3.1 Borel ideals in $\mathcal {P}({\mathbb N})$

We identify sets of natural numbers with their characteristic function in $2^{\mathbb N}$ and equip $\mathcal {P}({\mathbb N})$ with the topology resulting from this identification with the Cantor space. Alternatively, this is the compact topology induced by the metric $d(A,B) = 1/(\min (A \Delta B) +1)$ . An ideal ${\mathcal {I}}$ of the Boolean algebra $\mathcal {P}({\mathbb N})$ is Borel (analytic, $F_\sigma $ , etc.) if it is Borel (analytic, $F_\sigma $ , etc.) with respect to this topology.

Every Borel ideal ${\mathcal {I}}$ in the Boolean algebra $\mathcal {P}({\mathbb N})$ provides us with a textbook example of a Borel quotient structure, $\mathcal {P}({\mathbb N})/{\mathcal {I}}$ . Reformulating Conjecture Template 2.6 in this setting leads to the following problem, which in its most general form is still open (more on this in Section 6.1.2 and Section 7.2).

The following is the proper definition of algebraically trivial automorphisms for Borel quotients of $\mathcal {P}({\mathbb N})$ .

Definition 3.2. Let $\mathcal {I}$ and $\mathcal {J}$ be Borel ideals in $\mathcal {P}({\mathbb N})$ . A homomorphism

$$\begin{align*}\Phi\colon\mathcal{P}({\mathbb N})/\mathcal{I}\to\mathcal{P}({\mathbb N})/\mathcal{J} \end{align*}$$

is said to be algebraically trivial if it has a lifting $\Phi _{*} \colon \mathcal {P}({\mathbb N})\to \mathcal {P}({\mathbb N})$ which is a Boolean algebra homomorphism.

Rigidity problems for such quotients can be traced back at least to [Reference Erdös and Mauldin49, pp. 38–39], where Erdös and Ulam asked whether the quotients over the two ideals, known as the asymptotic density zero ideal ${\mathcal {Z}}_0$ and the logarithmic density zero ideal ${\mathcal {Z}}_{\log }$ , are isomorphic, where

(3.1) $$ \begin{align} \kern-13pt {\mathcal{Z}}_0 &= \left\{ A \subseteq {\mathbb N} \mid \lim_{n \to \infty} \frac{| A \cap n |}{n} = 0\right\}, \end{align} $$

(3.2) $$ \begin{align} {\mathcal{Z}}_{\log} &= \left\{ A \subseteq {\mathbb N} \mid \lim_{n \to \infty} \frac{\sum_{i \in A \cap n } 1 /i}{\sum_{i < n} 1/i} = 0 \right\}. \end{align} $$

The unexpected (at the time) answer to this question will be discussed in Section 6.1.2 and Section 7.2, where we will have something interesting to say also for the classes of analytic P-ideals and of generalised density ideals, to which both ${\mathcal {Z}}_0$ and ${\mathcal {Z}}_{\log }$ belong, and which we briefly anticipate here.

Verifying that ${\mathcal {Z}}_0$ and ${\mathcal {Z}}_{\log }$ are Borel P-ideals is a nice exercise, to which the following definition provides a hint.

For a lower semicontinuous submeasure $\mu $ on ${\mathbb N}$ let

$$\begin{align*} \operatorname{\mathrm{Fin}}(\mu)=\{X\subseteq {\mathbb N}\mid \mu(X)<\infty\},\qquad\quad \end{align*}$$

$$\begin{align*}\operatorname{\mathrm{Exh}}(\mu)=\{X\subseteq {\mathbb N}\mid \lim_{n \to \infty} \mu(X\setminus n)=0\}. \end{align*}$$

The requirement that $\mu $ is finite on finite sets gives that both $\operatorname {\mathrm {Fin}}(\mu )$ and $\operatorname {\mathrm {Exh}}(\mu )$ contain the ideal $\operatorname {\mathrm {Fin}}$ .

Density ideals were introduced in [Reference Farah52, Definition 1.13.1], and it is not difficult to see that both ${\mathcal {Z}}_0$ and ${\mathcal {Z}}_{\log }$ are density ideals ([Reference Farah52, Theorem 1.13.3(a)]; for a fix to part (b) of this theorem see [Reference Tryba169]). We also have the following results of Mazur [Reference Mazur114] and Solecki [Reference Solecki161].

Theorem 3.6. An ideal is $F_\sigma $ if and only if it is of the form $\operatorname {\mathrm {Fin}}(\mu )$ for a lower semicontinuous submeasure $\mu $ . An analytic ideal is a P-ideal if and only if it is of the form $\operatorname {\mathrm {Exh}}(\mu )$ for a lower semicontinuous submeasure $\mu $ . In this case,

$$ \begin{align*} \textstyle d_\mu(A,B)=\lim_n \mu((A\Delta B)\setminus n) \end{align*} $$

is a complete metric on $\mathcal {P}({\mathbb N})/{\mathcal {I}}$ .

(The last line is the easy [Reference Farah52, Lemma 1.2.3].) Note that if ${\mathcal {I}}$ is not an $F_\sigma $ ideal, then there is a decreasing sequence of ${\mathcal {I}}$ -positive sets $A_n$ such that $\lim _n \mu (A_n)=0$ . This shows a failure of the countable saturation of $\mathcal {P}({\mathbb N})/{\mathcal {I}}$ as a discrete structure, and we shall see why this is relevant to us in Section 6.1.2.

An important problem in the framework of Borel quotients of $\mathcal {P}({\mathbb N})$ is whether topologically trivial homomorphisms are automatically algebraically trivial. Submeasures from Definition 3.4 will be used in Section 5 to give partial answers to Question 3.7 below, which in its most general form is still open (see Theorem 5.13 for partial results).

3.2 Čech–Stone remainders

Given a locally compact space X, its Čech–Stone compactification $\beta X$ is the ‘largest’ compact space in which X sits densely as an open set. $\beta X$ is defined as the space with the universal property that if K is a compact Hausdorff space and $f\colon X\to K$ is continuous, one can find a continuous extension $\beta f\colon \beta X\to K$ .Footnote ⁹ The compact space $\beta X\setminus X$ is the Čech–Stone remainder (or the corona) of X. Čech–Stone remainders of zero-dimensional spaces and their homeomorphisms from a set theoretic point of view were analysed in [Reference Farah52] with the aid of Boolean algebraic methods.

We already mentioned how the Stone Duality relates $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ to ${\beta {\mathbb N} \setminus {\mathbb N}}$ . To study homeomorphisms of Čech–Stone remainders of other spaces, we need to take a closer look at the duality between homeomorphisms of remainders and homomorphisms of quotient algebras.

Fix a locally compact noncompact space X. When X is zero-dimensional, autohomeomorphisms of $\beta X\setminus X$ are completely determined by the image of clopen subsets. By identifying clopen sets with their characteristic functions, one endows the Boolean algebra of clopen subsets of X, $\mathrm {Clop}(X)$ , with the topology of pointwise convergence. If X is in addition second countable, this is a Polish topology. Since the Boolean algebra $\mathrm {Clop}(\beta X\setminus X)$ is isomorphic to the quotient of $\mathrm {Clop}(X)$ by the ideal of compact-open sets, $\mathrm {Clop}_c(X)$ , a continuous map from $\beta X\setminus X$ to $\beta Y\setminus Y$ corresponds to a Boolean algebra homomorphism from the Borel quotient structure $\mathrm {Clop}(Y)/\mathrm {Clop}_c(Y)$ to $\mathrm {Clop}(X)/\mathrm {Clop}_c(X)$ . As homeomorphisms of Čech–Stone remainders correspond to isomorphisms of Boolean algebra quotients, we have a notion of topologically trivial homeomorphism (where the induced isomorphism admits a Borel lifting), and one of algebraically trivial (admitting a lifting which is a Boolean algebra homomorphism).

For non zero-dimensional spaces, one cannot rely on any Boolean algebraic underlying structure, even though the remainder of $\mathbb R$ was studied using topological tools in [Reference Dow, Hart, Hart, Nagata and Vaughan42] and [Reference Hart89]. To study these objects, we enter the realm of $\mathrm {C}^{*}$ -algebras. The appropriate versions of Conjecture Template 2.6 and Question 3.7 will be stated in Section 4.2.

