  Our principal result is the following.
  Let $X$ and $Y$ be Banach spaces, let $G$ be a locally compact abelian group,
and let $K$ be an operator valued kernel defined on $G$ with values in the
space of bounded linear operators from $X$ to $Y$. Suppose that $R$ and
$\tilde{R}$ are representations of $G$ on $X$ and $Y$ respectively that
intertwine the values of $K$. Then, under suitable boundedness conditions on
$R, \tilde{R}$ and $K$, the formula $$T_Kx = \int_GK(u)R_{-u}xdu $$ defines a
bounded linear operator $T_K$ from $X$ to $Y$ with norm controlled by norm of
convolution by $K$ as a mapping from $L^p_X(G)$ into $L^p_Y(G)$, (for all
values of $p$ in the range $1\le p < \infty$.)
  A number of applications to the geometry of Banach spaces are given. Several
results are proved in the setting of abstract commutative harmonic analysis. We
outline the proof of the affirmative resolution of a conjecture of Rubio de
Francia. This technique of transference is used to obtain dimension free
estimates for certain operators in an $\Bbb{R}^n$ setting.
