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I am reading the paper "UNITARY EQUIVALENCE MODULO THE COMPACT OPERATORS AND EXTENSIONS OF C*-ALGEBRAS" by BDF and I seem to have some misunderstandings.

In the followng I don't understand why in the last line it says that every operator in $E$ is essentially normal? I thought the decision if an operator is esentially normal is either if $\pi(T)$ is normal in the Calkin algebra (with $\pi$ being the quotient map) or $T^{*}T-TT^{*}$ is compact, but how does $\phi$ tell us anything about the essential normal property?

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Also the following part confuses me. It says in the end that $S-S'$ compact (on the same argument as before that $\phi(S-S')=0$) which I am not sure about since also the identity $I$ and some operator $T$ is part of $E$, not only the compact operators.

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$T^*T-TT^*$ is compact by design. Since $\phi$ is a $*$-homomorphism onto a commutative algebra, $$ \phi(T^*T-TT^*)=\phi(T)\phi(T^*)-\phi(T^*)\phi(T)=0. $$ Which means that $T^*T-TT^*\in\ker\phi=K (H)$.

For the second part, you get $S-S'\in \ker\phi=K (H)$ so $R=S-S'\in K (H)\subset E'$. Then $S=R+S'\in E'$.

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  • $\begingroup$ Ah yes Thank you! $\endgroup$
    – craaaft
    Jun 28, 2023 at 17:00

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