On masas of the Calkin algebra generated by projections

TL;DR

Under the continuum hypothesis, the paper classifies maximal abelian self-adjoint subalgebras (masas) of the Calkin algebra generated by projections, revealing new types with unexpected properties.

Contribution

It provides a complete classification of certain masas in the Calkin algebra under CH and constructs new examples with properties unlike previously known types.

Findings

Constructed masas isomorphic to C(K) for specific compact spaces K.

Showed existence of many non-isomorphic masas without additional set-theoretic assumptions.

Demonstrated that some set-theoretic hypotheses are necessary for such classifications.

Abstract

Assuming the continuum hypothesis CH, we obtain complete $*$-isomorphic classification of maximal abelian self-adjoint subalgebras (masas) of the Calkin algebra $\mathcal Q(\ell_2)$ (bounded operators on a separable Hilbert space modulo compact operators) generated by projections. In particular, for any compact totally disconnected Hausdorff space $K$ of weight not exceeding the continuum and not admitting $G_\delta$ points we construct under CH a masa of $\mathcal Q(\ell_2)$ which is $*$-isomorphic to the algebra $C(K)$ of complex-valued continuous functions on $K$. This, among others, shows that masas of the Calkin algebra 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 $\ell_\infty/c_0\oplus L_\infty$. It can be shown that some additional set-theoretic hypothesis, like CH, is necessary for such results. However, without making any additional set-theoretic assumptions we still 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 and with properties unlike the three above examples.