Infinite dimensional geometric algebras

How much is known about infinite dimensional geometric algebras? It is sometimes mentioned in some papers and references; yet, I am not convinced that the Poincare-Hodge complement is well defined. Some of the products, like the exterior and interior products should still work in infinite dimensions, since an infinite dimensional Hilbert space theory exists. However, this may not work for all products. Does anyone have definitions for a complement or regressive product in infinite dimensions?