  In any dimension, the positive level generators of the very-extended
Kac-Moody algebra $E_{11}$ with completely antisymmetric spacetime indices are
associated to the form fields of the corresponding maximal supergravity. We
consider the local $E_{11}$ algebra, that is the algebra obtained enlarging
these generators of $E_{11}$ in such a way that the global $E_{11}$ symmetries
are promoted to gauge symmetries. These are the gauge symmetries of the
corresponding massless maximal supergravity. We show the existence of a new
type of deformation of the local $E_{11}$ algebra, which corresponds to the
gauging of the symmetry under rescaling of the fields. In particular, we show
how the gauged IIA theory of Howe, Lambert and West is obtained from an
eleven-dimensional group element that only depends on the eleventh coordinate
via a linear rescaling. We then show how this results in ten dimensions in a
deformed local $E_{11}$ algebra of a new type.
