Does the identity $\cos^2(x)+\sin^2(x)=1$ hold in a unital Banach algebra where $1$ is the unit?
Solution 1:
The Riesz functional calculus map $f\mapsto f(x)$ is an algebra homomorphism. So for any element $x$ in a unital Banach algebra and $f,g$ analytic functions on some open subset containing $\sigma(x)$ we have $(f\cdot g)(x) = f(x) g(x)$ and $(f+g)(x) = f(x) + g(x)$. So if $f^2+g^2 =1$, we will have $f^2(x) + g^2(x) = 1(x)$. Furthermore $1(x)$ is equal to the identity of the Banach algebra. So indeed, the Pythagorean trigonometric identity will still hold.