Infinite-dimensional calculus under weak spatial regularity of the processes

Two generalizations of It{\^o} formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations, when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an It{\^o} formula in a special class of Banach spaces, having a product structure with the noise in a Hilbertian component; again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by path-dependent It{\^o} calculus.