Bice's Question 7.7: is it consistent with ZFC that no tower of subsets of omega remains a tower of projections in the Calkin algebra?

Fix a separable infinite-dimensional Hilbert space $H$ with orthonormal basis $(e_n)$, let $\mathcal C(H)=\mathcal B(H)/\mathcal K(H)$ be the Calkin algebra (bounded operators modulo compact operators) with quotient map $\pi$, and for $X\subseteq\omega$ let $P_X$ be the diagonal projection onto $\overline{\operatorname{span}}\{e_n:n\in X\}$. A tower in $[\omega]^\omega$ is a $\subseteq^*$-descending reverse well-ordered chain of infinite sets with no pseudointersection. The map $X\mapsto\pi(P_X)$ embeds $\mathcal P(\omega)/\mathrm{Fin}$ into the projection lattice $\mathcal P(\mathcal C(H))$, and whether the image $\pi[P_{\mathcal X}]$ of a tower $\mathcal X$ is again a tower there is independent of ZFC in restricted forms: some towers die under extra hypotheses (CH, or the cardinal equality $\operatorname{non}(\mathcal M)=\sup_n\mathfrak m(\sigma\text{-}n\text{-linked})$), some tower survives whenever $\mathfrak b=\mathfrak t$, and in an iterated $\sigma$-centred forcing extension all towers survive. Question 7.7 of Bice's paper 'Filters in C*-Algebras' (2013) asks the mirror consistency question: is it relatively consistent with ZFC that $\pi[P_{\mathcal X}]$ is not a tower for every tower $\mathcal X\subseteq[\omega]^\omega$, i.e. that every classical tower admits a 'quantized pseudointersection' — an infinite-rank projection $Q$ with $\pi(Q)\le\pi(P_X)$ for all $X\in\mathcal X$? No resolution is known in the literature to date; the question is open.

Background

The Calkin algebra $\mathcal C(H)=\mathcal B(H)/\mathcal K(H)$ is the quotient of the algebra $\mathcal B(H)$ of all bounded linear operators on a separable infinite-dimensional Hilbert space $H$ by the closed ideal $\mathcal K(H)$ of compact operators (norm limits of finite-rank operators); write $\pi:\mathcal B(H)\to\mathcal C(H)$ for the quotient map. It is the prototypical corona (quotient) algebra in which set-theoretic independence phenomena of operator theory are studied. Fixing an orthonormal basis $(e_n)_{n\in\omega}$ of $H$, every set $X\subseteq\omega$ defines a diagonal projection $P_X$ onto $\overline{\operatorname{span}}\{e_n:n\in X\}$, and $X\mapsto\pi(P_X)$ embeds the Boolean algebra $\mathcal P(\omega)/\mathrm{Fin}$ of subsets of $\omega$ modulo finite sets into the lattice $\mathcal P(\mathcal C(H))$ of projections of the Calkin algebra — the diagonal embedding. In $\mathcal P(\mathcal C(H))$ the order is $\pi(P)\le\pi(Q)$ if and only if $P(1-Q)$ is compact, which for diagonal projections says exactly $X\subseteq^*Y$, where $\subseteq^*$ is almost inclusion ($X\setminus Y$ finite). Wofsey asked how much of the combinatorics of $\mathcal P(\omega)/\mathrm{Fin}$ survives this embedding, proposing 'quantized' analogues of the classical cardinal invariants of the continuum.

A tower in $[\omega]^\omega$ is a family $\mathcal X$ of infinite subsets of $\omega$ that is reverse well-ordered (well-ordered by the reverse order: every nonempty subfamily has a $\subseteq^*$-largest element, and there is no infinite strictly $\subseteq^*$-ascending sequence) and strictly descending modulo finite with respect to $\subseteq^*$, and has no pseudointersection, i.e. no infinite $X$ with $X\subseteq^*Y$ for every $Y\in\mathcal X$. The minimal length of a tower is the tower number $\mathfrak t$; one always has $\aleph_1\le\mathfrak t\le\mathfrak b\le\mathfrak c$, where $\mathfrak b$ is the bounding number (the least size of a family of functions $\omega\to\omega$ not all dominated by one function) and $\mathfrak c=2^{\aleph_0}$, and towers exist in ZFC. Bice's Definition 8.1(x) transfers the notion to any partially ordered set: a tower is a reverse well-ordered subset of the non-zero elements with no non-zero lower bound. Consequently, for $\mathcal X\subseteq[\omega]^\omega$ the image $\pi[P_{\mathcal X}]=\{\pi(P_X):X\in\mathcal X\}$ is a tower in $\mathcal P(\mathcal C(H))$ exactly when $\mathcal X$ is a tower in $[\omega]^\omega$ and, in addition, no infinite-rank projection $Q\in\mathcal B(H)$ (a projection whose range is infinite-dimensional, so that $\pi(Q)\ne 0$) satisfies $\pi(Q)\le\pi(P_X)$ for all $X\in\mathcal X$ — equivalently, $Q(1-P_X)$ is compact for every $X\in\mathcal X$. Such a $Q$ is a quantized pseudointersection of the tower: its range is almost contained in every $X\in\mathcal X$ in the sense of the Calkin order. A theorem of Steprāns, recorded in Bice's paper as Lemma 2.5.14 of Zamora-Avilés' thesis, shows that for every projection $P$ the family $\{X\subseteq\omega:\pi(P_X)\le\pi(P)\}$ is an analytic P-ideal — an ideal on $\omega$ closed under countable pseudo-unions (every countable subfamily is almost contained in a single member), which by Solecki's characterization has the form $\{X:\lim_n\varphi(X\setminus n)=0\}$ for a finite lower semicontinuous submeasure $\varphi$. Consequently a tower is captured by a projection $Q$ exactly when all its members lie in the dual filter of such an analytic P-ideal, namely $\{Y\subseteq\omega:\pi(P_{\omega\setminus Y})\le\pi(1-Q)\}$, which ties the question to the combinatorics of towers in filters.

