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edited Apr 13, 2017 at 12:58

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The projections $\mathcal{P}(\mathcal{C}(H))$ in the Calkin algebra have a similar property with the pair "on the other side", i.e. for every countable subset one can find a pair with exactly the same upper (or lower) bounds. This can be seen from the proof of Theorem 4.1 of my paper Filters in C*-algebras. Conversely, for any pair (or even countable subset) one can find an increasing sequence with exactly the same upper bounds, which applies more generally to the projections in any C*-algebra with real rank zero. As with Turing degrees, this can also be used to show that $\mathcal{P}(\mathcal{C}(H))$ is not a lattice. To see this first note that $\mathcal{P}(\mathcal{C}(H))$ has no $(\omega,1)$-gaps, by essentially the same argument used with $\mathscr{P}(\omega)/\mathrm{fin}$. Thus to show that $\mathcal{P}(\mathcal{C}(H))$ is not a lattice one simply needs to find a pair of projections that has a strictly increasing sequence with the same upper bounds, as done by Farah/Hadwin/Weaver (see this mathoverflow post this mathoverflow post)

The projections $\mathcal{P}(\mathcal{C}(H))$ in the Calkin algebra have a similar property with the pair "on the other side", i.e. for every countable subset one can find a pair with exactly the same upper (or lower) bounds. This can be seen from the proof of Theorem 4.1 of my paper Filters in C*-algebras. Conversely, for any pair (or even countable subset) one can find an increasing sequence with exactly the same upper bounds, which applies more generally to the projections in any C*-algebra with real rank zero. As with Turing degrees, this can also be used to show that $\mathcal{P}(\mathcal{C}(H))$ is not a lattice. To see this first note that $\mathcal{P}(\mathcal{C}(H))$ has no $(\omega,1)$-gaps, by essentially the same argument used with $\mathscr{P}(\omega)/\mathrm{fin}$. Thus to show that $\mathcal{P}(\mathcal{C}(H))$ is not a lattice one simply needs to find a pair of projections that has a strictly increasing sequence with the same upper bounds, as done by Farah/Hadwin/Weaver (see this mathoverflow post)

The projections $\mathcal{P}(\mathcal{C}(H))$ in the Calkin algebra have a similar property with the pair "on the other side", i.e. for every countable subset one can find a pair with exactly the same upper (or lower) bounds. This can be seen from the proof of Theorem 4.1 of my paper Filters in C*-algebras. Conversely, for any pair (or even countable subset) one can find an increasing sequence with exactly the same upper bounds, which applies more generally to the projections in any C*-algebra with real rank zero. As with Turing degrees, this can also be used to show that $\mathcal{P}(\mathcal{C}(H))$ is not a lattice. To see this first note that $\mathcal{P}(\mathcal{C}(H))$ has no $(\omega,1)$-gaps, by essentially the same argument used with $\mathscr{P}(\omega)/\mathrm{fin}$. Thus to show that $\mathcal{P}(\mathcal{C}(H))$ is not a lattice one simply needs to find a pair of projections that has a strictly increasing sequence with the same upper bounds, as done by Farah/Hadwin/Weaver (see this mathoverflow post)

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created Feb 26, 2016 at 15:02

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The projections $\mathcal{P}(\mathcal{C}(H))$ in the Calkin algebra have a similar property with the pair "on the other side", i.e. for every countable subset one can find a pair with exactly the same upper (or lower) bounds. This can be seen from the proof of Theorem 4.1 of my paper Filters in C*-algebras. Conversely, for any pair (or even countable subset) one can find an increasing sequence with exactly the same upper bounds, which applies more generally to the projections in any C*-algebra with real rank zero. As with Turing degrees, this can also be used to show that $\mathcal{P}(\mathcal{C}(H))$ is not a lattice. To see this first note that $\mathcal{P}(\mathcal{C}(H))$ has no $(\omega,1)$-gaps, by essentially the same argument used with $\mathscr{P}(\omega)/\mathrm{fin}$. Thus to show that $\mathcal{P}(\mathcal{C}(H))$ is not a lattice one simply needs to find a pair of projections that has a strictly increasing sequence with the same upper bounds, as done by Farah/Hadwin/Weaver (see this mathoverflow post)