Tensor algebras over Hilbert spaces. I

Abstract

operators. From a pure probability viewpoint the Clifford distribution relevant to the skew-symmetric tensors is closely related to the normal distribution relevant to the symmetric tensors. It is the only 'skew' distribution in which orthogonal manifolds of SC are stochastically independent, as the normal distribution is the only commutative one with this property. As in the case of the normal distribution, if x and y are vectors in a Hilbert space having independent Clifford distributions, then x + y again has the Clifford distribution. Of course, it is necessary to extend the notions of stochastic independence, conditional expectation, and integration itself to the generalized distributions that arise here. The relevant probabilistic concepts are introduced in the present paper, while the necessary integration theory is contained in [5].