On almost complex structures on abstract Wiener spaces (Q1917028)

Let \((B, H, \mu)\) be a real abstract Wiener space; \(B\) is a real separable Banach space, \(H\) is a real separable Banach space imbedded in \(B\) continuously and densely, and \(\mu\) is a Gaussian measure on \(B\). From a geometrical point of view based on the Malliavin calculus, the space \(H\) is thought of as a tangent space of \(H\). An isometry \(J : H \to H\) is said to be an almost complex structure if \(J^2 = -\text{Id}_H\), following the notion on finite-dimensional manifolds. A Newlander-Nirenberg type theorem on the almost complex structure on \(B\) is established; \(J\) admits a natural complex abstract Wiener space \((B_J, H_J, \mu_J)\) and \(B_J\) can be identified with \(B\) in a measure theoretical sense or an exact sense under a suitable condition on \(J\) as a mapping of \(B\) to \(B\). A set function which can distinguish if a point is in \(H\) or not is also introduced on an abstract Wiener space with a nice almost complex structure, and an Egorov theorem is shown with respect to the set function.