is an odd-dimensional anti-de Sitter spacetime AdS_{2n+1}, which can be
described as a U(1) bundle over \widetilde{CP}^n, a non-compact version of CP^n
corresponding to the coset manifold SU(n,1)/U(n). In particular, we study the
AdS_5\times S^5 and AdS_7\times S^4 solutions of type IIB supergravity and
eleven-dimensional supergravity. Applying a timelike Hopf T-duality
transformation to the former provides a new solution of the type IIA^* theory,
of the form \widetilde{CP}^2\times S^1\times S^5. We show how the Hopf-reduced
solutions provide further examples of ``supersymmetry without supersymmetry.''
We also present a detailed discussion of the geometrical structure of the
Hopf-fibred metric on AdS_{2n+1}, and its relation to the horospherical metric
that arises in the AdS/CFT correspondence.