4.1 Preliminaries

There are essentially two ways to present $\mathrm {C}^{*}$ -algebras, an abstract one and a concrete one. We start with the latter.

Let H be a complex Hilbert space, that is, a complex Banach space equipped with a sesquilinear inner product $\langle \cdot , \cdot \rangle $ and a norm given by $\lVert \xi \rVert := \langle \xi , \xi \rangle ^{1/2}$ . Note that the theory of Hilbert spaces is $\kappa $ -categorical (in the context of continuous logic) for every infinite cardinal $\kappa $ , indeed every infinite-dimensional Hilbert space H with density character $\kappa $ is isometric to the space

$$\begin{align*}\ell_2(\kappa) := \{ (z_j) \in \mathbb C^\kappa\mid \textstyle\sum_j | z_j |^2 < \infty \}. \end{align*}$$

Let $\mathcal B(H)$ denote the set of all linear bounded operators from H to itself. It is equipped with the structure $(\mathcal B(H), \| \cdot \|, +, \{\lambda _z \}_{z \in \mathbb C},\cdot , ^{*})$ , where $+$ is the usual sum, $\lambda _z$ is the scalar multiplication by z, $\cdot $ is the composition of operators, $^{*}$ is the adjoint, and the operator norm $\| \cdot \|$ for $a \in \mathcal B(H)$ is defined as

$$\begin{align*}\| a \| := \sup_{\xi \in H} \frac{\| a(\xi) \|}{\| \xi \|} = \sup_{\| \xi \| \le 1} \|a(\xi) \|. \end{align*}$$

A concrete $\mathrm {C}^{*}$ -algebra is a norm-closed complex subalgebra of $\mathcal B(H)$ closed under $^{*}$ .

Alternatively, $\mathrm {C}^{*}$ -algebras can be defined by providing an axiomatisation in the following way. A $^{*}$ -Banach algebra is a tuple $(A, \| \cdot \|, +, \cdot , ^{*})$ such that $(A, +, \cdot )$ is a $\mathbb C$ -algebra, the tuple $(A, \| \cdot \|, +)$ is a complex Banach space, $^{*}$ is a linear involution, and the following conditions hold:

1. (4.1.1) $(ab)^{*} = b^{*} a^{*}$ for all $a,b \in A$ ,

2. (4.1.2) $\| ab \| \le \| a \| \| b \|$ for all $a,b \in A$ ,

3. (4.1.3) $\| a^{*} \| = \| a \|$ for all $a \in A$ .

A (abstract) $\mathrm {C}^{*}$ -algebra is a $^{*}$ -Banach algebra satisfying the following additional condition, known as the $\mathrm {C}^{*}$ -equality.

1. (4.1.4) $ \| a^{*}a \| = \| a \|^2$ for all $a \in A$ .

Albeit seemingly mild, the $\mathrm {C}^{*}$ -equality is a powerful condition that tightly connects the topological nature of $\mathrm {C}^{*}$ -algebras with their algebraic properties. It implies, for instance, that in a $\mathrm {C}^{*}$ -algebra the norm is completely determined by its algebraic structure, and therefore it is unique [Reference Farah61, Corollary 1.3.3]. It moreover entails that $^{*}$ -homomorphisms between $\mathrm {C}^{*}$ -algebras, namely linear multiplicative maps preserving the involution $^{*}$ (the relevant arrows in this category), are automatically continuous [Reference Farah61, Lemma 1.2.10].

By the Gelfand–Naimark–Segal (GNS) construction (see [Reference Farah61, Section 1.10]) every $\mathrm {C}^{*}$ -algebra A is isometrically isomorphic to a concrete $\mathrm {C}^{*}$ -algebra in some $\mathcal B(H)$ . Equivalently, the GNS-construction shows that concrete $\mathrm {C}^{*}$ -algebras are axiomatised by the conditions listed above, allowing us to identify the two definitions.

4.1.1 Examples I: Abelian $\mathrm {C}^{*}$ -algebras

One of the cornerstones in the theory of $\mathrm {C}^{*}$ -algebras is the Gelfand Representation theorem [Reference Farah61, Theorem 1.3.1], which states that every abelian $\mathrm {C}^{*}$ -algebra is isomorphic to one of the form $C_0(X)$ , for some locally compact X, where

$$\begin{align*}C_0(X)=\{f\colon X\to\mathbb C\mid f\text{ is continuous and vanishes at infinity}\} \end{align*}$$

is considered with the supremum norm and pointwise operations. The Gelfand Representation theorem establishes a natural equivalence between the category of unital abelian $\mathrm {C}^{*}$ -algebras and the category of compact Hausdorff spaces (e.g., [Reference Farah61, Theorem 1.3.2]). This duality, usually referred to as the Gelfand–Naimark duality, motivates the leading philosophy in the subject, namely that $\mathrm {C}^{*}$ -algebras provide the noncommutative analog of topological spaces. Indeed, many definitions, techniques, and results in $\mathrm {C}^{*}$ -algebras arise as adaptations of ideas that stem from topology.

In the special case where X is moreover compact zero-dimensional, the Gelfand–Naimark duality can be combined with the Stone duality, allowing to interpret Boolean algebras as a particular instance of abelian $\mathrm {C}^{*}$ -algebras, as hinted at the beginning of Section 3.

If X is a locally compact space, consider the $\mathrm {C}^{*}$ -algebras

$$\begin{align*}C_b(X)=\{f\colon X\to\mathbb C\mid f\text{ is continuous and bounded}\} \end{align*}$$

with the sup norm and pointwise operations. Then, $C_0(X)$ is a norm-closed ideal in $C_b(X)$ ; furthermore, $C_b(X)$ is isomorphic to $C(\beta X)$ and the $\mathrm {C}^{*}$ -algebra quotient $C_b(X)/C_0(X)$ is isomorphic to $C(\beta X\setminus X)$ .

4.2 Multiplier and corona algebras

Multiplier algebras provide an abstract generalisation of $\mathcal B(H)$ and a noncommutative analog of the Čech–Stone compactification, while corona algebras similarly generalise the Calkin algebra and Čech–Stone remainders of topological spaces (Example 4.4).

An ideal J in a $\mathrm {C}^{*}$ -algebra A is essential if its annihilator,

$$\begin{align*}J^\perp=\{a\in A\mid ab=ba=0\text{ for all }b\in J\} \end{align*}$$

is trivial. For example, $C_0(X)$ is an essential ideal in $C_b(X)$ , and a direct summand is an inessential ideal in a direct sum of (nontrivial) $\mathrm {C}^{*}$ -algebras.

There are many ways in which a non-unital $\mathrm {C}^{*}$ -algebra A embeds as an essential ideal into a unital one. In the abelian case of $A = C_0(X)$ , by the Gelfand–Naimark duality these correspond to the ways a locally compact Hausdorff space X can be embedded as a dense open set in a compact Hausdorff space. Among these, the Čech–Stone compactification $\beta X$ is the maximal compactification of X, in the sense that every other compactification $\gamma X$ is the continuous image of $\beta X$ via the continuous extension of the identity map on X.

Given a $\mathrm {C}^{*}$ -algebra A, its multiplier algebra ${\mathcal {M}}(A)$ is the unital $\mathrm {C}^{*}$ -algebra such that when a unital $\mathrm {C}^{*}$ -algebra B contains A as an essential ideal, then the identity map on A extends uniquely to a $^{*}$ -homomorphism from B to ${\mathcal {M}}(A)$ [Reference Blackadar12, Chapter II.7.3.1]. The correspondence with the Čech–Stone compactification is due to the fact that when $A \cong C_0(X)$ then ${\mathcal {M}}(C_0(X)) \cong C(\beta X)$ . We refer to [Reference Blackadar12, Chapter II.7.3] and [Reference Farah61, Section 13] for a rigorous presentation and equivalent definitions of ${\mathcal {M}}(A)$ . The following one is illuminating [Reference Farah61, Corollary 13.2.2].

