I wounder if I ask you(Dear Mohamed El Naschie) to explain your answer

"cross product": a special subalgebra of R3: a(ijk)x(j)y(k) = z(i)

this construction works in all dimension, for example: 

det(x,a,b,c, ...):

a "cross product" of n-1 n-tupels a,b,c, ... resulting in projection on x (n-tuple)

then there are several extensions possible to R(n), n to infinity. 

(here: Hilbert-space as sequence of square summable numbers)

trivial example: this cross-product of the unit-vectors (0, ... , 0,1, 0 ...) is equal 1, as it should be.

When you define a cross product among vectors one gets two kinds of vectors: Polar vectors and axial vectors. Axial vectors are pseudo-vectors: they do not change sign upon inversion of the crossed vectors. 

Now in case of states if a cross product were to be defined then one would get pseudo-states too. Now I don't know what they are, they might probably even exist, just emphasizing the fact that a cross product involving states should take this into consideration.

@Mohamed: I didn't get what you were trying to say pertaining to the question, could you please answer in a concise form?