A dichotomy for derivations and automorphisms of C*-algebras

TL;DR

This paper establishes a dichotomy in the complexity of relations between derivations and automorphisms of separable C*-algebras, linking their properties to descriptive set theory classifications.

Contribution

It introduces a dichotomy for the relations of outer derivations and automorphisms, connecting algebraic properties with descriptive set theoretic complexity classifications.

Findings

Outer derivations are trivial iff inner automorphisms are low complexity in automorphism group.

The space of inner derivations is norm-closed iff inner automorphisms are low complexity.

Provides a complexity-theoretic characterization of C*-algebras with only inner derivations.

Abstract

Building on previous work of Kadison--Ringrose, Elliott, Akemann--Pedersen, and this author, we prove a dichotomy for the relation of outer equivalence of derivations and unitary equivalence of derivable automorphisms for a separable C*-algebra $A$: either such relations are trivial, or the relation $E_{0}^{\mathbb{N}}$ of tail equivalence of countably many binary sequences is reducible to them. When $A$ is furthermore \emph{unital}, this implies that $A$ has no outer derivation if and only if the group $\mathrm{Inn}\left( A\right) $ of inner automorphisms is $\boldsymbol{\Sigma }_{2}^{0}$ in $\mathrm{Aut}\left( A\right) $, if and only if it is $\boldsymbol{\Sigma }_{3}^{0}$ in $\mathrm{Aut}\left( A\right) $. Furthermore, one has that the space of inner derivations is norm-closed if and only if \textrm{Inn}$\left(A\right) $ is norm-closed, if and only if $\mathrm{Inn}\left( A\right) $ is $\boldsymbol{\Pi }_{3}^{0}$ in $\mathrm{\mathrm{Aut}}\left( A\right) $. This provides a complexity-theoretic characterization of C*-algebras with only inner derivations, which as a by-product rules out $D(\boldsymbol{\Pi }_{2}^{0})$ as a possible complexity class for $\mathrm{Inn}\left( A\right) $ in $\mathrm{\mathrm{Aut}}\left( A\right) $ for a separable unital C*-algebra $A$.

Equations

$${\mathrm{H}}^1\left(A\right):=\frac{\mathrm{D}\left(A\right)}{\mathrm{DInn}\left(A\right)}$$

$${\mathrm{Ph}}^{\alpha }\left({\mathrm{Ph}}^{\beta }\left(G/N\right)\right)={\mathrm{Ph}}^{\alpha +\beta }\left(G/N\right)$$

$${\Vert \mathbf{t}\Vert }_{\infty }:=\underset{n}{\sup }\left|t_n\right|$$

$$f:{\left(0,1\right)}^{\mathbb{N}}\to X$$

$$f_0:{\left(0,1\right)}^{\mathbb{N}}/c_0\to X/E\text{,}$$

$$\phi {\left(x\right)}_{⟨n,m⟩}=2^{-n}(x_n{)}_m\text{.}$$

$$\phi \left(x\right)\in c_0\iff \forall n\exists m\forall k\ge m\text{, }{x}_{n,k}=0\text{;}$$

$${\phi }_0:{\left(2^{\mathbb{N}}\right)}^{\mathbb{N}}/E_0^{\mathbb{N}}\to [0,1{]}^{\mathbb{N}}/c_0\text{.}$$

$$f:{\left(2^{\mathbb{N}}\right)}^{\mathbb{N}}\to Y$$

$$f_0:{\left(2^{\mathbb{N}}\right)}^{\mathbb{N}}/E_0^{\mathbb{N}}\to Y/F\text{.}$$

$$g_0:{\left(2^{\mathbb{N}}\right)}^k/E_0^k\to Y/E$$

$$f_0{|}_{X/E_0^{\mathbb{N}}}=(g_0\circ {\pi }_k){|}_{X/E_0^{\mathbb{N}}}$$

$${\pi }_k:{\left(2^{\mathbb{N}}\right)}^{\mathbb{N}}/E_0^{\mathbb{N}}\to {\left(2^{\mathbb{N}}\right)}^k/E_0^k$$

$$h:{\left(-2,2\right)}^{\mathbb{N}}\to Y$$

$$h_0:{\left(-2,2\right)}^{\mathbb{N}}/c_0\to Y/F\text{.}$$

$$g:{\left(2^{\mathbb{N}}\right)}^k\to Y$$

$$g_0:{\left(2^{\mathbb{N}}\right)}^k/E_0^k\to Y/F$$

$$h_0{|}_{X/E_0^{\mathbb{N}}}=(g_0\circ {\pi }_k){|}_{X/E_0^{\mathbb{N}}}\text{.}$$

$$\left(x_0,z,w_z\right)\in X\text{.}$$

$$X_0:=\left\{\left(x_0,z,w_z\right):z\in Z\right\}\text{.}$$

$${\Vert \phi \left(x\right)-\phi \left(x^{\prime }\right)\Vert }_{\infty }\ge 2^{-\left(k+1\right)}\text{.}$$

$$\left\{\phi \left(x\right):x\in X_0\right\}$$

$$\left(h_0\circ {\phi }_0\right)\left(x\right)=\left(g_0\circ {\pi }_k\right)\left(x\right)=\left(g_0\circ {\pi }_k\right)\left(x^{\prime }\right)=\left(h_0\circ {\phi }_0\right)\left(x\right)\text{.}$$

$$\delta \left(xy\right)=x\delta \left(y\right)+\delta \left(x\right)y$$

$$\Vert pa\Vert =\frac{1}{2}\Vert \delta {|}_{pM}\Vert \text{;}$$

$$f:\mathrm{Bal}\mathrm{l}\left(\mathrm{D}\left(A\right)\right)\times \mathrm{Ball}\left(\mathrm{D}\left(A\right)\right)\to \mathrm{Ball}(\mathrm{D}\left(A\right))$$

$$\left(\delta ,{\delta }^{\prime }\right)\mapsto \frac{1}{2}\left(\delta -{\delta }^{\prime }\right)\text{.}$$

$$\left\{\left(\delta ,{\delta }^{\prime }\right)\in \mathrm{Ball}\left(\mathrm{D}\left(A\right)\right)\times \mathrm{Ball}\left(\mathrm{D}\left(A\right)\right):\delta \text{ and }{\delta }^{\prime }\text{ are outer equivalent}\right\}$$

$$\mathrm{Ball}\left(\mathrm{D}\left(A\right)\right)\cap \mathrm{DInn}\left(A\right)$$

Full text

The author was partially supported by the Marsden Fund Fast-Start Grant VUW1816 and the Rutherford Discovery Fellowship VUW2002 “Computing the Shape of Chaos” from the Royal Society of New Zealand, the Starting Grant 101077154 “Definable Algebraic Topology” from the European Research Council, the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA) of the Istituto Nazionale di Alta Matematica (INDAM), and the University of Bologna.