The state of knowledge is as follows. Wofsey proved that under the Continuum Hypothesis (CH) some tower dies — there is a tower $\mathcal X$ for which $\pi[P_{\mathcal X}]$ is not a tower, built from an explicit 'thinness criterion' — and, dually, that towers added generically with finite conditions do survive; under Martin's Axiom (the assertion that for any poset without uncountable antichains and any family of fewer than $\mathfrak c$ dense subsets there is a filter meeting them all) the classical picture transfers to $\mathcal P(\mathcal C(H))$. Bice strengthened both directions. His Theorem 7.2 produces a dying tower under the cardinal equality $\operatorname{non}(\mathcal M)=\sup_n\mathfrak m(\sigma\text{-}n\text{-linked})$, where $\operatorname{non}(\mathcal M)$ is the least size of a non-meager set of reals and $\mathfrak m(\mathcal P)$ is the least number of dense subsets of a poset $\mathcal P$ for which a filter meeting them all need not exist, $\mathcal P$ ranging over countable unions of $n$-linked (every $n$ elements compatible) posets; the witnessing tower is bounded below by a block projection whose block sizes tend to infinity. His Theorem 7.3 produces a surviving tower whenever $\mathfrak b=\mathfrak t$. His Theorem 7.6 proves that in the canonical finite-support iteration of all $\sigma$-centred posets (posets that are countable unions of centred subsets, where a subset is centred if every finite subfamily has a common lower bound — a model of $\mathfrak m(\sigma\text{-centred})=\mathfrak c$, a weak form of Martin's Axiom) every tower of the extension survives. It follows that any model in which every tower dies must satisfy $\mathfrak b>\mathfrak t$ and cannot satisfy $\mathfrak m(\sigma\text{-centred})=\mathfrak c$. Bice describes the circle of questions as a possible 'tower analog of the Kadison–Singer conjecture' — his paper's Section 5 had given new equivalent formulations of Kadison–Singer, a classical 1959 problem about extensions of pure states (norm-one positive functionals that are not proper convex combinations) from the diagonal to all of $\mathcal B(H)$, settled positively in 2015 by Marcus, Spielman and Srivastava with no effect on the tower problem — and closes the section with the question below, which he states he cannot answer.

Problem Statement

Let $H=\ell^2(\omega)$ with standard orthonormal basis $(e_n)$, let $\mathcal C(H)=\mathcal B(H)/\mathcal K(H)$ be the Calkin algebra, and let $\pi:\mathcal B(H)\to\mathcal C(H)$ be the quotient map. For $X\subseteq\omega$ write $P_X$ for the projection onto $\overline{\operatorname{span}}\{e_n:n\in X\}$, and for $\mathcal X\subseteq[\omega]^\omega$ write $\pi[P_{\mathcal X}]=\{\pi(P_X):X\in\mathcal X\}$. Recall that a tower in $[\omega]^\omega$ is a family that is reverse well-ordered and strictly descending modulo finite with respect to almost inclusion $\subseteq^*$ ($X\subseteq^*Y$ meaning $X\setminus Y$ finite) and has no pseudointersection (no infinite $X_0\subseteq\omega$ with $X_0\subseteq^*Y$ for every $Y\in\mathcal X$), and that a tower in the projection lattice $\mathcal P(\mathcal C(H))$ is a reverse well-ordered set of non-zero projections with no non-zero lower bound, where $\pi(P)\le\pi(Q)$ if and only if $P(1-Q)$ is compact (for diagonal projections, exactly $X\subseteq^*Y$).

Question (Bice, Question 7.7): is it relatively consistent with ZFC that $\pi[P_{\mathcal X}]$ is not a tower in $\mathcal P(\mathcal C(H))$ for every tower $\mathcal X\subseteq[\omega]^\omega$? (The quantifier may be read over all $\mathcal X\subseteq[\omega]^\omega$ or only over towers; the two readings are equivalent, because $\pi[P_{\mathcal X}]$ can be a tower only if $\mathcal X$ is itself a tower modulo finite.) Equivalently: is it consistent that every $\subseteq^*$-tower $\mathcal X$ admits a quantized pseudointersection — an infinite-rank projection $Q$ (one with infinite-dimensional range, so $\pi(Q)\ne 0$) with $\pi(Q)\le\pi(P_X)$, i.e. $Q(1-P_X)$ compact, for all $X\in\mathcal X$?

A resolution in the opposite direction — a construction in ZFC of a single tower $\mathcal X$ for which $\pi[P_{\mathcal X}]$ is a tower — would answer the consistency question negatively; either answer settles it.

References

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