  We introduce a class of non-commutative Heisenberg like infinite dimensional
Lie groups based on an abstract Wiener space. The Ricci curvature tensor for
these groups is computed and shown to be bounded. Brownian motion and the
corresponding heat kernel measures, $\{\nu_t\}_{t>0},$ are also studied. We
show that these heat kernel measures admit: 1) Gaussian like upper bounds, 2)
Cameron-Martin type quasi-invariance results, 3) good $L^p$ -- bounds on the
corresponding Radon-Nykodim derivatives, 4) integration by parts formulas, and
5) logarithmic Sobolev inequalities. The last three results heavily rely on the
boundedness of the Ricci tensor.