1. Introduction

Automorphisms are fundamental objects associated with a given C*-algebra. The automorphisms of a C*-algebra representing a quantum physical system can be thought of as the symmetries of such a system. Every unitary element of the multiplier algebra of the C*-algebra induces an automorphism defined by conjugation by such a unitary. These automorphisms can be though of as “trivial”. From the physical standpoint, they represent quantum symmetrics corresponding to “observables”.

The study of automorphisms of C*-algebras, and the problem of determining which ones are inner, go back to the origin of C*-algebra theory. Among the works on this topic, the series of papers [9, 10] by Kadison and Ringrose play a very important role, both on their own right and due to their influence on later works. In these papers, the authors fix a separable C*-algebra $A$, and consider the group $\mathrm{\mathrm{Aut}}\left(A\right)$ of automorphisms of $A$ as a topological group. Actually, there are two natural topologies on $\mathrm{Aut}\left(A\right)$: the point-norm topology, i.e., the topology of pointwise convergence, and the norm-topology, i.e. the topology induced by the uniform distance. The point-norm topology renders $\mathrm{\mathrm{Aut}}\left(A\right)$ a *Polish group [11, Section 9], while the norm topology is, generally, not separable. The point-norm topology together with the uniform distance endow $\mathrm{\mathrm{\mathrm{Aut}}}\left(A\right)$ with the structure of Polish topometric group [2]. In the following, all the topological notions concerning $\mathrm{\mathrm{Aut}}\left(A\right)$ will be understood to refer to the point-norm topology, unless otherwise specified. Inner automorphism of $A$ form a subgroup $\mathrm{Inn}\left(A\right)$ which is always a Borel set, although not necessarily closed, nor norm-closed. The problem of determining when $\mathrm{Inn}\left(A\right)$ is norm-closed in $\mathrm{\mathrm{Aut}}\left(A\right)$ is investigated in [9, 10]. As observed below, when $A$ is a unital C-algebra, $\mathrm{Inn}\left(A\right)$ is norm-closed in $\mathrm{\mathrm{Aut}}\left(A\right)$ if and only if it is $\boldsymbol{\Pi}_{3}^{0}$ in $\mathrm{\mathrm{Aut}}\left(A\right)$.

In general, $\mathrm{Inn}\left(A\right)$ is a Polish subgroup of $\mathrm{\mathrm{Aut}}\left(A\right)$. This means that $\mathrm{Inn}\left(A\right)$ is itself a Polish group, with respect to a (necessarily unique) Polish group topology that makes the inclusion $\mathrm{Inn}\left(A\right)\rightarrow\mathrm{\mathrm{Aut}}\left(A\right)$ continuous. Indeed, the group $U\left(A\right)$ of unitary multipliers of $A$ is Polish with respect to the strict topology, and $\mathrm{Inn}\left(A\right)$ is the image of $U\left(A\right)$ under the continuous homomorphism that maps a unitary multiplier to the corresponding inner automorphism. In particular, $\mathrm{Inn}\left(A\right)$ is always a Borel set, and as remarked above the problem of determining its Borel complexity can be seen as a generalization of the problem of determining when it is norm-closed (when $A$ is unital at least).

Automorphisms of C*-algebras are closely connected with derivations and -derivations. A (-)derivation of a C*-algebra is a (self-adjoint) bounded linear operator that satisfies the Leibnitz identity. Derivations and -derivations arise naturally in the study of uniformly-continuous one-parameter groups of automorphisms of C-algebras, as these all admit “infinitesimal generators” that are -derivations. A (-)derivation is *inner *if it is induced by a (selfadjoint) element $a$ of the multiplier algebra, i.e., it is of the form $x\mapsto[ia,x]$. This corresponds to the fact that the corresponding uniformly-continuous one-parameter group of automorphisms is induced by the unitary $\exp\left(ia\right)$.

The problem of determining which derivations are inner has attracted considerable attention in the early days of C*-algebra theory. Particularly, Elliott and Akemann–Pedersen characterized the separable C*-algebras where all the derivations are inner [6, 1]. More recently, Somerset has characterized the separable unital C*-algebras where the space of derivations is *closed *with respect to the topology induced by the uniform distance [20]. (The space of inner derivations is always dense in the topology of pointwise convergence.)

Much like $\mathrm{\mathrm{Aut}}\left(A\right)$, $\mathrm{D}\left(A\right)$ can be seen as a “Polish-Banach space”. This means that $\mathrm{D}\left(A\right)$ is a Banach space with respect to the operator norm, such that for every $n\in\mathbb{N}$ the ball $n\mathrm{\mathrm{Ball}}(\mathrm{D}\left(A\right))$ of radius $n$ is a Polish space with respect to the topology of pointwise convergence, such that the norm is lower semicontinuous and the vector space operations are continuous. As before, topological notions pertaining to $\mathrm{D}\left(A\right)$ will refer to the Polish topology on bounded sets, unless otherwise specified. Inner derivations form a Borel subspace $\mathrm{DInn}\left(A\right)$ of $\mathrm{D}\left(A\right)$. The space

[TABLE]

can be seen as a particular instance of Banach algebra cohomology [8]. Precisely, $\mathrm{H}^{1}\left(A\right)$ is the cohomology $\mathrm{H}^{1}\left(M\left(A\right),M\left(A\right)\right)$ in degree $1$ of the multiplier algebra $M\left(A\right)$ of $A$ with coefficients in $M\left(A\right)$. One can consider $\mathrm{H}^{1}\left(A\right)$ as a seminormed space with respect to the quotient seminorm. Clearly, such a seminorm is actually a norm, and $\mathrm{H}^{1}\left(A\right)$ is a Banach space, if and only if $\mathrm{DInn}\left(A\right)$ is norm-closed in $\mathrm{D}\left(A\right)$.

A *-derivation $\delta$ on $A$ induces an automorphism $\mathrm{\exp}\left(\delta\right)$ of $A$ [16, Section 8.7]. Automorphisms of this form are called derivable. Two derivations are outer equivalent if their difference is inner. We say that a derivation is contractive if it has norm less than or equal to $1$.

The main results of [6, 1] yield the following characterization of separable C*-algebras with only inner derivations.

Theorem 1.1** (Akemann–Elliott–Pedersen).**

Let $A$ be a separable C-algebra. The following assertions are equivalent:*

1. (1)

$\mathrm{H}^{1}\left(A\right)=0$, i.e., every derivation of $A$ is inner; 2. (2)

every summable central sequence of $A$ is trivial; 3. (3)

$A=B\oplus C$* where $B$ has continuous trace and $C$ is a direct sum of simple C*-algebras.*

We will give a *complexity-theoretic *refinement of such a characterization, in the context of Borel complexity of sets and equivalence relations [7]. Recall that $E_{0}$ denotes the relation of *tail equivalence *on the space $2^{\mathbb{N}}$ of binary sequences. The relation $E_{0}^{\mathbb{N}}$ on $\left(2^{\mathbb{N}}\right)^{\mathbb{N}}$ is the countable product of copies of $E_{0}$. It can thus be seen as the relation of tail equivalence of countably many binary sequences. In terms of such a relation, we obtain the following Dichotomy Theorem for the relation of outer equivalence of derivations and unitary equivalence of derivable automorphisms.

