On masas of the Calkin algebra generated by projections
Authors
Piotr Koszmider
Abstract
First, assuming the continuum hypothesis CH, for every compact totally disconnected Hausdorff space $K$ of weight not exceeding the continuum and without $G_δ$ points, we construct a masa of the Calkin algebra $\mathcal Q(\ell_2)$ which is $*$-isomorphic to the algebra $C(K)$ of complex-valued continuous functions on $K$. This is sharp in two ways: (1) there cannot be other $*$-isomorphic types of masas of $\mathcal Q(\ell_2)$ generated by projections and so, this result gives a complete $*$-isomorphic classification of masas of $\mathcal Q(\ell_2)$ generated by projections, (2) some additional set-theoretic hypothesis, like CH, is necessary to have all these C*-algebras as masas of $\mathcal Q(\ell_2)$.
This shows that masas of the Calkin algebras could have rather unexpected properties compared to the previously known three $*$-isomorphic types of them generated by projections: $\ell_\infty/c_0$, $L_\infty$ and $\ell_\infty/c_0\oplus L_\infty$.
Secondly, without making any additional set-theoretic assumptions we construct a family of maximal possible cardinality (of the power set of $\mathbb R$) of pairwise non-$*$-isomorphic masas of $\mathcal Q(\ell_2)$ generated by projections which (a) are not SAW*-algebras unlike the liftable masas (Gelfand spaces in this group of our masas are not $F$-spaces) (b) do not admit conditional expectations. This improves the results which required additional set-theoretic hypotheses to construct a single masa of $\mathcal Q(\ell_2)$ generated by projections without a commutative lift.
