  Riesz space (non-pointwise) generalizations for iterative processes are given
for the concepts of recurrence, first recurrence and conditional ergodicity.
Riesz space conditional versions of the Poincar\'{e} Recurrence Theorem and the
Kac formula are developed. Under mild assumptions, it is shown that every
conditional expectation preserving process is conditionally ergodic with
respect to the conditional expectation generated by the Ces\`aro mean
associated with the iterates of the process. Applied to processes in
$L^1(\Omega,{\mathcal A},\mu)$, where $\mu$ is a probability measure, new
conditional versions of the above theorems are obtained.