Lemma 4.1. For a non-unital $\mathrm {C}^{*}$ -algebra A, and any nondegenerateFootnote ¹¹ faithful representation $\Phi \colon A\to \mathcal B(H)$ there is an isomorphism between ${\mathcal {M}}(A)$ and the idealiser

$$\begin{align*}\{b\in \mathcal B(H)\mid b\Phi[A]\cup \Phi[A]b\subseteq \Phi[A]\}. \end{align*}$$

The multiplier algebra of a non-unital $\mathrm {C}^{*}$ -algebra is always nonseparable in the norm topology. It is however possible to make up for this for separable $\mathrm {C}^{*}$ -algebras by considering the strict topology.

Definition 4.2. Let A be a non-unital $\mathrm {C}^{*}$ -algebra. The strict topology on ${\mathcal {M}}(A)$ is the topology generated by the seminorms

$$\begin{align*}\lambda_a(b) := \lVert ab \rVert \text{ and } \rho_a(b) := \lVert ba \rVert, \ \forall a \in A, b \in {\mathcal{M}}(A). \end{align*}$$

The multiplier algebra ${\mathcal {M}}(A)$ is the completion of A with respect to the uniformity associated with the strict topology [Reference Farah61, Lemma 13.1.5]. Moreover, if A is separable, the strict topology is Polish on the unit ball of ${\mathcal {M}}(A)$ , giving it the structure of a standard Borel space [Reference Farah61, Lemma 13.1.8]. If X is a locally compact second countable topological space, this is the topology on $C(\beta X)$ given by uniform convergence on compact subsets of X.

Corona algebras constitute the main family of noncommutative quotient structures considered in this survey. We list some examples that are discussed in the following sections.

As mentioned before, if A is separable the unit ball of $\mathcal {M}(A)$ is equipped with a natural Polish topology (the strict topology) containing the unit ball of A as a Borel subset where all considered operations are Borel. This endows the unit ball of the corona of A with a Borel quotient structure as in Definition 2.1.

Definition 4.5. Let A and B be separable $\mathrm {C}^{*}$ -algebra. A homomorphism $\Phi \colon \mathcal {Q}(A)\to \mathcal {Q}(B)$ is topologically trivial if there is a map $\Phi _{*}\colon {\mathcal {M}}(A)\to {\mathcal {M}}(B)$ that is Borel when restricted to the unit ball of $\mathcal {M}(A)$ considered with the strict topology, and makes the following diagram commute.

The following, stated in [Reference Coskey and Farah29], is an instance of Conjecture Template 2.6 in the current setting.

4.3 The BDF Question

The interest in the existence of outer automorphisms of the Calkin algebra was sparked in the groundbreaking paper [Reference Brown, Douglas and Fillmore21], which was the apex of a long series of efforts aiming to classify elements of $\mathcal B(H)$ up to essential unitary equivalence (we refer to [Reference Davidson30] for an excellent survey on the subject).

The search for an invariant able to completely classify elements of $\mathcal B(H)$ up to essential unitary equivalence goes back to the early investigations on operator algebras by Weyl and von Neumann, where the relation $\sim _{\mathcal K(H)}$ naturally came out in the study of the stability of pseudodifferential operators under compact perturbations. In 1909 Weyl proved in [Reference Weyl181] that every self-adjoint operator a on a separable Hilbert space can be written as $a = d + c$ , where d is a diagonalisable operator and $c \in \mathcal K(H)$ . Weyl also observed that if $a \sim _{\mathcal K(H)} b$ then the essential spectra of a and b are the same. Given an operator $a \in \mathcal B(H)$ , its essential spectrum $\sigma _e(a)$ is the subset of $\mathbb C$ consisting of all the elements in the spectrum of a which are either eigenvalues of infinite multiplicity or accumulation points. The set $\sigma _e(a)$ is known to be equal to $\sigma (\dot a)$ , the spectrum of $\dot a$ in the Calkin algebra [Reference Higson and Roe91, Proposition 2.2.2].

More than 20 years after Weyl’s work, in [Reference Von Neumann180] von Neumann proved the converse, and much harder, implication: if two self-adjoint operators a and b have the same essential spectrum, then $a \sim _{\mathcal K(H)} b$ .

In Problem 4 of his Ten Problems in Hilbert Space [Reference Halmos88], Halmos asked whether every normal operator is the sum of a diagonalisable operator and a compact one, pushing towards an extension of Weyl’s studies. A positive answer to his question was found shortly after by Berg [Reference Berg10] and Sikonia [Reference Sikonia159], who independently generalised Weyl and von Neumann’s results to normal operators. This collective effort is nowadays summarised by the so called Weyl–von Neumann–Berg–Sikonia Theorem.

This latter characterisation also applies to all compact perturbations of normal operators in $\mathcal B(H)$ . In [Reference Brown, Douglas and Fillmore20] Brown, Douglas and Fillmore further extended the scope of this classification to all essentially normal operators. These are all the elements $a \in \mathcal B(H)$ such that $aa^{*}- a^{*}a \in \mathcal K(H)$ or, equivalently, such that $\dot a$ is normal in $\mathcal {Q}(H)$ .

Note that the family of essentially normal operators is strictly larger than the set of compact perturbations of normal operators. The most notorious witness of this fact (and possibly ‘the most important single operator’ in $\mathcal B(H)$ , [Reference Blackadar12, Example I.2.4.3(ii)]) is the unilateral shift s.

The operator $s^{*}s - ss^{*}$ is a projection whose range is 1-dimensional, hence compact. Therefore s is essentially normal, in particular $\dot s$ is a unitary in $\mathcal {Q}(H)$ . We remark that, although the definition of the unilateral shift depends on the choice of a basis, it is unique up to unitary equivalence.

The Fredholm index can be used to check that s is not a compact perturbation of a normal operator. The Fredholm index of an operator $a \in \mathcal B(H)$ is defined as

$$\begin{align*}\operatorname{\mathrm{ind}}(a) := \dim(\operatorname{\mathrm{Ker}}(a)) - \dim(\operatorname{\mathrm{Ker}}(a^{*})), \end{align*}$$

whenever the two quantities on the right-hand side are finite, in which case we say that a is Fredholm. Atkinson’s theorem [Reference Blackadar12, Theorem I.8.3.6] states that an operator a is Fredholm if and only if $\dot a$ is invertible in $\mathcal {Q}(H)$ . Hence the unilateral shift s is Fredholm.

The Fredholm index is an invariant for the relation of essential unitary equivalence, and direct computations show that it is always zero on Fredholm normal operators. Since $\operatorname {\mathrm {ind}}(s) =\ {-}1$ , it follows that s cannot be a compact perturbation of a normal operator. Furthermore, the Fredholm index shows that the essential spectrum alone is not sufficiently fine to completely classify essentially normal operators up to essential unitary equivalence. Indeed there exist unitaries in $\mathcal B(H)$ whose essential spectrum is equal to $\sigma _e(s) = \mathbb T$ (one example is the bilateral shift, which is defined as a shift on some orthonormal basis of H indexed over ${\mathbb Z}$ ), but none of these operators is essentially unitarily equivalent to s, as their Fredholm index is equal to 0.

Brown, Douglas and Fillmore gave a full characterisation of unitary equivalence up to compact perturbation on essentially normal operators in [Reference Brown, Douglas and Fillmore20] by adding the Fredholm index to the invariant.

If moreover a and b are such that $\dot a$ and $\dot b$ are unitaries in $\mathcal {Q}(H)$ , then the classification is even cleaner, as the two operators are equivalent if and only if $\sigma _e(a) = \sigma _e(b)$ and $\operatorname {\mathrm {ind}}(a) = \operatorname {\mathrm {ind}}(b)$ [Reference Brown, Douglas and Fillmore20, Theorem 3.1].

