"Spectra of maximal almost orthogonal families of projections in the Calkin algebra"Godziszewski, MichałLet $H$ be an infinite dimensional separable complex Hilbert space with inner product $\langle \cdot | \cdot \rangle$. Let $\mathcal{B}(H)$ be a Banach space of bounded linear operators on $H$ with the operator norm. In case when $H = \ell^2(\omega)$, we can distinguish a particular subalgebra of the Banach space $\mathcal{B}(H)$: we define $\mathcal{K}(H)$ as the smallest Banach subalgebra of $\mathcal{B}(H)$ containing all finite-dimensional operators, and we call its elements compact operators. So, $T \in \mathcal{B}(H)$ is compact if it is a limit of finite-rank operators.\footnote{Equivalently, an operator $T \in \mathcal{B}(H)$ is compact if the image of the closed unit ball $B \subset H$ under $T$ is precompact, which in turn is equivalent to $T$ being weak-norm continuous when restricted to $B$.} The collection $\mathcal{K}(H)$ has the structure of a $\mathrm{C}^\ast$-algebra and is a ring-theoretical ideal in $\mathcal{B}(H)$. The Calkin algebra is the quotient $\mathrm{C}^\ast$- algebra $$\mathcal{C}(H) = \mathcal{B}(H) / \mathcal{K}(H),$$ where the quotient mapping is denoted by $\pi: \mathcal{B}(H) \rightarrow \mathcal{C}(H).$ Every separable $\mathrm{C}^\ast$-algebra is isomorphic to a $\mathrm{C}^\ast$-subalgebra of the Calkin algebra. We are interested in the set of projections in the Calkin algebra, i.e., in the set: $$P(\mathcal{C}(H)) = \{ p \in \mathcal{C}(H): p = p^\ast = p^2\}.$$ For a set $A \subseteq \omega$, let $P_A$ be the projection onto $\ell^2(A) \subseteq \ell^2(\omega)$. The map $A \mapsto P_A$ embeds the Boolean algebra $\mathcal{P}(\omega)$ into the space of projections $P(H)$. The map $A \mapsto \pi(P_A)$ defines an embedding of $\mathcal{P}(\omega)/\setminus fin$ into $P(\mathcal{C}(H))$. This map is called the diagonal embedding. A family of projections $A \subseteq P(\mathcal{C}(H))$ is almost orthogonal if the product of any two elements $p, q \in A$ is the zero of the algebra $\mathcal{C}(H)$. In this paper we investigate the possible spectra of maximal almost orthogonal families of projections in the Calkin algebra. The collection of projections $P(\mathcal{C}(H))$ is a natural object to study, as it can be identified with the lattice of projections on $\mathcal{B}(H)$ modulo a natural equivalence relation, so we can identify elements of $P(\mathcal{C}(H))$ with closed subspaces of $\mathcal{B}(H)$. An important result by Wofsey 2007 states: $$\text{Let $A$ be a family of disjoint uncountable sets. Then $\mathbb{P}_A \Vdash \forall X \in A \: \exists Y (|Y| = |X| \: \& \: Y \text{ is a m.a.o.f. } ).$} $$ In other words, for any family of cardinals $C$ there is a forcing notion such that $C$ is included in the spectrum of m.a.o.f.'s. Wofsey's result is an operator-theoretic counterpart of the (positive) result of Hechler concerning spectra of maximal almosts disjoint families of sets. We have been searching for an operator-theoretic counterpart of the (negative) strengthening of Hechler's result on spectra of mad families given by Blass. Thus, our main question in this paper is: can we isolate conditions, under which a specific set of cardinals $C$ can be not only included, but actually equal to the spectrum of maximal almost orthogonal family of projections in a given model of set theory? $$ \:\: $$ Theorem. Assume $GCH$. Let $C$ be a set of cardinals satisfying the following conditions: 1) $\forall \kappa \in C \: \: \kappa$ is uncountable; 2) $C$ is closed; 3) $\forall \kappa \in [\aleph_1, |C|] \: \: \kappa \in C$; and 4) $\forall \kappa \in C \: cf(\kappa) = \omega \Rightarrow \kappa^+ \in C$. Then there exists a forcing notion $\mathbb{P}$ such that it satisfies the countable chain condition and forces the spectrum of maximal almost orthogonal families to be exactly $C$. $$ \:\: $$ This is joint work with Vera Fischer. |
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