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Problem

Given a C*-algebra $\mathcal{A}$.

Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$.
(More precisely, strongly continuous one-parameter groups.)

Denote their derivations by $\delta:\mathcal{D}\to\mathcal{A}$ and $\delta':\mathcal{D}'\to\mathcal{A}$.

Then one has: $$\delta=\delta'\implies\tau=\tau'$$ (Here, equality is meant in terms of operators resp. maps.)

How do I check this?

For dynamics over Hilbert spaces I would proceed by something along: $$i\frac{\mathrm{d}}{\mathrm{d}t}\|\varphi(t)\|^2=\langle H\varphi(t),\varphi(t)\rangle-\langle\varphi(t),H\varphi(t)\rangle=0$$ But for the C*-algebra case this path is not directly available.

Disclaimer

I hope to get a hint from here.
(I haven't got any respond yet from stack exchange.)

Problem

Given a C*-algebra $\mathcal{A}$.

Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$.
(More precisely, strongly continuous one-parameter groups.)

Denote their derivations by $\delta:\mathcal{D}\to\mathcal{A}$ and $\delta':\mathcal{D}'\to\mathcal{A}$.

Then one has: $$\delta=\delta'\implies\tau=\tau'$$ (Here, equality is meant in terms of operators resp. maps.)

How do I check this?

For dynamics over Hilbert spaces I would proceed by something along: $$i\frac{\mathrm{d}}{\mathrm{d}t}\|\varphi(t)\|^2=\langle H\varphi(t),\varphi(t)\rangle-\langle\varphi(t),H\varphi(t)\rangle=0$$

Disclaimer

I hope to get a hint from here.
(I haven't got any respond yet from stack exchange.)

Problem

Given a C*-algebra $\mathcal{A}$.

Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$.
(More precisely, strongly continuous one-parameter groups.)

Denote their derivations by $\delta:\mathcal{D}\to\mathcal{A}$ and $\delta':\mathcal{D}'\to\mathcal{A}$.

Then one has: $$\delta=\delta'\implies\tau=\tau'$$ (Here, equality is meant in terms of operators resp. maps.)

How do I check this?

For dynamics over Hilbert spaces I would proceed by: $$i\frac{\mathrm{d}}{\mathrm{d}t}\|\varphi(t)\|^2=\langle H\varphi(t),\varphi(t)\rangle-\langle\varphi(t),H\varphi(t)\rangle=0$$ But for the C*-algebra case this path is not directly available.

Disclaimer

I hope to get a hint from here.
(I haven't got any respond yet from stack exchange.)

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Source Link

Problem

Given a C*-algebra $\mathcal{A}$.

Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$.
(More precisely, strongly continuous one-parameter groups of automorphisms.)

Denote their derivations by $\delta:\mathcal{D}\to\mathcal{A}$ and $\delta':\mathcal{D}'\to\mathcal{A}$.

Then one has: $$\delta=\delta'\implies\tau=\tau'$$ (Here, equality is meant in terms of operators resp. maps.)

How do I check this?

For dynamics over Hilbert spaces I would proceed by something along: $$i\frac{\mathrm{d}}{\mathrm{d}t}\|\varphi(t)\|^2=\langle H\varphi(t),\varphi(t)\rangle-\langle\varphi(t),H\varphi(t)\rangle=0$$

Disclaimer

I hope to get a hint from here.
(I haven't got any respond yet from stack exchange.)

Problem

Given a C*-algebra $\mathcal{A}$.

Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$.
(More precisely, strongly continuous one-parameter groups of automorphisms.)

Denote their derivations by $\delta:\mathcal{D}\to\mathcal{A}$ and $\delta':\mathcal{D}'\to\mathcal{A}$.

Then one has: $$\delta=\delta'\implies\tau=\tau'$$ (Here, equality is meant in terms of operators resp. maps.)

How do I check this?

For dynamics over Hilbert spaces I would proceed by something along: $$i\frac{\mathrm{d}}{\mathrm{d}t}\|\varphi(t)\|^2=\langle H\varphi(t),\varphi(t)\rangle-\langle\varphi(t),H\varphi(t)\rangle=0$$

Disclaimer

I hope to get a hint from here.
(I haven't got any respond yet from stack exchange.)

Problem

Given a C*-algebra $\mathcal{A}$.

Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$.
(More precisely, strongly continuous one-parameter groups.)

Denote their derivations by $\delta:\mathcal{D}\to\mathcal{A}$ and $\delta':\mathcal{D}'\to\mathcal{A}$.

Then one has: $$\delta=\delta'\implies\tau=\tau'$$ (Here, equality is meant in terms of operators resp. maps.)

How do I check this?

For dynamics over Hilbert spaces I would proceed by something along: $$i\frac{\mathrm{d}}{\mathrm{d}t}\|\varphi(t)\|^2=\langle H\varphi(t),\varphi(t)\rangle-\langle\varphi(t),H\varphi(t)\rangle=0$$

Disclaimer

I hope to get a hint from here.
(I haven't got any respond yet from stack exchange.)

Source Link

C*-Algebras: Dynamics vs. Derivations

Problem

Given a C*-algebra $\mathcal{A}$.

Consider dynamics $\tau:\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$ and $\tau':\mathbb{R}\to\mathrm{Aut}(\mathcal{A})$.
(More precisely, strongly continuous one-parameter groups of automorphisms.)

Denote their derivations by $\delta:\mathcal{D}\to\mathcal{A}$ and $\delta':\mathcal{D}'\to\mathcal{A}$.

Then one has: $$\delta=\delta'\implies\tau=\tau'$$ (Here, equality is meant in terms of operators resp. maps.)

How do I check this?

For dynamics over Hilbert spaces I would proceed by something along: $$i\frac{\mathrm{d}}{\mathrm{d}t}\|\varphi(t)\|^2=\langle H\varphi(t),\varphi(t)\rangle-\langle\varphi(t),H\varphi(t)\rangle=0$$

Disclaimer

I hope to get a hint from here.
(I haven't got any respond yet from stack exchange.)