At the very end of [Reference Brown, Douglas and Fillmore20] the authors introduce a weakening of essential unitary equivalence. Given $a,b \in \mathcal B(H)$ we say that $a \approx b$ if and only if there exists an automorphism $\alpha \in \operatorname {\mathrm {Aut}}(\mathcal {Q}(H))$ such that $\alpha (\dot a) = \dot b$ . When restricting to essentially normal operators, the relation $\sim _{\mathcal K(H)}$ is obtained from $\approx $ by requiring that $\alpha $ is innerFootnote ¹² . In [Reference Brown, Douglas and Fillmore20] it is asked whether $\approx $ and $\sim _{\mathcal K(H)}$ coincide on such operators. Note that $a \approx b$ implies $\sigma _e(a) = \sigma _e(b)$ , thus the two relations are different only if there is an automorphism of $\mathcal {Q}(H)$ which moves an element of $\mathcal {Q}(H)$ to another one with a different Fredholm index. Since a unitary of Fredholm index $n\neq 0$ has an mth root if and only if m divides n, every automorphism of $\mathcal {Q}(H)$ induces a group automorphism of ${\mathbb Z}$ (the range of the Fredholm indices) and this question boils down to the following.

Since inner automorphisms preserve the Fredholm index, an automorphism witnessing a positive answer to Question 4.16 cannot be inner. As a test question, in [Reference Brown, Douglas and Fillmore21, Section 1.6(ii)] the authors asked whether outer automorphisms of the Calkin algebra exist at all.

It is unlikely that Brown, Douglas, and Fillmore expected that the answer to their question might involve set theory, although a few years after Shelah started a subject that would eventually lead to the resolution of the second problem (Section 7.1). The question of the existence of outer automorphisms of the Calkin algebra will be thoroughly discussed in Section 6.5 and Section 7.3.

We remark that it is still not known whether Question 4.16 has a positive answer consistently with $\mathrm {ZFC}$ . Indeed, the restriction of every known outer automorphism of $\mathcal {Q}(H)$ to a separable $\mathrm {C}^{*}$ -subalgebra is implemented by a unitary, and therefore preserves the Fredholm index.

The methods developed by Brown, Douglas, and Fillmore in [Reference Brown, Douglas and Fillmore20] had much deeper implications than simply classifying normal operators in $\mathcal {Q}(H)$ up to unitary equivalence. Their work led to the introduction of a new $\mathrm {C}^{*}$ -algebraic invariant $\operatorname {\mathrm {Ext}}$ , which is defined as follows (we refer to [Reference Blackadar11, Chapter VII] for a textbook on the topic).

With the language of extensions, the classification problem we are discussing can be investigated from a much broader perspective. Note first that, given an essentially normal element $a \in \mathcal B(H)$ , by the Gelfand–Naimark Duality the unital $\mathrm {C}^{*}$ -algebra generated by $\dot a$ in $\mathcal {Q}(H)$ is isomorphic to the abelian $\mathrm {C}^{*}$ -algebra $C(\sigma _e(a))$ . The isomorphism sends the identity map on $\sigma _e(a)$ to $\dot a$ , and can be thought as a unital extension of $C(\sigma _e(a))$ . Given two essentially normal operators $a,b \in \mathcal B(H)$ such that $X := \sigma _e(a) = \sigma _e(b)$ , asking whether $a \sim _{\mathcal K(H)} b$ translates into verifying that the two unital extensions of $C(X)$ by $\dot a$ and $\dot b$ defined above are unitarily equivalent. Hence, the classification of essentially normal operators up to $\sim _{\mathcal K(H)}$ is reduced to the study of $\operatorname {\mathrm {Ext}}(C(X))$ (usually simply denoted by $\operatorname {\mathrm {Ext}}(X)$ ), in the particular case where X is a compact subset of $\mathbb {C}$ .

In [Reference Brown, Douglas and Fillmore21] the authors extend their study to $\operatorname {\mathrm {Ext}}(X)$ for X a general compact metrisable space, which by the Gelfand–Naimark duality corresponds to the analysis of $\operatorname {\mathrm {Ext}}$ for separable, abelian, unital $\mathrm {C}^{*}$ -algebras. They prove that $\operatorname {\mathrm {Ext}}(X)$ is always an abelian group, and that it in this setting $\operatorname {\mathrm {Ext}}$ is a homotopy invariant, covariant functor, which is moreover a K-homology. Their work effectively led to the birth of a brand new subject combining $\mathrm {C}^{*}$ -algebras and algebraic topology, known as extension theory (standard references for this subject are [Reference Blackadar11] and [Reference Higson and Roe91]).

Note that the group $\operatorname {\mathrm {Aut}}(\mathcal {Q}(H))$ acts by composition on $\operatorname {\mathrm {Ext}}(A)$ , for every unital separable $\mathrm {C}^{*}$ -algebra A. From this perspective, Question 4.16 interrogates about the triviality of the action of $\operatorname {\mathrm {Aut}}(\mathcal {Q}(H))$ on $\operatorname {\mathrm {Ext}}(\mathbb T)$ , since $\sigma _e(s) =\sigma _e(s^{*}) = \mathbb T$ . More specifically, the function that maps each (class of an) extension $\varphi \in \operatorname {\mathrm {Ext}}(\mathbb T)$ to the Fredholm index $\operatorname {\mathrm {ind}}(\varphi (\operatorname {\mathrm {id}}_{\mathbb T}))$ , is an isomorphism between $\operatorname {\mathrm {Ext}}(\mathbb T)$ and ${\mathbb Z}$ . The extensions of $C(\mathbb T)$ which are mapped by this isomorphism to $-1$ and $1$ are exactly those that send $\operatorname {\mathrm {id}}_{\mathbb T}$ to (something which is unitarily equivalent to) $\dot s$ and $\dot s^{*}$ , respectively. Since the only nontrivial automorphism of the group ${\mathbb Z}$ is the one moving $1$ to $-1$ , understanding whether the action of $\operatorname {\mathrm {Aut}}(\mathcal {Q}(H))$ on $\operatorname {\mathrm {Ext}}(\mathbb T)$ is trivial reduces to answering Question 4.16. We conclude this section with a more general problem suggested by Nigel Higson.

In the case when $A = C(\mathbb T)$ , Higson’s question reduces to Question 4.16. In general, every inner automorphism acts trivially on $\operatorname {\mathrm {Ext}}(A)$ , and both questions are $\Sigma ^2_1$ statements hence Woodin’s theorem (see Section 12) suggests that CH provides an optimal setting for a possible positive answer to either of the two questions.

4.4 Sakai’s question

In this section we introduce an old open problem which originated from a series of papers due to Sakai ([Reference Sakai142], [Reference Sakai143] and [Reference Sakai144]), and which has been abundantly overlooked over the last decades. Similarly to some other topics discussed in the current section, this question has a purely $\mathrm {C}^{*}$ -algebraic flavour and was motivated by the investigations on inner derivations of simple $\mathrm {C}^{*}$ -algebras contained in Sakai’s aforementioned works.

Both simplicity and separability are necessary in Question 4.19. The former is due to the elementary example $B = A \oplus C$ , where A is any non-unital $\mathrm {C}^{*}$ -algebra and C is any unital $\mathrm {C}^{*}$ -algebra, which gives $\mathcal {Q}(A) \cong \mathcal {Q}(B)$ .

For what concerns separability, in [Reference Sakai144] Sakai shows that there exist multiple nonseparable simple $\mathrm {C}^{*}$ -algebras whose corona is equal to any prescribed finite-dimensional $\mathrm {C}^{*}$ -algebra (see also [Reference Ghasemi and Koszmider85] for examples of non-simple nonseparable $\mathrm {C}^{*}$ -algebras whose corona is one-dimensional).

Question 4.19 was attributed to Sakai and was explicitly stated for the first time in Elliott’s work [Reference Elliott47], where (a strengthening of) the following theorem is proved. Recall that a $\mathrm {C}^{*}$ -algebra is UHF if it can be written as the direct limit of matrix algebras.