Theorem 1.2**.**

Let $A$ be a separable C-algebra. Let $E$ be either relation of outer equivalence of (contractive) derivations of $A$, or the relation of unitary equivalence of derivable automorphisms of $A$. Then either $E$ is trivial, or the relation $E_{0}^{\mathbb{N}}$ of tail equivalence of countably many binary sequenes is Borel reducible to $E$.*

In the proof of Theorem 1.2 we will provide several equivalent characterizations of separable C*-algebras with outer derivations; see Theorem 11.1. One of these is Property APE, a variation of Property AEP introduced by [13]. A major ingredient in the proof of Theorem 1.2 is Shani’s Dichotomy Theorem for the relation $E_{0}^{\mathbb{N}}$ [19, Theorem 1.1].

In the* unital* case, one can rephrase our main result in terms of the complexity of $\mathrm{Inn}\left(A\right)$ as a Borel subset of $\mathrm{\mathrm{Aut}}\left(A\right)$. Recall that if $\Gamma$ is a complexity class for Borel subsets of Polish spaces that is not self-dual, and $A$ is a subset of a Polish space $X$, then we say that $A$ is $\Gamma$ in $X$ if $A\in\Gamma\left(X\right)$, and $\Gamma$ is the complexity class of $A$ in $X$ if and only if $A\in\Gamma\left(X\right)$ and $X\setminus A\notin\Gamma\left(X\right)$. The complexity class $D(\boldsymbol{\Pi}_{2}^{0})$ comprises the sets that one can write as intersection between a $\boldsymbol{\Pi}_{2}^{0}$ set and a $\boldsymbol{\Sigma}_{2}^{0}$ set. The possible *complexity classes *of Polish subgroups of Polish group have been completely classified in [15]. In particular, it is shown in [15] that if $H$ is a $\boldsymbol{\Pi}_{3}^{0}$ and not closed Polish subgroup of a Polish group of $G$, then its complexity class is either $\boldsymbol{\Sigma}_{2}^{0}$, or $D(\boldsymbol{\Pi}_{2}^{0})$, or $\boldsymbol{\Pi}_{3}^{0}$.

The following result characterizing the separable unital C*-algebras $A$ such that $\mathrm{Inn}\left(A\right)$ is closed is a consequence of [18, Theorem 0.8], [13, Theorem 1.2], and [7, Theorem 6.3.1]; see also [17, Theorem 3.1].

Theorem 1.3**.**

Let $A$ be a separable unital C-algebra. The following assertions are equivalent:*

1. (1)

$A$* has continuous trace;* 2. (2)

$\mathrm{Inn}\left(A\right)$* is closed in $\mathrm{\mathrm{Aut}}\left(A\right)$;* 3. (3)

the relation of unitary equivalence of automorphisms of $A$ is classifiable by countable structures; 4. (4)

$E_{0}$* is not Borel reducible to the relation of unitary equivalence of automorphisms of $A$.*

We extend this by characterizing the separable unital C*-algebras $A$ such that $\mathrm{Inn}\left(A\right)$ is $\boldsymbol{\Sigma}_{2}^{0}$ or $\boldsymbol{\Pi}_{3}^{0}$, and ruling out $D(\boldsymbol{\Pi}_{2}^{0})$ as a possible complexity class.

Theorem 1.4**.**

Let $A$ be a separable unital C-algebra. The following assertions are equivalent:*

1. (1)

$\mathrm{H}^{1}\left(A\right)=0$, i.e., every derivation of $A$ is inner; 2. (2)

$\mathrm{Inn}\left(A\right)$* is $\boldsymbol{\Sigma}_{2}^{0}$ in $\mathrm{\mathrm{Aut}}\left(A\right)$;* 3. (3)

$\mathrm{Inn}\left(A\right)$* is $\boldsymbol{\Sigma}_{3}^{0}$ $\mathrm{\mathrm{Aut}}\left(A\right)$;* 4. (4)

$E_{0}^{\mathbb{N}}$* is not Borel reducible to the coset relation of $\mathrm{Inn}\left(A\right)$ in $\mathrm{\mathrm{Aut}}\left(A\right)$.*

Furthermore, the following assertions are equivalent:

1. (a)

$\mathrm{H}^{1}\left(A\right)$* is a Banach space, i.e., $\mathrm{DInn}\left(A\right)$ is norm-closed in $\mathrm{D}\left(A\right)$;* 2. (b)

$\mathrm{Inn}\left(A\right)$* is norm-closed in $\mathrm{\mathrm{Aut}}\left(A\right)$;* 3. (c)

$\mathrm{Inn}\left(A\right)$* is $\boldsymbol{\Pi}_{3}^{0}$ in $\mathrm{Aut}\left(A\right)$.*

Proof.

The implication (1)$\Rightarrow$(4) follows from Theorem 1.2.

(4)$\Rightarrow$(1) Since $E_{0}^{\mathbb{N}}$ is the coset relation of a Polish subgroup $H$ of a Polish group $G$ of complexity class $\boldsymbol{\Pi}_{3}^{0}$ [14, Proposition 4.14]. Here, $G$ is the countable product of copies of $\left(\mathbb{Z}/2\right)^{\mathbb{N}}$, and $H$ is the countable product of copies of $\left(\mathbb{Z}/2\right)^{\left(\mathbb{N}\right)}$. By [14, Proposition 4.13] this implies that $\mathrm{Inn}\left(A\right)$ is not $\boldsymbol{\Sigma}_{3}^{0}$ in $\mathrm{\mathrm{Aut}}\left(A\right)$.

(2)$\Rightarrow$(1) If $\alpha$ is an automorphism of $A$ such that $\left\|\alpha-\mathrm{id}_{A}\right\|<2$, then $\alpha$ is *derivable *[16, Theorem 8.7.7]. Thus, there exists a *-derivation $\delta$ of $A$ such that $\alpha=\exp\left(\delta\right)$. Since by assumption $\delta$ is inner, also $\alpha$ must be inner. Let $\left(u_{n}\right)$ be a dense sequence in $U\left(A\right)$. Then we have that an automorphism $\alpha$ of $A$ is inner if and only if there exists $n\in\omega$ such that $\left\|\mathrm{Ad}\left(u_{n}\right)-\alpha\right\|\leq 1$. This shows that $\mathrm{Inn}\left(A\right)$ is $\boldsymbol{\Sigma}_{2}^{0}$ in $\mathrm{Aut}\left(A\right)$.

