Eric Forgy Noncommutative Stochastic Calculus

Given any first order calculus $(d,\Omega^1(A))$ on a commutative algebra $A$ , we have

d(f g) = d(g f)\implies [d f,g] = [d g,f].

According to

H. C. Baehr, A. Dimakis, F. Müller-Hoissen, Differential Calculi on Commutative Algebras,

this is enough to show that the commutator is a symmetric $A$ -bilinear function of $d f$ and $d g$ , i.e.

[d f,g] = C(d f,d g).

The symmetric $A$ -bilinear map $C$ extends to a product

C:\Omega^1(A)\times\Omega^1(A)\to\Omega^1(A),

since $\Omega^1(A)$ is generated by elements of the form $a(d b)$ for $a,b\in A$ . We will denote this product

\alpha\bullet\beta \coloneqq C(\alpha,\beta).

This allows us to write the product rule in “left component form” as

d(f g) = f(d g) + g(d f) + d f\bullet d g,

in “right component form” as

d(f g) = (d g)f + (d f)g - d f\bullet d g,

or the “symmetrized form” as

d(f g) = \frac{1}{2}\left[f(d g) + (d g)f\right] + \frac{1}{2}\left[g(d f) + (d f)g\right].

Created on January 4, 2010 at 15:23:35 by Eric Forgy