The methods used to prove this result involve the study of the connected components of the unitary group of $\mathcal {Q}(A)$ , which are shown to be directly related to the $K_0$ -group $K_0(A)$ , which in turn characterises $A\otimes \mathcal K(H)$ up to isomorphism. We refer to [Reference Rørdam and Størmer139, Chapter 1] for further details and proper context on this foundational classification result, and to [Reference White182] for an up-to-date overview of the Elliott classification program. In [Reference Farah65, Corollary D] it was proven that $\mathcal {Q}(H)$ is not isomorphic to $\mathcal {Q}(A\otimes \mathcal K(H))$ for any Elliott-classifiable $\mathrm {C}^{*}$ -algebra A, thus giving support to the conjecture that $\mathcal {Q}(H)$ is not isomorphic to $\mathcal {Q}(A)$ for a simple A not isomorphic to the algebra of compact operators. This result was extended in [Reference Farah and Szabó76, Corollary 5.3] where it was proven that there are $2^{\aleph _0}$ non-isomorphic coronas of simple separable $\mathrm {C}^{*}$ -algebras.

We remark that the analog of Question 4.19 for multiplier algebras has been fully answered in [Reference Brown19], where it is proved that if A and B are separable $\mathrm {C}^{*}$ -algebras such that ${\mathcal {M}}(A) \cong {\mathcal {M}}(B)$ , then $A \cong B$ .

It is not clear whether set theory could play a role in further progress on Question 4.19. Nevertheless, this problem highlights how little we know about the general structure theory of corona algebras. The paper [Reference Sakai144] ends with a series of open problems aiming at a better understanding on this matter; we include a sample.

For an arbitrary $\mathrm {C}^{*}$ -algebra C there exists a $\mathrm {C}^{*}$ -algebra A such that $\mathcal {Q}(A)\cong C$ (see [Reference Farah and Vignati78, Proposition 3.2]), but it is not known whether such A can be chosen to be simple. As we said, it is possible to have $\mathcal {Q}(A)\cong \mathbb C$ for a simple A [Reference Sakai144, Theorem 1], and also for an AF (but not simple) A [Reference Ghasemi and Koszmider85]. Since for a finite-dimensional $\mathrm {C}^{*}$ -algebra F and any A we have $F\otimes \mathcal {Q}(A)\cong \mathcal {Q}(F\otimes A)$ , the answer to Question 4.21 is positive for a finite-dimensional C. Also, for every compact metric space K there is a separable Banach space X such that the ‘Calkin algebra’ $\mathcal B(X)/\mathcal K(X)$ is isomorphic to $C(K)$ ([Reference Motakis121]; the case when K is a singleton was a major open problem until [Reference Argyros and Haydon2]).

5.1 Approximate homomorphisms

The following is probably the most studied context for Ulam-stability. For $\varepsilon>0$ , a function $f\colon G_1\to G_2$ between groups such that $G_2$ is equipped with a metric d is said to be an $\varepsilon $ -homomorphism if

(5.1) $$ \begin{align} d(f(gh),f(g)f(h))<\varepsilon \end{align} $$

for all g and h in $G_1$ . The question is, under what conditions one can conclude that f is near a true group homomorphism?

The following convention clearly does not cover $\mathrm {C}^{*}$ -algebras, to which we will return later.

There are numerous variations of this definition—one can talk about a specific class of maps being stable, or specify different classes for the structures in the domain and those in the range (note that the former need not be metric structures).

Example 5.3. The category of all finite subgroups of the unit circle, with the induced metric, is not Ulam-stable. Define $f_n\colon {\mathbb Z}/(2n+1){\mathbb Z}\to {\mathbb Z}/(2n){\mathbb Z}$ by

$$\begin{align*}f_n\left(e^{i 2\pi \frac{k}{2n+1}}\right)=e^{i\pi \frac{k}{n}} \end{align*}$$

for $0\leq k<2n+1$ . Since for all $k,k'<2n+1$ the difference (modulo $2n$ ) between $[k+k']_{2n+1}$ and $[k+k']_{2n}$ is at most $1$ , we have that

$$\begin{align*}\textstyle \left| f_n\left(e^{i 2\pi \frac{k}{2n+1}}\right) f_n\left(e^{i 2\pi \frac{k'}{2n+1}}\right)- f_n\left(e^{i 2\pi \frac{k+k'}{2n+1}}\right)\right| =|1-e^{i2\pi\frac{1}{2n+1}}|<\frac {2\pi}{2n+1}, \end{align*}$$

hence $f_n$ is a $2\pi /(2n+1)$ -homomorphism. Since every homomorphism between these two groups is trivial, $f_n$ cannot be 1-approximated by a homomorphism.

A clever way to remedy Example 5.3 is to allow passing to a larger group in the range. This approach is taken in [Reference Gowers and Hatami86], where it was shown that an approximate representation of a finite group $\Gamma $ in $M_n(\mathbb C)$ (with respect to the Hilbert–Schmidt norm) is near a representation of $\Gamma $ in a larger matrix algebra in which $M_n(\mathbb C)$ embeds unitally; see also [Reference De Chiffre, Ozawa and Thom36] for a von Neumann-algebraic adaptation of the Gowers–Hatami result. Also, Kazhdan [Reference Kazhdan104] showed that approximate representations of amenable groups (where the range is the unitary group on an infinite dimensional Hilbert space) are uniformly close to representations, while the same fails for the free group $\mathbb F_2$ , or any group which contains it [Reference Burger, Ozawa and Thom23].

5.2 From Borel to asymptotically additive

Ulam-stability is directly connected with the existence of liftings for homomorphisms between reduced products, the definition of which we quickly recall.

Definition 5.4. Fix $\mathcal {G}$ as in Convention 5.1. If $(A_n,d_n)$ , for $n\in {\mathbb N}$ , is a sequence of metric structures in $\mathcal {G}$ and ${\mathcal {I}}$ is an ideal on ${\mathbb N}$ , the reduced product $\prod _n A_n/\bigoplus _{\mathcal {I}} A_n$ is the quotient of the product $\prod _n A_n$ modulo the congruence that identifies $(x_n)$ and $(y_n)$ if and only if

$$\begin{align*}\lim_{n \to {\mathcal{I}}} d_n(x_n, y_n) = \inf_{X\in {\mathcal{I}}} \sup_{n\notin X} d_n(x_n,y_n)=0. \end{align*}$$

The following convention will help to simplify the notation (see (5.2) below).

In the case of groups, or rings, $+$ and $0$ have the obvious interpretation. In Boolean algebras, $+$ will stand for $\cup $ .

For a fixed sequence of structures $A_n$ the following notation will come handy. For $I\subseteq {\mathbb N}$ , we write $A_I=\prod _{n\in I} A_n$ (thus $A_{{\mathbb N}}=\prod _n A_n$ ) and consider the natural projection map $\pi _I\colon A_{{\mathbb N}}\to A_I$ . We identify $A_I$ with

(5.2) $$ \begin{align} \{a\in A_{{\mathbb N}}\mid a_n=0\text{ for all }n\notin I\}. \end{align} $$

This convention identifies a quotient $A_I$ of $A_{\mathbb N}$ with a subalgebra thereof.

In the following lemma $\mathcal {P}({\mathbb N})$ is identified with $\prod _{\mathbb N}\{0,1\}$ .

In the case when ${\mathcal {I}}=\operatorname {\mathrm {Fin}}$ , Lemma 5.7 was extracted from Shelah’s proof (Section 7.1) in [Reference Veličković174]. Its proof and the proof of the following (taken from [Reference Farah54, Section 4]) use the method of stabilisers, isolated by Just in [Reference Just97].

Lemma 5.8. Suppose that $\mathcal {G}$ is a category of Boolean algebras or groups as in Convention 5.1 and $A_n$ and $B_n$ belong to $\mathcal {G}$ . Let ${\mathcal {I}}$ and ${\mathcal {J}}$ be ideals in $\mathcal {P}({\mathbb N})$ , both containing $\operatorname {\mathrm {Fin}}$ . If

$$\begin{align*}\textstyle \Phi\colon \prod_n A_n/\bigoplus_{{\mathcal{I}}} A_n\to \prod_n B_n/\bigoplus_{{\mathcal{J}}} B_n \end{align*}$$

is a topologically trivial homomorphism, then $\Phi $ has an asymptotically additive lifting.

In the proof, the assumption that all $A_n$ are finite is needed in order to use the classical characterisation of comeager subsets of the product $\prod _n A_n$ (see, e.g., [Reference Farah61, Theorem 9.9.1]). The analog of Lemma 5.8 is true in some other categories (see Section 7.4).