The equivalence of (a) and (b) is the equivalence of (i) and (iv) in [9, Theorem 5.3]. The equivalence of (b) and (c) follows from Proposition 6.2. ∎

Corollary 1.5**.**

*The complexity class $D(\boldsymbol{\Pi}_{2}^{0})$ cannot arise as the complexity class of $\mathrm{Inn}\left(A\right)$ within $\mathrm{\mathrm{Aut}}\left(A\right)$ for a separable unital C-algebra $A$. Indeed, if $\mathrm{Inn}\left(A\right)$ is $\boldsymbol{\Pi}_{3}^{0}$ and not closed in $\mathrm{\mathrm{Aut}}\left(A\right)$ then its complexity class is either $\boldsymbol{\Sigma}_{2}^{0}$ or $\boldsymbol{\Pi}_{3}^{0}$.

Proof.

By [15, Theorem 1.1], if $\mathrm{Inn}\left(A\right)$ is $\boldsymbol{\Pi}_{3}^{0}$ and not closed, then its complexity class is either $\boldsymbol{\Sigma}_{2}^{0}$, or $D(\boldsymbol{\Pi}_{2}^{0})$, or $\boldsymbol{\Pi}_{3}^{0}$. By Theorem 1.4, it cannot be $D(\boldsymbol{\Pi}_{2}^{0})$. ∎

The conclusion of Corollary 1.5 does not hold if $A$ is not unital. An example of a separable continuous trace C*-algebra such that $\mathrm{Inn}\left(A\right)$ has complexity class $D(\boldsymbol{\Pi}_{2}^{0})$ in $\mathrm{\mathrm{Aut}}\left(A\right)$ is produced in [3].

Acknowledgments

We are grateful to Eusebio Gardella, Ilja Gogic, and Assaf Shani for helpful conversations and useful remarks.

2. Polish groups and their heart

Recall that a *Polish space *is a topological space with a countable basis and a compatible complete metric. A Polish group is a topological group whose topology is Polish. A subgroup $N$ of a Polish group $G$ is *Polish *if it is endowed with a (necessarily unique) Polish topology that makes the inclusion $N\rightarrow G$ continuous. A homogeneous space with a Polish cover is a space explicitly presented as $G/N$ where $G$ is a Polish group and $N$ is a Polish subgroup. When $N$ is normal in $G$, $G/N$ is also a group, called group with a Polish cover. The groups with a Polish cover $G/N$ where $G$ and $N$ are abelian Polish groups are the objects of the *left heart *of the category of abelian Polish groups [14].

A subspace with a Polish cover of a homogeneous space with a Polish cover $G/N$ is a subspace of the form $H/N$ where $H$ is a Polish subspace of $G$. As $H$ is a Borel subset of $G$, one can define the *complexity *of $H/N$ in $G/N$ to be the complexity of $H$ in $G$ (as a Borel set). Thus, for example we say that $H/N$ is $\Gamma_{\alpha}$ in $G/N$ if $H$ is $\Gamma_{\alpha}$ in $G$.

A subgroup with a Polish cover of an abelian group with a Polish cover $G/N$ is a subgroup of the form $H/N$ where $H$ is a Polish subgroup of $G$. As $H$ is a Borel subset of $G$, one can define the *complexity *of $H/N$ in $G/N$ to be the complexity of $H$ in $G$ (as a Borel set). Thus, for example we say that $H/N$ is $\Gamma_{\alpha}$ in $G/N$ if $H$ is $\Gamma_{\alpha}$ in $G$, and $\Gamma_{\alpha}$ is the complexity class of $H/N$ in $G/N$ if $H$ is $\Gamma_{\alpha}$ in $G$ and $G\setminus H$ is not $\Gamma_{\alpha}$ in $G$.

A homogeneous space with a Polish cover $G/N$ admits a canonical chain of subspaces $\mathrm{Ph}^{\alpha}\left(G/N\right)=G_{\alpha}/N$ for $\alpha<\omega_{1}$, called phantom (or Solecki) subspaces, characterized by their complexity. Thus, $\mathrm{Ph}^{\alpha}\left(G/N\right)$ is the smallest $\Gamma_{\alpha}$ subspace with a Polish cover of $G/N$. Such subspaces can be explicitly defined by recursion on $\alpha$. Thus, $G_{0}$ is the closure of $N$ in $G$, while $G_{1}$ comprises the $x\in G$ such that for every identity neighborhood $V$ of $N$ there exist $z,w\in N$ such that $xz$ and $wx$ belong to the closure of $V$ in $G$. One then has that $\mathrm{Ph}^{\alpha+1}\left(G/N\right)=\mathrm{Ph}^{1}\left(\mathrm{Ph}^{\alpha}\left(G/N\right)\right)$, while at limits one takes intersections. By induction, one sees that

[TABLE]

for all countable ordinals $\alpha,\beta$. The *Solecki length *of $G/N$ is the least $\alpha<\omega_{1}$ such that $\mathrm{Ph}^{\alpha}\left(G/N\right)$ is trivial. It is proved in [15, Theorem 7.2] that the Solecki length of $G/N$ is the least $\alpha<\omega_{1}$ such that for every open subset $U$ of $N$, there exists a $\boldsymbol{\Sigma}_{1+\alpha}^{0}$ subset $X$ of $G$ such that $U=X\cap N$. This is equivalent to the assertion that the identity map of $N$ is $\boldsymbol{\Sigma}_{1+\alpha}^{0}$-measurable as a function from $N$ with its subspace inherited from $G$ to $N$ with its Polish topology. One can explicitly characterize the elements of the Polish subgroup $G_{\alpha}$ of $G$ such that $\mathrm{Ph}^{\alpha}\left(G/N\right)=G_{\alpha}/N$.

Definition 2.1**.**

Suppose that $G$ is a Polish group, and $N$ is a normal Polish subgroup of $G$. We denote by $V$ an identity neighborhoods of $N$. An element $x$ of $\hat{G}$ is:

1. (1)

$V$-phantom of order zero if, for every identity neighborhood $A$ in $G$ there exists $g\in V$ such that $gx\in A$; 2. (2)

$V$-phantom of order $\beta+1$ if, for every identity neighborhood $W$ in $N$ there exists $g\in V$ such that $gx$ is $W$-phantom of order $\beta$; 3. (3)

$V$-phantom of order a limit ordinal $\lambda$ if it is $V$-phantom of order $\beta$ for every $\beta<\lambda$; 4. (4)

phantom of order $\alpha$ if and only if it is $N$-phantom of order $\alpha$.

Then one can show by induction on $\alpha$ that an element $x$ of $G$ is $V$-phantom of order $\alpha$ if and only if it belongs to the closure of $V$ inside $G_{\alpha}$. In particular, since $V$ is dense in $G_{\alpha}$, an element $x$ of $G$ is phantom of order $\alpha$ if and only if $x+N$ belongs to $\mathrm{Ph}^{\alpha}\left(G/N\right)$.

