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I have come across the two following relations between subrepresentations and projections in two slightly different situations and want to clarify the differences:

  • Let $(\mathcal{H},\pi)$ be a representation of a $C^*$-algebra $\mathfrak{A}$.

    There is a one-to-one correspondence between (closed) subrepresentations $(\mathcal{K},\sigma)$ and projections $P_{\sigma}\in\pi(\mathfrak{A})^{'}$ on to the subrepresentations space.

  • Let $(\mathcal{H},\pi)$ be a normal representation of a von Neumann algebra $\mathfrak{M}$.

    $\mathrm{Ker}(\pi)$ is an ultraweakly closed two sided ideal in $\mathfrak{M}$, hence there exists a central projection $Q\in\mathfrak{M}$ such that $$ \mathrm{Ker}(\pi)= \mathfrak{M}Q = Q \mathfrak{M} $$ (cf. e.g. C*-algebras and W*-algebras, Sakai, Proposition 1.10.5 p.25)

For a $C^*$-algebra, one may consider the central projection in the universal envelopping von Neumann algebra but it is clearly not the same as the previous $P_{\sigma}$.

Is there however a link between the two situations and maybe even a link between the two projections, shtg like "the second is the central carrier/support of the first"?

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  • The central carrier/support of $P_{\sigma}$ projects onto the smallest subrepresentation that is also stable under $\pi(\mathfrak{A})'$ (whereas $P_{\sigma}\mathcal{H}$ is only stable under $\pi(\mathfrak{A})$)

Symmetrically, a projection $P$ that belongs to a concrete von Neumann algebra $\mathfrak{M}\subseteq \mathcal{B}(\mathcal{H})$ (e.g. $\pi(\mathfrak{A})''$) just projects on some random closed subspace (one says that "the von Neumann algebra knows about" that space which is furthermore stable under $\mathfrak{M}'$). The central carrier/support of $P$ projects on the smallest closed subspace that contains $P\mathcal{H}$ and is invariant under $\mathfrak{M}$. cf. for ex. "Von Neumann algebra", J. Dixmier (1981 edition), Corollary 1 bottom of p.6.

  • Recall that a von Neumann algebra is always unital: $P_{\sigma}$ is precisely the unit of $\sigma(\mathfrak{A})^{''}$. It is a central projection of the von Neumann algebra $\pi(\mathfrak{A})''$ but may actually not belong to the C*-algebra $\pi(\mathfrak{A})$.

  • This $P_{\sigma}$ does not correspond to the $Q$ (cf. question) but rather to $(1_{\mathfrak{M}}-Q)$, the unit of the complementary von Neumann algebra $I$ (which is s.t. $\mathfrak{M} =\mathrm{Ker}(\pi)\oplus I $, cf. this or Proposition 5.2.4 p.116 from Dixmier's "C∗-algebras".

Remark: what unconsciously surprised me is actually the fact that in one case one has a projection depending on the data of a Hilbert space in the first case but not in the second. This is related to the fact that the "commutant"requires a concrete representation while the "center" can be defined intrinsically.

(Related property to check: a representation selects a (two-sided) ideal, conversely, a representation of an ideal can be extended to a representation of the whole algebra, cf. "An invitation to C*-algebras" Arveson p.15 and the theorem p. 16)

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