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A little wish...: is there a theorem that corresponds or implies the following

Let $A$ be a $C^*$ algebra with the data of a representation in $B(H)$. Let $x,y$ be two vectors and call $S(x,y)$ the set of states induced by vectors of $P(span(x,y))$, where $P$ stands for projective space.

The map from $P(span(x,y))$ to $S(x,y)$ is injective iff $x$ and $y$ belong to the unitarily equivalent irreducible subrepresentation of $A$.

or maybe it is not a "iff" but only one direction.

I want to write such a result as a motivation to introduce superselection sector as irreducible representation of the algebra of observables in physics.

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  • $\begingroup$ a little cue: The motivation comes from physics. I abstracted the prototype example of something that is not observable: the superposition of the wavefunction of a fermion and a boson, the "relative phase" of the combination is not obvervable. $\endgroup$
    – Noix07
    Commented Oct 1, 2013 at 13:56
  • $\begingroup$ Finally it is just awkward reformulation of: two linearly independant vectors from an irreducible representation of a $C^*$ algebra induce different states. $\endgroup$
    – Noix07
    Commented Oct 7, 2013 at 17:07

1 Answer 1

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Given a vector, a state is defined as

$$\omega_{\lambda\mathbf{x}+\mu\mathbf{y}}: A\ni a\mapsto \langle \lambda\mathbf{x}+\mu\mathbf{y}|\pi(a)|\lambda\mathbf{x}+\mu\mathbf{y}\rangle / \lVert \lambda\mathbf{x}+\mu\mathbf{y} \rVert^2\\ =\lvert \lambda\rvert^2 \omega_{\mathbf{x}}+ \lvert \mu\rvert^2 \omega_{\mathbf{y}} + \lambda^* \mu \langle\mathbf{x}|\pi(a) |\mathbf{y}\rangle + \lambda \mu^*\langle\mathbf{y}|\pi(a) |\mathbf{x}\rangle $$

Assuming the representation decomposes into direct sum (a sufficient condition: semi-simple algebra, are there other conditions?) one sees that the two last term vanish if x,y are in different irreducible subrepresentations. And hence one cannot observe the argument of $\lambda^* \mu$

There is not to worry about unitarily equivalent irreducible representation of anything. The usual situation is that we have a direct sum of irreducible representation of the gauge group (first kind) and in that case the algebra can contain intertwiners, it is not the case here.

However that is not a proof, we've just seen that we definitely don't have an injective map from the projective plane to the subset of states, if $\mathbf{x}$ and $\mathbf{y}$ are in different representations.

A proof can be found in Dixmier's book on $C^*$ algebra, proposition 2.5.6

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