3. $c_{0}$-orthogonality

To simplify the statement and proofs of our main results, we introduce a new notion concerning equivalence relations on Polish spaces. Denote by $c_{0}$ the separable Banach space of vanishing sequences of real numbers, endowed with the supremum norm. We regard $c_{0}$ as a subspace of $\mathbb{R}^{\mathbb{N}}$. For a subset $X$ of $\mathbb{R}^{\mathbb{N}}$, we let $X/c_{0}$ be the quotient of $X$ by the restriction of the coset relation of $c_{0}$ on $\mathbb{R}^{\mathbb{N}}$ to $X$. For $\boldsymbol{t}\in\mathbb{R}^{\mathbb{N}}$ we define

[TABLE]

to be its supremum norm. A subset $Y$ of $\mathbb{R}^{\mathbb{N}}$ is:

• •

$\delta$-separated if, for every distinct $\boldsymbol{t},\boldsymbol{s}\in Y$, $\left\|\boldsymbol{t}-\boldsymbol{s}\right\|_{\infty}\geq\delta$;

• •

well-separated if it is $\delta$-separated for some $\delta>0$.

To simplify the statement of the main results, we consider the following natural notion:

Definition 3.1**.**

An equivalence relation $E$ on a Polish space $X$ is $c_{0}$-orthogonal if for every Borel function

[TABLE]

that induces a function

[TABLE]

there exists an uncountable well-separated $Y\subseteq\left(0,1\right)^{\mathbb{N}}$ such that $f_{0}$ is constant on $Y/c_{0}$.

Clearly, in the definition of $c_{0}$-orthogonal one can replace $\left(0,1\right)$ with any other open interval.

4. Tail equivalence and dichotomy

We now establish a relation between the notion of $c_{0}$-orthogonality and the relation of tail equivalence of countably many binary sequences. Recall that $E_{0}$ denotes the relation on $2^{\mathbb{N}}$ of *tail equivalence *of binary sequences. Let $\left\langle\cdot,\cdot\right\rangle:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ be a bijection such that $\left\langle n,m\right\rangle\leq\left\langle n^{\prime},m^{\prime}\right\rangle$ for all $n,m,n^{\prime},m^{\prime}\in\mathbb{N}$ such that $n\leq n^{\prime}$ and $m\leq m^{\prime}$. Define $\varphi:\left(2^{\mathbb{N}}\right)^{\mathbb{N}}\rightarrow\mathbb{R}^{\mathbb{N}}$ by setting

[TABLE]

Then $\varphi$ is a continuous homomorphism such that

[TABLE]

see [7, Lemma 8.5.3]. Thus, $\varphi$ induces an injective function

[TABLE]

We recall the statement of Shani’s Dichotomy Theorem for the relation $E_{0}^{\mathbb{N}}$ on $\left(2^{\mathbb{N}}\right)^{\mathbb{N}}$ [19, Theorem 1.1].

Theorem 4.1** (Shani).**

Let $F$ be an analytic equivalence relation on a Polish space $Y$. Suppose that the relation $E_{0}^{\mathbb{N}}$ of tail equivalence of countably many binary sequences is not Borel reducible to $F$. Let

[TABLE]

be a Borel function that induces a function

[TABLE]

Then there exist $k\in\omega$, a comeager $X\subseteq\left(2^{\mathbb{N}}\right)^{\mathbb{N}}$ and a Borel function $g:(2^{\mathbb{N}})^{k}\rightarrow Y$ that induces a Borel function

[TABLE]

and such that

[TABLE]

where

[TABLE]

is the projection on the first $k$ coordinates.

We isolate the following consequence of Shani’s $E_{0}^{\mathbb{N}}$ Dichotomy Theorem.

Corollary 4.2**.**

Let $F$ be an analytic equivalence relation on $Y$. If $F$ is not $c_{0}$-orthogonal, then the relation $E_{0}^{\mathbb{N}}$ is Borel reducible to $F$.

Proof.

We prove the contrapositive. Suppose that the relation $E_{0}^{\mathbb{N}}$ of tail equivalence of countably many binary sequences is not reducible to $F$. We verify that $F$ is $c_{0}$-orthogonal. Let

[TABLE]

be a Borel function that induces a function

[TABLE]

Consider the Borel function $h\circ\varphi:\left(2^{\mathbb{N}}\right)^{\mathbb{N}}\rightarrow Y$. By Shani’s Dichotomy Theorem, there exist $k\in\mathbb{N}$, a comeager subset $X\subseteq\left(2^{\mathbb{\mathbb{N}}}\right)^{\mathbb{N}}$ and a Borel function

[TABLE]

that induces a Borel function

[TABLE]

such that

[TABLE]

By the Kuratowski–Ulam theorem [11, Theorem 8.41], there exists $x_{0}\in\left(2^{\mathbb{N}}\right)^{k}$, a comeager subset $Z\subseteq 2^{\mathbb{N}}$, such that for every $z\in Z$ there exists $w_{z}\in\left(2^{\mathbb{N}}\right)^{\mathbb{N}}$ such that

[TABLE]

Consider

[TABLE]

Observe that, for distinct $x,x^{\prime}\in X_{0}$,

[TABLE]

Thus, we have that

[TABLE]

is an uncountable well-separated subset of $\left(-2,2\right)^{\mathbb{N}}$. Furthermore, for $x,x^{\prime}\in X_{0}/E_{0}^{\mathbb{N}}$ we have that

[TABLE]

This concludes the proof. ∎

5. Derivations

Suppose that $A$ is a C*-algebra. A *derivation *of $A$ is a (necessarily bounded) linear map $\delta$ on $A$ satisfying the derivation identity

[TABLE]

for all $x,y\in A$ [16, Section 8.6]. A *-derivation is a derivation $\delta$ such that $\delta\left(x\right)$ is self-adjoint whenever $x\in A$ is self-adjoint. The operator norm $\left\|\delta\right\|$ of a *-derivation is equal to its spectral radius [16, Theorem 8.6.5]. For $a,b\in A$ we define their commutator $[a,b]:=ab-ba$. Every self-adjoint element of $A$ determines a *-derivation $\mathrm{ad}\left(ia\right):x\mapsto[ia,x]$. The *-derivations of this form are called inner.

If $M$ is a von Neumann algebra that contains $A$, then every *-derivation $\delta$ of $A$ is $\sigma$-weakly continuous and extends to a *-derivation of $M$, which we still denote by $\delta$. Furthermore, there exists a unique $a\in M_{\mathrm{sa}}$ with $\delta=\mathrm{ad}\left(ia\right)$ and such that for every projection $p\in Z\left(M\right)$,

[TABLE]

see [9, Theorem 3.1].

For a separable C*-algebra $A$, we let $\mathrm{D}\left(A\right)$ be the Banach space of derivations of $A$. We let $\mathrm{DInn}\left(A\right)$ be the subspace consisting of inner *-derivations. For every $n\in\mathbb{N}$, the ball $n\mathrm{\mathrm{Ball}}\left(\mathrm{D}\left(A\right)\right)$ is a Polish space when endowed with the topology of pointwise convergence. We regard $\mathrm{D}\left(A\right)$ as a standard Borel space endowed with the Borel structure that renders it the colimit of $\left(n\mathrm{\mathrm{Ball}}\left(\mathrm{D}\left(A\right)\right)\right)_{n\in\mathbb{N}}$ in the category of standard Borel spaces and Borel functions. We call outer equivalence the coset relation of $\mathrm{DInn}\left(A\right)$ within $\mathrm{D}\left(A\right)$. The quotient of $\mathrm{D}\left(A\right)$ by $\mathrm{DInn}\left(A\right)$ is denoted by $\mathrm{H}^{1}\left(A\right)$.