Every progress on Problem 5.9 connects Ulam-stability to the question when topologically trivial homomorphisms are algebraically trivial for certain quotient structures. In the following, $\mathcal {G}$ is a class of structures as in Convention 5.1. In addition, we assume that $\mathcal {G}$ is closed under taking finite products.

A sketch of a proof

(2) implies (1): Assume (1) fails. Fix $\varepsilon>0$ such that for every $n\geq 1$ there is a $1/n$ -homomorphism $f_n\colon A_n\to B_n$ that cannot be uniformly $\varepsilon $ -approximated by a homomorphism. Then $(a_n)\mapsto (f_n(a_n))$ is a lifting of an algebraically nontrivial homomorphism from $\prod _n A_n/\bigoplus _{\operatorname {\mathrm {Fin}}} A_n$ into $\prod _n B_n/\bigoplus _{\operatorname {\mathrm {Fin}}} B_n$ .

(1) implies (2): Fix an asymptotically additive lifting of a homomorphism $\Phi \colon \prod _n A_n/\bigoplus _{\mathcal {I}} A_n\to \prod _n B_n/\bigoplus _{\mathcal {J}} B_n$ . Since $\mathcal {G}$ is closed under finite products, we can assume $I(n)=J(n)=\{n\}$ (see Definition 5.6) for all n. Thus there are $f_n\colon A_n\to B_n$ such that $\Phi _{*}((a_n))=(f_n(a_n))$ lifts $\Phi $ . Since the signature is finite and the structures are finite, a pigeonhole argument shows that for every $m\geq 1$ the set

$$\begin{align*}X_m=\{n\mid f_n\text{ is not a } 1/m\text{-homomorphism}\} \end{align*}$$

belongs to ${\mathcal {I}}$ . Choose a sequence $(\varepsilon _m)$ such that every $1/m$ -homomorphism can be $\varepsilon _m$ -approximated by a true homomorphism and $\lim _m \varepsilon _m=0$ . Let $X_0={\mathbb N}$ and for $j\in X_n\setminus X_{n+1}$ replace $f_n$ with a true homomorphism $g_n$ that $\varepsilon _m$ -approximates it (for n that does not belong to $\bigcup _n X_n$ , if any, $f_n$ is already a homomorphism). The asymptotically additive lift given by the sequence $(g_n)$ is a true homomorphism, as required.

5.3 Boolean algebras

A submeasure (Definition 3.4) on a set I is strictly positive if $\varphi (X)>0$ for every nonempty $X\subseteq I$ . In this case, the Boolean algebra $\mathcal {P}(I)$ is equipped with the distance $d_\varphi (X,Y)=\varphi (X\Delta Y)$ . Of interest to us is the following.

Lemma 5.8, Proposition 5.10, and Proposition 5.11 together imply that, for homomorphisms between quotients over analytic P-ideals (and especially for generalised density ideals), the equivalence of algebraic triviality and topological triviality hinges on Ulam-stability of approximate homomorphisms between metric Boolean algebras of the form $(\mathcal {P}({\mathbb N}), d_\varphi )$ where $d_\varphi =\varphi (X\Delta Y)$ for a strictly positive submeasure $\varphi $ on ${\mathbb N}$ . The failure of Ulam-stability for such class was proved in [Reference Farah51].

It makes therefore sense to isolate the following property.

A submeasure is said to be nonpathological if it is the supremum of measures dominated by it.Footnote ¹⁴ As suggested by the adjective, pathological submeasures are not easy to come by, and for the naturally occurring $F_\sigma $ and analytic P-ideals, the submeasure $\varphi $ associated with them (see Theorem 3.6) can usually be chosen to be nonpathological. Such ideals are called nonpathological. In [Reference Farah51] and [Reference Farah53] it was proved that the approximate Boolean algebra homomorphisms and group homomorphisms with respect to nonpathological submeasures are Ulam-stable. Thus Lemma 5.7 and Proposition 5.10 together imply that nonpathological analytic P-ideals have the Radon–Nikodym property. The other parts of the following partial answer to Question 3.7 are proved using similar ideas.

The first four items in Theorem 5.13 apply even when these structures are considered as just groups, with the appropriate notion of Radon–Nikodym property; see [Reference Kanovei and Reeken103] and [Reference Kanovei and Reeken102] for additional information. Analytic, and even $F_\sigma $ , ideals without the Radon–Nikodym property exist by [Reference Farah52, Theorem 1.9.5], but the following is open.

A version of this question for analytic P-ideals appears in [Reference Farah52, Question 1.14.3], and its reformulation in terms of Ulam-stability for approximate isomorphisms is [Reference Farah52, Question 1.14.4].

5.4 $\mathrm {C}^{*}$ -algebras

We now describe the situation in the case of $\mathrm {C}^{*}$ -algebras and how Ulam-stability phenomena affect the relation between topologically trivial and algebraically trivial automorphisms of corona $\mathrm {C}^{*}$ -algebras (see Question 4.11).

For Banach space-based structures, such as $\mathrm {C}^{*}$ -algebras, an $\varepsilon $ - $^{*}$ -homomorphism is a map between their unit balls which satisfies the analog of (5.1) for the algebraic operations and for multiplication by scalars of modulus at most $1$ .Footnote ¹⁵ By $A_1$ we denote the unit ball of a $\mathrm {C}^{*}$ -algebra A.

Definition 5.15. A function $f\colon A_1\to B_1$ between unit balls of $\mathrm {C}^{*}$ -algebras is an $\varepsilon $ - $^{*}$ -homomorphism if for all x and y in $A_1$ and every $\lambda \in \mathbb C$ with $|\lambda |\leq 1$ each of the following has norm at most $\varepsilon $ :

$$ \begin{align*} f(a{\kern-1pt}+{\kern-1pt}b){\kern-1pt}-{\kern-1pt}f(a){\kern-1pt}-{\kern-1pt}f(b), \ f(ab){\kern-1pt}-{\kern-1pt}f(a)f(b),\ f(\lambda a){\kern-1pt}-{\kern-1pt}\lambda f(a),\ f(a^{*}){\kern-1pt}-{\kern-1pt}f(a)^{*}. \end{align*} $$

We abuse notation, and say that a map between $\mathrm {C}^{*}$ -algebras is an $\varepsilon $ - $^{*}$ -homomorphism if its restriction to the unit ball is such.

In the following we deviate from the convention established in Definition 5.2 and consider the stability of approximate morphisms between specified ‘source objects’ and ‘target objects.’ This is done partly in order to efficiently handle the Calkin algebra and partly out of despair for the current lamentable lack of sharper results.

Apart from the following two results, the question of Ulam-stability for other classes of $\mathrm {C}^{*}$ -algebras is wide open.

Proof In [Reference Šemrl146] an $\varepsilon $ -homomorphism between $\mathrm {C}^{*}$ -algebras was defined by requiring

(5.3) $$ \begin{align} \begin{aligned} \|f(x+y)-f(x)-f(y)\|&\leq \varepsilon(\|x\|+\|y\|),\\ \|f(xy)-f(x)f(y)\|&\leq \varepsilon(\|x\|\|y\|),\\ \|f(x^{*})-f(x)^{*}\|&\leq \varepsilon\|x\| \end{aligned} \end{align} $$

for all x and y in the domain, and proved that for such a function f between abelian $\mathrm {C}^{*}$ -algebras there is a $^{*}$ -homomorphism $\Phi $ such that $\|\Phi -f\|\leq ((4+\frac 32 (1+\pi ))\varepsilon +1-\sqrt {1-4\varepsilon })\|f\|$ [Reference Šemrl146, Theorem 5.1].

A simple computation (regrettably absent from [Reference Vignati178]) shows that if $\varphi \colon A_1\to B_1$ is an $\varepsilon $ - $^{*}$ -homomorphism in the sense of Definition 5.15, then $f\colon A\to B$ defined by $f(x)=\|x\|\varphi (x/\|x\|)$ for $x\neq 0$ and $f(0)=0$ is a $4\varepsilon $ -homomorphism in the sense of (5.3) and that $\|f(x)-\varphi (x)\|\leq \varepsilon $ for all $x\in A_1$ . The thesis follows by connecting the dots.

