This actually should operate on spaces and should create a new space.
Closure of a space under the relevant operator means that the space
contains the resultant space and that the operator at least makes sense
for the elements of the domain (meaning that the domain spaces are
subspaces of a larger space which is endowed with the relevant operator).
Commutativity depends on the relevant domain and operator.

The tensor product (te'oi'i) of vector spaces (producing a new tensor
space) is of this form.