Proposition 5.1**.**

Let $A$ be a separable C-algebra. The relation of outer equivalence of derivations on $A$ is Borel.*

Proof.

It suffices to prove that, for every $n\in\mathbb{N}$, the relation of outer equivalence of derivations of norm at most $n$ is Borel. By homogeneity, it suffices to consider the case $n=1$. Consider the continuous function

[TABLE]

[TABLE]

Then we have that

[TABLE]

is the preimage of

[TABLE]

under the continuous function $f$. Thus, it suffices to prove that

[TABLE]

is a Borel subset of $\mathrm{\mathrm{Ball}}\left(\mathrm{D}\left(A\right)\right)$. To this purpose, it suffices to prove that, for every $n\in\mathbb{N}$, the set

[TABLE]

is a Borel subset of $n\mathrm{\mathrm{Ball}}\left(\mathrm{D}\left(A\right)\right)$. By homogeneity, it suffices to consider the case $n=1$. Let $Z\left(A\right)=A^{\prime}\cap M\left(A\right)$ be the center of $M\left(A\right)$. Then $Z\left(A\right)\cap\mathrm{\mathrm{\mathrm{Ball}}}\left(M\left(A\right)\right)$ is strictly closed in $\mathrm{\mathrm{\mathrm{Ball}}}\left(M\left(A\right)\right)$. Thus the coset relation $F$ of $Z\left(A\right)$ on $\mathrm{\mathrm{Ball}}\left(M\left(A\right)\right)$ is smooth, and the quotient $X:=\mathrm{\mathrm{Ball}}\left(M\left(A\right)\right)/F$ is a standard Borel space. The continuous function

[TABLE]

induces an injective Borel function

[TABLE]

whose image is

[TABLE]

By [11, Theorem 15.1], this shows that such a set is Borel, concluding the proof. ∎

6. Automorphisms

Suppose that A\is a separable C*-algebra. Then $\mathrm{Aut}\left(A\right)$ is a Polish group endowed with the topology of pointwise convergence. Let $M\left(A\right)$ be the multiplier algebra of $A$, and $U\left(A\right)$ be the unitary group of $M\left(A\right)$. Then $\mathrm{\mathrm{Ball}}\left(M\left(A\right)\right)$ is a Polish space with respect to the strict topology, generated by the seminorms

[TABLE]

for $a\in A$. Since the $U\left(A\right)$ is norm-closed in $\mathrm{Ball}(M\left(A\right))$ and the norm is strictly lower-semicontinuous, it follows that $U\left(A\right)$ is $G_{\delta}$ in $\mathrm{\mathrm{Ball}}\left(M\left(A\right)\right)$, and hence Polish [11, Theorem 3.11]. The subgroup $\mathrm{Inn}\left(A\right)$ of *inner *automorphisms is a Polish subgroup, being the image of the Polish group $U\left(A\right)$ of unitary multipliers under the continuous homomorphism

[TABLE]

We can define the homogeneous space with a Polish cover

[TABLE]

We have the corresponding phantom subspaces

[TABLE]

from $\alpha<\omega_{1}$.

Definition 6.1**.**

Let $A$ be a separable C*-algebra. The *automorphism length *of $A$ is the least $\alpha<\omega_{1}$ such that $\mathrm{Ph}^{\alpha}\mathrm{Out}\left(A\right)$ is trivial.

It is easily seen that a unital separable C-algebra has automorphism length zero if and only if every central sequence is trivial. This is equivalent to the assertion that the C-algebra has continuous trace [1, Theorem 2.4]. In general, a separable C*-algebra has automorphism length zero if and only if it has continuous trace, and its spectrum $X$ is such that the Čech cohomology group $\mathrm{H}^{2}\left(X\right)$ is a Polish group; see [3].

Lemma 6.2**.**

Let $A$ be a separable unital C-algebra. We have that $\mathrm{Ph}^{1}\mathrm{\mathrm{Aut}}\left(A\right)$ is the norm-closure of $\mathrm{Inn}\left(A\right)$. The Polish group topology on $\mathrm{Ph}^{1}\mathrm{\mathrm{Aut}}\left(A\right)$ is induced by the norm.*

Proof.

For $k\in\omega$ let $V_{k}$ be the group of inner automorphisms $\beta$ of $A$ such that $\left\|\beta-\iota\right\|\leq 2^{-k}$. Then $\left(V_{k}\right)$ is a basis of zero neighborhoods of $\mathrm{Inn}\left(A\right)$. Furthermore, the closure $\overline{V}_{k}^{\mathrm{\mathrm{Aut}}\left(A\right)}$ of $V$ in $\mathrm{\mathrm{Aut}}\left(A\right)$ is contained in the set of automorphisms $\beta$ of $A$ such that $\left\|\beta-\iota\right\|\leq 2^{-k}$.

Denote by $H$ the closure of $\mathrm{Inn}\left(A\right)$ with respect to the uniform distance. Let $\left(u_{n}\right)_{n\in\omega}$ be a dense sequence in $U\left(A\right)$. Then we have that $\alpha\in H$ if and only if for every $k\in\mathbb{N}$ there exists $n\in\omega$ such that $\left\|\alpha-\mathrm{Ad}\left(u_{n}\right)\right\|\leq 2^{-k}$. This shows that $H$ is $\boldsymbol{\Pi}_{3}^{0}$ in $\mathrm{\mathrm{Aut}}\left(A\right)$. This implies that $H\subseteq\mathrm{Ph}^{1}\mathrm{\mathrm{Aut}}\left(A\right)$.

Conversely, suppose that $\alpha\in\mathrm{Ph}^{1}\mathrm{\mathrm{Aut}}\left(A\right)$. By definition of $s_{1}\left(\mathrm{Out}\left(A\right)\right)$, for every $k\in\mathbb{N}$ there exists $u\in U\left(A\right)$ such that $\mathrm{Ad}\left(u\right)\circ\alpha$ belongs to $\overline{V}_{k}^{\mathrm{\mathrm{Aut}}\left(A\right)}$. This implies that $\left\|\mathrm{Ad}\left(u\right)\circ\alpha-\iota\right\|\leq 2^{-k}$. As this holds for every $k\in\mathbb{N}$, $\alpha\in H$.