The proof of the following uses the Gelfand–Naimark duality, the topological definition of algebraic triviality (Definition 4.8), a weakening of Lemma 5.8, and Theorem 5.18.

5.5 Reduced products of $\mathrm {C}^{*}$ -algebras

In the noncommutative case, the route towards proving algebraic triviality from Ulam-stability is somewhat oblique (and in fact, not yet completely understood in the general case), unless the algebras are reduced products. A unital $\mathrm {C}^{*}$ -algebra A does not have central projections if it cannot be written as $A=B\oplus C$ where B and C are nonzero $\mathrm {C}^{*}$ -algebras; this assumption is used only to mitigate notational issues. The reduced product $\prod _n A_n/\bigoplus _n A_n$ was defined in Example 4.4 (3).

The proof of Theorem 5.20 roughly follows the line of the proofs of Lemma 5.8—Proposition 5.10, with some technical wrinkles. First, one applies Lemma 5.8 to the projections in the center of $\prod _nA_n/\bigoplus _nA_n$ , which is isomorphic to $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ , to get an asymptotically additive lift, and then one runs similar arguments as in Proposition 5.10.

Definition 5.21. For an infinite $X\subseteq {\mathbb N}$ let

$$\begin{align*}\textstyle {\mathcal{M}}_X=\prod_{n\in X} M_n(\mathbb C)/\bigoplus_{n\in X} M_n(\mathbb C). \end{align*}$$

Corollary 5.22 applies only to reduced products, and $\mathcal {Q}(H)$ isn’t one; see Section 6.4 for a remedy.

We conclude with a very general question and two of its very specific instances. Each one of these questions corresponds to a particular instance of the lifting problem for $*$ -homomorphisms between quotient $\mathrm {C}^{*}$ -algebras.

5.6 The Kadison–Kastler conjecture

Since our main interest is in isomorphisms, it is natural to consider the following notion [Reference McKenney and Vignati116, Definition 1.2]. Recall that by $A_1$ we denote the unit ball of a $\mathrm {C}^{*}$ -algebra A.

Ulam-stability of $\mathrm {C}^{*}$ -algebras is related to the Kadison–Kastler perturbation theory. In [Reference Kadison and Kastler100], Kadison and Kastler initiated the study of perturbations of algebras of operators. If A and B are two $\mathrm {C}^{*}$ -algebras sitting in the same $\mathcal B(H)$ , the Kadison–Kastler distance is the Hausdorff distance between their unit balls, that is

$$\begin{align*}d_{\textrm{KK}}(A,B)=\max\{\sup_{a\in A_1}\inf_{b\in B_1}\left\lVert a-b \right\rVert,\sup_{b\in B_1}\inf_{a\in A_1}\left\lVert a-b \right\rVert\}. \end{align*}$$

Kadison and Kastler asked what properties are preserved at a small $d_{\textrm {KK}}$ -distance. They implicitly conjectured that, given a $\mathrm {C}^{*}$ -algebras A, there exists $\varepsilon>0$ such that if $d_{\textrm {KK}}(A,B)<\varepsilon $ then A and B are isomorphic. (A stronger version of this conjecture asks for the $\mathrm {C}^{*}$ -algebras to be conjugated by a unitary close to the identity.)

We say that A satisfies the Kadison–Kastler conjecture if such an $\varepsilon $ exists. A class $\mathcal C$ of $\mathrm {C}^{*}$ -algebras is said to be Kadison–Kastler stable (or KK stable) if each algebra in $\mathcal C$ satisfies the Kadison–Kastler conjecture and the same $\varepsilon $ witnesses stability uniformly on elements of $\mathcal C$ . The class of nonseparable $\mathrm {C}^{*}$ -algebras is not KK stable, and there is even a nonseparable $\mathrm {C}^{*}$ -algebra that does not satisfy the Kadison–Kastler conjecture ([Reference Choi and Christensen27], see also [Reference Farah61, Section 14.4]). The class of unital separable amenableFootnote ¹⁶ $\mathrm {C}^{*}$ -algebras is KK stable [Reference Christensen, Sinclair, Smith, White and Winter28], but it is not known whether the class of all separable unital $\mathrm {C}^{*}$ -algebras is KK stable.

Fix $\varepsilon>0$ , and suppose that A and B are two separable $\mathrm {C}^{*}$ -algebras such that $d_{\textrm {KK}}(A,B)<\varepsilon $ . If to each element of $A_1$ we associate an element of $B_1$ at distance $<\varepsilon $ , we obtain an $\varepsilon $ - $^{*}$ -isomorphism between A and B. Therefore, if A is stable under approximate isomorphisms, A satisfies the Kadison–Kastler conjecture. Similarly, if a class of $\mathrm {C}^{*}$ -algebras is Ulam-stable, it is KK stable. We do not know whether either of the two implications above can be reversed.

6.1 Saturation

If there is no obvious isomorphism between two structures, then a standard method for constructing one uses ‘back-and-forth’ methods. These techniques generally consist of an inductive process building an increasing sequence (transfinite, if need be) of partial isomorphismsFootnote ¹⁷ between the objects involved, and whose domains become larger and larger as the induction goes on. These constructions often require a considerable amount of bookkeeping to keep track of the partial maps defined at each step, especially if the structures are nonseparable. Nevertheless, a theorem of Keisler states that if two structures are saturated and have the same cardinality, then a back-and-forth system between them exists if and only if they are elementarily equivalent (Theorem 6.2 below). This readily extends to metric structures of same density character. Since many quotients such as ultraproducts and some corona algebras are saturated under $\mathrm {CH}$ , Keisler’s theorem often reduces the question of isomorphism to the question of elementary equivalence. Analogous back-and-forth methods can be used to prove, again under CH, the existence of topologically nontrivial isomorphisms.

In this subsection we review the effects of countable saturation on the isomorphism questions for coronas of $\mathrm {C}^{*}$ -algebras under the assumption of $\mathrm {CH}$ . The non-expert reader may refer to [Reference Yaacov, Berenstein, Henson, Usvyatsov and Chatzidakis9] for all basic definitions and for an introduction to the model theory of metric structures, and to [Reference Farah, Hart and Sherman69] for a reference focusing specifically on operator algebras.

6.1.6 The Palyutin–Palmgren approach

An elegant alternative approach to proving countable saturation of reduced products associated with countably generated ideals that does not use the Feferman–Vaught theorem is given in [Reference Palmgren124]. It uses analysis of reduced products due to Palyutin [Reference Palyutin125, Reference Palyutin126] that deserves to be better known. Palyutin defined the set of h-formulas of a language $\mathcal {L}$ (not to be confused with Horn formulas) to be the smallest set of $\mathcal {L}$ -formulas that includes all atomic formulas and such that if $\varphi $ and $\psi $ are h-formulas then so are $\varphi \wedge \psi $ , $(\forall x)\varphi $ , $(\exists x)\varphi $ , and $(\exists x)\varphi \wedge (\forall x)(\varphi \rightarrow \psi )$ . By induction on complexity, one proves the following analog of Łoś’s theorem for arbitrary reduced products.

Theorem 6.12. If $M_i$ , for $i\in I$ , are structures of the same language, ${\mathcal {I}}$ is an ideal on I, $M=\prod _i M_i/\bigoplus _{\mathcal {I}} M_i$ , $\varphi (\bar x)$ is an h-formula, and $\bar a=[\bar a_i]_{\mathcal {I}}$ in M is of the appropriate sort then $M\models \varphi (\bar a)$ if and only if

$$\begin{align*}\{i\in I \mid M_i\models \neg \varphi(\bar a_i)\}\in {\mathcal{I}}. \end{align*}$$

The key to Palmgren’s proof is the observation that in every reduced power associated with $\operatorname {\mathrm {Fin}}$ (or any other ideal such that the quotient Boolean algebra is atomless) every formula is equivalent to a Boolean combination of h-formulas. Therefore every type over such reduced product is axiomatised by a set consisting of h-formulas and negations of h-formulas. This proof was extended to prove countable saturation of reduced powers over a larger class of the so-called layered ideals in [Reference De Bondt, Farah and Vignati31, Theorem 4.1].

