Suppose that $A$ is a finite-dimensional commutative $\mathbb{C}$-algebra, and one has two analytic functions $f,g: A\rightarrow A$. Then is it true that if $f$ and $g$ coincide $\mathbb{R}\subseteq\mathbb{C}^N\cong A$, then they must be identical on all of $A$?
This came up when I was reading pages 151-152 of the book Heat Kernels and Dirac Operators, where the algebra $A$ is the (commutative) algebra of even forms on a manifold and it is claimed (top of page 152) that the $A$-analyticity of the functions $f = dp_t/dt$ and $g = -Hp_t$ means that to check their equality it is sufficient to prove that they coincide as functions on the reals. (Here $dp_t/dt$ and $-Hp_t$ are somewhat complicated polynomials of differential forms.)
This step also seems to appear at the top of page 158 of John Roe's book (available freely at http://www.maths.ed.ac.uk/~aar/papers/roeindex.pdf).
If this is indeed true (or if there is another reason for this "analytic continuation" step), any explanations or references would be greatly appreciated.
For anyone interested, here is the relevant reference that answers this question, page 29 of Liviu Nicolaescu's excellent notes:
http://www3.nd.edu/~lnicolae/ind-thm.pdf
It isn't as simple as the identity-type theorem I thought might exist, but applies to power series, in particular polynomials, and answers precisely the question that is needed to make sense of the "analytic continuation" in the heat symbol proof above.