This shows that $\mathrm{Ph}^{1}\mathrm{\mathrm{Aut}}\left(A\right)$ is equal to the closure of $\mathrm{Inn}\left(A\right)$ with respect to the uniform distance. We have that setting $W_{k}$ to be the intersection of $\overline{V}_{k}^{\mathrm{\mathrm{Aut}}\left(A\right)}$ with $\mathrm{Ph}^{1}\mathrm{\mathrm{\mathrm{Au}}t}\left(A\right)$, one has that $\left(W_{k}\right)$ is a basis of identity neighborhoods in $\mathrm{Ph}^{1}\mathrm{\mathrm{Aut}}\left(A\right)$; see [15, Section 4]. It is easily seen that $W_{k}$ comprises the elements $\beta$ of $\mathrm{Ph}^{1}\mathrm{\mathrm{Aut}}\left(A\right)$ such that $\left\|\beta-\iota\right\|\leq 2^{-k}$. This shows that the Polish group topology on $\mathrm{Ph}^{1}\mathrm{\mathrm{\mathrm{Au}}t}\left(A\right)$ is induced by the uniform distance. Alternatively, this follows from the uniqueness of the Polish group topology on $\mathrm{Ph}^{1}\mathrm{\mathrm{Aut}}\left(A\right)$ that makes the inclusion $\mathrm{Ph}^{1}\mathrm{\mathrm{Aut}}\left(A\right)\rightarrow\mathrm{\mathrm{Aut}}\left(A\right)$ continuous. ∎

Proposition 6.3**.**

Let $A\subseteq B\left(H\right)$ be a concrete C-algebra, and $M$ its w*-closure. Suppose that the center $ZM$ of $M$ is contained in $M\left(A\right)$. Then a -derivation $\delta$ of $A$ satisfying $\left\|\delta\right\|<2\pi$ is inner if and only if $\mathrm{\exp}\left(\delta\right)$ is an inner automorphism.

Proof.

The proof is similar to the one of [9, Theorem 4.6]. Let $U\left(A\right)$ be the unitary group of $M\left(A\right)$. Let $\delta$ be a *-derivation of $A$ such that $\mathrm{\exp}\left(\delta\right)$ is an inner automorphism. Since $\left\|\delta\right\|<2\pi$, there exists a selfadjoint element $b$ of $M$ such that $\delta=\mathrm{ad}\left(ib\right)$ and $\left\|b\right\|<\pi$, whence $\exp\left(\delta\right)=\mathrm{Ad}\left(\exp\left(ib\right)\right)$. If $\exp\left(\delta\right)$ is inner, there exists a unitary multipleir $u$ of $A$ such that $\exp\left(\delta\right)=\mathrm{Ad}\left(u\right)$. This shows that

[TABLE]

belongs to $U\left(ZM\right)\subseteq U\left(A\right)$. Thus,

[TABLE]

Since $\left\|b\right\|<\pi$ we have that $-1$ does not belong to the spectrum of $\exp\left(ib\right)$. Hence, by continuous functional calculus we have $b\in M\left(A\right)$, concluding the proof. The other implication is obvious. ∎

Corollary 6.4**.**

Let $A$ be a separable primitive C-algebra. Then a derivation $\delta$ of $A$ satisfying $\left\|\delta\right\|<2\pi$ is inner if and only if $\mathrm{\exp}\left(\delta\right)$ is inner.*

7. The APE Property

Property AEP for a separable C*-algebra $A$ have been introduced in [13, Definition 4.4]. This notion is inspired by the work of Elliott and Akemann–Pedersen [6, 1] characterizing C*-algebras with only inner derivations. It is proved in [13, Theorem 4.5] that Property AEP is equivalent to the property of having an outer derivation.

Property AEP was defined in terms of the relation of $\ell_{1}$-equivalence of sequences. It was motivated by the goal of showing that, for C*-algebras with outer derivations, the relation of unitary equivalence of automorphisms of C*-algebras is not classifiable by countable structures. We now introduce a variant, which we term Property Akemann–Pedersen–Elliott (APE, Latin root of “bee”).

Suppose that $A$ is a separable C*-algebra, $\boldsymbol{a}=\left(a_{n}\right)$ is a dense sequence in $\mathrm{\mathrm{Ball}}\left(A\right)$. Suppose that $\boldsymbol{x}:=\left(x_{n}\right)$ is a sequence of pairwise orthogonal positive contractions of $A$ such that, for every $n\in\mathbb{N}$ and $i\leq n$,

[TABLE]

As remarked in [13, Theorem 4.5], if $\boldsymbol{t}\in\left(0,1\right)^{\mathbb{N}}$, then the series

[TABLE]

converges in the strong operator topology to a selfadjoint element $x_{\boldsymbol{t}}$ of $A^{\ast\ast}$.

For $n\in\mathbb{N}$, let $\alpha_{\boldsymbol{t},n}$ be the inner automorphism associated with the unitary

[TABLE]

Then the sequence $\left(\alpha_{\boldsymbol{t},n}\right)_{n\in\mathbb{N}}$ converges to an automorphism $\alpha_{\boldsymbol{t}}$ of $A$. This is a *derivable *automorphism, associated with the *-derivation $\delta_{\boldsymbol{t}}:=\mathrm{ad}\left(ix_{\boldsymbol{t}}\right)$ induced by the positive contraction $x_{\boldsymbol{t}}\in A^{\ast\ast}$.

Since the $x_{n}$’s are pairwise orthogonal, if $\boldsymbol{t},\boldsymbol{s}\in\left(0,1\right)^{\mathbb{N}}$ are such that $\boldsymbol{t}-\boldsymbol{s}\in c_{0}$, then the series

[TABLE]

converges to a positive contraction $x_{\boldsymbol{t}-\boldsymbol{s}}\in A$. Then we have that

[TABLE]

and

[TABLE]

Definition 7.1**.**

Let $A$ be a separable C*-algebra, and $\boldsymbol{a}=\left(a_{n}\right)$ be a dense sequence in $\mathrm{\mathrm{Ball}}\left(A\right)$. A sequence $\boldsymbol{x}=\left(x_{n}\right)$ as above is an APE sequence for $\boldsymbol{a}$ if for every uncountable well-separated $Y\subseteq\left(0,1\right)^{\mathbb{N}}$ there exist $\boldsymbol{s},\boldsymbol{t}\in Y$ such that $\alpha_{\boldsymbol{t}}$ and $\alpha_{\boldsymbol{s}}$ are not unitarily equivalent.

Definition 7.2**.**

Let $A$ be a separable C*-algebra. Then $A$ satisfies the APE Property if for every dense sequence $\boldsymbol{a}$ in $\mathrm{\mathrm{Ball}}\left(A\right)$ there exists an APE sequence for $A$.

We now consider the implications that the APE Property has for the relations of outer equivalence of derivations and unitary equivalence of automorphisms.

Lemma 7.3**.**

Let $A$ be a separable C-algebra. Consider the following conditions:*

1. (1)

$A$* satisfies Property APE;* 2. (2)

the relations of outer equivalence of (contractive) derivations and derivable automorphisms of $A$ is not $c_{0}$-orthogonal; 3. (3)

the relation $E_{0}^{\mathbb{N}}$ of tail equivalence of countably many binary sequences is Borel reducible to the relations of unitary equivalence of contractive derivations and derivable automorphisms of $A$.

