Properties of positive linear maps

Question

This is from the paper "A Note on Essentially Normal Operators" by Arveson. I am wondering how property number of 3 and 4 are come to be here. Property 1 is just by the theorem saying that it preserves identities, property 2 is because of the image of A being in the Calkin Algebra which are exactly the operators of the form $A+K$ with $K$ being compact (I think?). But I can't figure out how to compute the last 2 properties (3+4)?

Answer 1

The Calkin algebra is not what you say it is. Rather, by definition, it is the quotient of the algebra of bounded operators on $\mathscr{H}$, denoted $\mathscr{L}(\mathscr{H})$ in Arveson's notation, by the ideal of compact operators on $\mathscr{H}$, denoted $\mathscr{C}(\mathscr{H})$ in Arveson's notation.

As for the numbered statements, these are all just algebraic observations about quotients, more to do with abstract ring theory than anything operator theoretic, per se. Let $I$ be an ideal in a unital ring $R$ with $\pi : R \to R/I$ the canonical quotient map. Let $\phi:R/I\to R$ be a function (but not necessarily a ring homomorphism) satisfying $\pi \circ \phi = \mathrm{id}$ and preserving identities. Fix $A \in R$ and denote $z = \pi(A)$. Then,

1. $\phi(1)=1$,
2. $\phi(z) = A+K$, where $K \in I$,
3. $\phi(fg)-\phi(f)\phi(g) \in I$ for all $f,g \in R/I$,
4. $\phi(f) \in I$ if and only if $f=0$.

All of the statements 2-4 come down to the fact that $B \in I$ if and only if $\pi(B)=0$.