6.1.7 Asymptotic sequence algebras

We write $A_\infty $ for $\prod _n A/\bigoplus _{\mathrm {Fin}} A$ for any structure A, metric or discrete (in case of $\mathrm {C}^{*}$ -algebra, this is the asymptotic sequence algebra). The following is an immediate consequence of Theorems 6.8, 6.10, 6.11, and 6.2.

Corollary 6.13. Assume $\mathrm {CH}$ . If $(A_n)_{n \in {\mathbb N}}$ and $(B_n)_{n \in {\mathbb N}}$ are structures of the same language of density character $\leq \aleph _1$ such that $A_n\equiv B_n$ for all but finitely many n, then $\prod _n A_n/\bigoplus _{\operatorname {\mathrm {Fin}}} A_n$ and $\prod _n B_n/\bigoplus _{\operatorname {\mathrm {Fin}}} B_n$ are isomorphic.

In particular, if A and B are elementarily equivalent and of density character $\leq \aleph _1$ , then $A_\infty \cong B_\infty $ , and both structures are isomorphic to the ultrapower of $C(K,A)$ associated with any nonprincipal ultrafilter on ${\mathbb N}$ , K being the Cantor space.

For discrete structures ‘density character’ is of course a synonym for cardinality. For more see [Reference Farah63, Theorems A–F].

This brings us to the question of elementary equivalence of $\mathrm {C}^{*}$ -algebras. Finite-dimensional $\mathrm {C}^{*}$ -algebras and separable UHF-algebras are elementarily equivalent if and and only if they are isomorphic [Reference Carlson, Cheung, Farah, Gerhardt-Bourke, Hart, Mezuman, Sequeira and Sherman24, Theorem 3]. However, even for (separable) AF-algebras the relation of elementary equivalence is ‘smooth’ while the isomorphism of AF-algebras is a ‘non-smooth’ relation (it is classifiable by countable structures [Reference Farah, Toms and Törnquist77]). Hence, there are non-isomorphic elementarily equivalent unital AF-algebras whose asymptotic sequence algebras, under $\mathrm {CH}$ , as well as their ultrapowers are isomorphic (see [Reference Eagle and Vignati45, Reference De Bondt, Vaccaro, Veličković and Vignati34] for examples of such AF-algebras).

We finish this section by another application of the metric Feferman–Vaught theorem which was used in [Reference Ghasemi84] to provide examples of ‘genuinely’ noncommutative $\mathrm {C}^{*}$ -algebras A and B such that the existence of an isomorphisms between them is independent from $\mathrm {ZFC}$ .

Example 6.14. Fix a sequence $(A_n)_{n \in {\mathbb N}}$ of pairwise non-isomorphic unital separable $\mathrm {C}^{*}$ -algebras (for example, each $A_n$ could be $M_n(\mathbb C)$ ). Suppose $\mathcal {L}$ is the language of $\mathrm {C}^{*}$ -algebras and fix a countable uniformly dense set $\{\psi _i\mid i\in {\mathbb N}\}$ of $\mathcal {L}$ -sentences. That is, for every $\varepsilon>0$ , for every $\mathcal {L}$ -sentence $\theta $ and for every $\mathcal {L}$ -structure A there is $\psi _i$ such that

$$\begin{align*}|\psi_i^A - \theta^A|<\varepsilon. \end{align*}$$

The set $\{\psi _i\mid i\in {\mathbb N}\}$ exists by [Reference Yaacov, Berenstein, Henson, Usvyatsov and Chatzidakis9, Proposition 6.6]. Since the range of each $\psi _i$ is a bounded subset of $\mathbb R$ , by compactness arguments we can find a sequence of natural numbers $(k(n))_{n\in {\mathbb N}}$ such that every i we have $\lim _n \psi _i^{A_{k(n)}} = r_i$ for some $r_i\in \mathbb R$ . As $\psi _i$ are uniformly dense, for every $\mathcal {L}$ -sentence $\theta $ there is a real number $r_\theta $ such that

(*) $$ \begin{align} \lim_n \theta^{A_{k(n)}}= r_\theta. \end{align} $$

Suppose $X=\{g(n) \mid n\in {\mathbb N}\}$ and $Y=\{h(n) \mid n\in {\mathbb N}\}$ are disjoint infinite subsets of $\{k(n) \mid n\in {\mathbb N}\}$ . Let

$$\begin{align*}A:=\prod_n A_{g(n)}/\bigoplus_{\operatorname{\mathrm{Fin}}} A_{g(n)} \quad \text{and} \quad B:=\prod_n A_{h(n)}/\bigoplus_{\operatorname{\mathrm{Fin}}} A_{h(n).} \end{align*}$$

For every $\mathcal {L}$ -sentence $\theta $ from $(*)$ we have $\lim _n \theta ^{A_{g(n)}}= \lim _n \theta ^{A_{h(n)}}$ and therefore Theorem 6.10 implies that $A\equiv B$ . Since A and B are countably saturated (Theorem 6.8) CH implies that $A \cong B$ . However, $X\cap Y= \emptyset $ , which means $A_{g(n)} \not \cong A_{h(m)},$ for all $n,m\in {\mathbb N}$ . This shows that any isomorphism between A and B, constructed under CH, cannot be algebraically trivial (see Corollary 7.21).

6.2 Full saturation: The dividing line

In Example 2.7 we presented a counterexample to the second part of Conjecture 2.6. The gist of this example is that the reduced product $(\prod _n {\mathbb Z}/2{\mathbb Z})/(\bigoplus _{\mathbb N}{\mathbb Z}/2{\mathbb Z})$ is fully saturated, regardless of whether CH holds or not. The fact that this does not apply to $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ , the grandfather of all structures that satisfy the second part of Conjecture 2.6, is a classical result of Hausdorff who in [Reference Hausdorff90] famously constructed an $(\omega _1,\omega _1)$ -gap showing that $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ is not an $\aleph _2$ -saturated Boolean algebra.Footnote ²⁰ By Theorem 6.8 every reduced product associated with $\operatorname {\mathrm {Fin}}$ is countably saturated. Are there other possibilities for saturation of reduced products associated with $\operatorname {\mathrm {Fin}}$ ? In Question 10.1 in the original draft of the present paper we asked a vague question along these lines, answered by [Reference De Bondt, Farah and Vignati31, Theorems 1 and 2] as follows (‘stable’ refers to the model-theoretic notion, see, e.g., [Reference Pillay131]).

A result analogous to (6.2.1) for ultrapowers appears in [Reference Farah, Hart and Sherman69, Theorem 5.6], and the original proof of (6.2.1) closely followed the proof of the latter. This proof was considerably simplified by using ideas of Palyutin and Palmgren (see Section 6.1.6). Item (6.2.2) is proven by injecting $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ into the quotient in a gap-preserving manner. Its proof applies in a wider context, showing for example that if M is a reduced product over the asymptotic density zero ideal ${\mathcal {Z}}_0$ (see Section 3.1), or any other ${\mathcal {I}}$ such that $\mathcal {P}({\mathbb N})/{\mathcal {I}}$ is not countably saturated, whose theory is not stable then M is not even countably saturated [Reference De Bondt, Farah and Vignati31, Corollary 2.3].

6.3 The failure of countable saturation

In hindsight, the two factors that make the construction of nontrivial automorphisms of $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ so easy (by today’s standards) are its countable saturation and the elimination of quantifiers in atomless Boolean algebras. The latter bit can be easily dispensed with: In a structure whose theory does not admit elimination of quantifiers, one just needs to consider only partial isomorphisms that are elementary. On the other hand, saturation makes all the difference. In hindsight, it is not very surprising that, unlike ‘commutative’ quotients, the Calkin algebra and many other coronas are not countably saturated. Known instances of the failure of countable saturation in coronas of $\sigma $ -unital $\mathrm {C}^{*}$ -algebras are associated with obstructions of K-theoretic nature.