  We study the infinite spatial dimensionality limit of the recently developed
Extremely Correlated Fermi Liquid (ECFL) theory for the t-J model. We directly
analyze the Schwinger equations of motion for the Gutzwiller projected electron
Green's function. From simplifications arising in this limit, we are able to
make several exact statements about the theory. The ECFL Green's function is
shown to have a momentum independent Dyson (Mori) self energy. For practical
calculations we introduce a partial projection parameter \lambda, and obtain
the complete set of ECFL integral equations to second order in \lambda. In a
related publication, these equations are compared in detail with the dynamical
mean field theory for the large U Hubbard model. Paralleling the well known
mapping for the Hubbard model, we find that the infinite dimensional t-J model
can be mapped to the infinite-U Anderson impurity model with a
self-consistently determined set of parameters. This mapping extends
individually to the auxiliary Green's function and the caparison factor of the
ECFL theory. Additionally, the optical conductivity is shown to be obtainable
from the Green's function with negligibly small vertex corrections. These
results are shown to hold to each order in \lambda.
