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Let X be a Banach Lattice Algebra and $X_+=\{f\in X: f>0\}$. Let $f:X_+\rightarrow X$ be continuously differentiable.

Question: When does the expression $\frac{f'(x)}{x}$ for $x\in X_+$, make sense? why is it allowed to divide by the positive element?
I saw such expression in an article, I want a justification for defining it.

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Adapting reasonable axioms for a Banach lattice algebra, it follows that $X_+$ is open. As long as $X$ is commutative, this means $f^\prime(x) \cdot x^{-1}$ and it does make perfect sense since $f$ is differentiable.

Here positive most likely means strictly positive and such element should be invertible (again, it is not clear what is a Banach lattice algebra).

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  • $\begingroup$ But if X is not particularly mentioned to be commutative, does it make sense? $\endgroup$ – Shinning Star Apr 12 '17 at 12:50
  • $\begingroup$ @ShinningStar does $\frac{x}{y}$ make sense for invertible matrices? $\endgroup$ – Tomek Kania Apr 12 '17 at 13:20
  • $\begingroup$ yes in that case $\frac{x}{y}=x.y^{-1}$ $\endgroup$ – Shinning Star Apr 12 '17 at 15:14
  • $\begingroup$ @ShinningStar why not $y^{-1}x$? $\endgroup$ – Tomek Kania Apr 12 '17 at 15:51
  • $\begingroup$ yes can be so. thank you so much. $\endgroup$ – Shinning Star Apr 13 '17 at 8:23

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