Reputation
1,724
Next privilege 2,000 Rep.
Edit questions and answers
Badges
1 9 18
Impact
~20k people reached

  • 0 posts edited
  • 0 helpful flags
  • 129 votes cast
7h
comment Unifying (& “justifying”) the various definitions for differential operators
Ah, yes! $\phantom{*}$
7h
comment Unifying (& “justifying”) the various definitions for differential operators
You can add another definition: a linear operator $D:C^\infty(X)\to C^\infty(X)$ is a differential operator of order $n$ if for any sequence of functions $f_0,f_1,...,f_n\in C^\infty(X)$ the commutator with the operators of multiplication by $f_i$ vanishes: $[...[[D,f_0],f_1]...,f_n]=0$.
7h
comment Unifying (& “justifying”) the various definitions for differential operators
No, you don't need the topology on ${\mathcal O}_X$. You can look at the proof of J.Peetre's theorem in S.Helgason's book: amazon.com/Geometric-Invariant-Differential-Operators-Spherical/… He also gives an equivalent definition: the operator $D:C^\infty(X)\to C^\infty(X)$ must not extend the support of function: $\text{supp}Df\subseteq \text{supp}f$. And these operators indeed have locally finite order.
Feb
3
awarded  Good Question
Feb
3
revised Stone-Weierstrass theorem for holomorphic functions?
deleted 23 characters in body
Feb
3
revised Stone-Weierstrass theorem for holomorphic functions?
added 528 characters in body
Feb
2
revised Stone-Weierstrass theorem for holomorphic functions?
added 331 characters in body
Feb
2
comment Stone-Weierstrass theorem for holomorphic functions?
@AlexM. I edited. There is no need to take into account the stereotype theory, when you define topology on ${\mathcal C}^\infty(M)$. This is equivalent to your condition with vector fields $X_i$.
Feb
2
revised Stone-Weierstrass theorem for holomorphic functions?
I added the description of topology on ${\mathcal C}^\infty(M)$
Dec
11
comment Stone-Weierstrass theorem for holomorphic functions?
Simon, I don't understand you. Why meromorphic functions? You can consider a narrower class, the class of entire functions on ${\mathbb C}$, and even in this situation the problem is unsolved: if $A$ is a subalgebra in ${\mathcal O}({\mathbb C})$, nobody knows which conditions $A$ should satisfy for being dense in ${\mathcal O}({\mathbb C})$.
Dec
10
comment Stone-Weierstrass theorem for holomorphic functions?
Necessary, but I would be surprised if it were essential in the answer. I believe, there must be a much more strong condition instead of separating tangent vectors.
Dec
10
comment Stone-Weierstrass theorem for holomorphic functions?
David, actually, I would not expect that the criteria for ${\mathcal O}(M)$ and for ${\mathcal C}^\infty(M)$ will be similar. I would suppose that separating points will be their common part, but separating tangent vectors sounds strange for me.
Dec
10
comment Stone-Weierstrass theorem for holomorphic functions?
David, your example is interesting, but I think, you did not understand my question. First, I have no hypotheses on what the answer should be. Second, separating points does not mean the equality of spectra. For example, from the Stone-Weierstrass theorem it follows that the algebra $A$ of almost periodic functions is dense in ${\mathcal C}({\mathbb R})$, but $\text{Spec}(A)\cong\text{Bohr compactification of} \ {\mathbb R}\ne{\mathbb R}\cong\text{Spec}({\mathcal C}({\mathbb R}))$ (if $\text{Spec}$ is defined somewhat reasonably, say, if it means continuous involutive characters).
Dec
10
comment Stone-Weierstrass theorem for holomorphic functions?
Yes, of course!
Dec
10
comment Stone-Weierstrass theorem for holomorphic functions?
@SimonHenry, as far as I understand, even in the case of $M={\mathbb C}$ there is no answer.
Dec
10
revised Stone-Weierstrass theorem for holomorphic functions?
edited body
Dec
9
awarded  Nice Question
Dec
9
comment Stone-Weierstrass theorem for holomorphic functions?
Ah, I see. Maybe...
Dec
9
comment Stone-Weierstrass theorem for holomorphic functions?
Runge's theorem is about a concrete subalgebra $A$, the algebra of rational functions, but I am asking about an arbitrary subalgebra $A\subseteq{\mathcal O}(M)$. I don't know, maybe there were generalizations...
Dec
9
revised Stone-Weierstrass theorem for holomorphic functions?
added 1 character in body