Then (1)$\Rightarrow$(2)$\Rightarrow$(3).

Proof.

The implication (1)$\Rightarrow$(2) is immediate from the definition of Property APE and of $c_{0}$-orthogonality. The implication (2)$\Rightarrow$(3) follows from Corollary 4.2 and the fact that the relations of unitary equivalence of contractive derivations and derivable automorphism are Borel; see Proposition 5.1. ∎

8. The primitive case

Recall that an ideal of a C*-algebra is primitive if it is the kernel of an irreducible representation. A C*-algebra is primitive if the trivial ideal is primitive, i.e., if it has a faithful irreducible representation.

Lemma 8.1**.**

Let $A$ be a separable primitive nonsimple infinite-dimensional C-algebra. Then $A$ satisfies Property APE.*

Proof.

We follow the ideas of [13, Lemma 4.6]. Since $A$ is primitive, we can fix a faithful irreducible representation $\pi$ of $A$ on a separable Hilbert space. Then $\pi$ has a unique extension to a $\sigma$-weakly continuous representation of $A^{\ast\ast}$ on $H$, which we still denote by $\pi$

Let $\boldsymbol{a}=\left(a_{n}\right)$ be a dense sequence in $\mathrm{\mathrm{Ball}}\left(A\right)$, and fix a strictly positive contraction $b_{0}$ of $A$ [4, Proposition II.4.2.1]. Following the argument of [1, Lemma 3.2], one can define a sequence $\boldsymbol{x}=\left(x_{n}\right)$ of pairwise orthogonal positive contractions such that for every $\varepsilon>0$ and every $k,n\in\mathbb{N}$ such that $k\leq n$, one has that

[TABLE]

and

[TABLE]

We claim that $\boldsymbol{x}$ is an APE sequence for $\boldsymbol{a}$. Fix $\delta>0$ and a $\delta$-separated uncountable subset $Y\subseteq\left(0,1\right)^{\mathbb{N}}$. Suppose by contradiction hat for every $\boldsymbol{s},\boldsymbol{t}\in Y$, the automorphisms $\alpha_{\boldsymbol{t}}$ and $\alpha_{\boldsymbol{s}}$ are unitary equivalent. This means that $\alpha_{\boldsymbol{t}}\circ\alpha_{\boldsymbol{s}}^{-1}$ is inner. Since the latter automorphism is equal to $\exp\left(\delta_{\boldsymbol{s}}-\delta_{\boldsymbol{t}}\right)$, by Proposition 6.3 this implies that

[TABLE]

is inner. Thus, there exist $z_{\boldsymbol{t},\boldsymbol{s}}$ in the center of $A^{\ast\ast}$ such that

[TABLE]

Since $\pi$ is irreducible, it is in particular a factor representation. Thus, the relative commutant of $\pi[A]$ in $B\left(H\right)$ consists only of scalar multiples of the identity. In particular, we have that $\pi\left(z_{\boldsymbol{t,s}}\right)\in\mathbb{C}1$ for every $\boldsymbol{s},\boldsymbol{t}\in Y$. This implies that

[TABLE]

and hence

[TABLE]

Fix $\boldsymbol{s}\in Y$. For distinct $\boldsymbol{t},\boldsymbol{t}^{\prime}\in Y$ there exists $m\in\mathbb{N}$ such that

[TABLE]

Hence, we have that

[TABLE]

As $Y$ is uncountable, this contradicts the separability of $\pi[A]$. ∎

9. Lifting

A crucial step in the proof of the main result of [13] consists in showing that Property AEP is liftable. Here we prove that the same holds for Property APE.

Lemma 9.1**.**

Let $A$ and $B$ be separable C-algebras, and $\pi:A\rightarrow B$ be a surjective -homomorphism. If $B$ satisfies Property APE, then so does $A$.

Proof.

Denote by $J$ the kernel of $\pi$. Let $\boldsymbol{a}=\left(a_{n}\right)$ be a dense sequence in $\mathrm{\mathrm{Ball}}\left(A\right)$. Observe that $\pi\left(\boldsymbol{a}\right)=\left(\pi\left(a_{n}\right)\right)$ is a dense sequence in $\mathrm{\mathrm{Ball}}\left(B\right)$. Since by hypothesis $B$ satisfies Property APE, we can consider an APE sequence $\boldsymbol{y}=\left(y_{n}\right)$ for $\pi\left(\boldsymbol{a}\right)$. Then by [12, Lemma 10.1.12] we can find a sequence $\boldsymbol{z}=\left(z_{n}\right)$ of pairwise orthogonal positive contractions in $A$ such that $\pi\left(z_{n}\right)=y_{n}$ for every $n\in\omega$.

Let $\left(e_{n}\right)$ be an *increasing approximate unit *for $J$ that is quasicentral in $A$. This means that $\left(e_{n}\right)$ is an increasing sequence of positive elements of $J$ such that, for every $b\in J$ and $a\in A$, the sequence $\left([e_{n},a]\right)_{n\in\mathbb{N}}$ converges to [math] and the sequences $\left(e_{n}b\right)$ and $\left(be_{n}\right)$ converge to $b$.

By [4, II.5.1.1], for every $x\in A$ one has that

[TABLE]

Applying this to $x=z_{n}a_{i}-a_{i}z_{n}$ for $i\leq n$ implies that, for every $n\in\mathbb{N}$ one can find $k_{n}\in\mathbb{N}$ such that, setting

[TABLE]

one has that

[TABLE]

for $i\leq n$. Suppose that $\boldsymbol{s},\boldsymbol{t}\in\left(0,1\right)^{\mathbb{N}}$ are such that the corresponding automorphisms $\alpha_{\boldsymbol{s}}$ and $\alpha_{\boldsymbol{t}}$ defined in terms of the sequence $\boldsymbol{x}$ are unitarily equivalent. Then the automorphisms $\beta_{\boldsymbol{s}}$ and $\beta_{\boldsymbol{t}}$ obtained from $\boldsymbol{s}$ and $\boldsymbol{t}$ in terms of the sequence $\boldsymbol{y}$ are also unitarily equivalent. Since $\boldsymbol{y}$ is an APE sequence for $\pi(\boldsymbol{a})$, this implies that $\boldsymbol{x}$ is an APE sequence for $\boldsymbol{a}$. ∎

Corollary 9.2**.**

If $A$ is a C-algebra whose primitive spectrum $\mathrm{Prim}\left(A\right)$ is not $T_{1}$, then $A$ has Property APE.*

Proof.

Since the primitive spectrum of $A$ is not $T_{1}$, $A$ has a nonsimple primitive quotient. The conclusion follows from Lemma 8.1 and liftability of Property APE (Lemma 9.1). ∎

10. A stronger property

As in the proof of the main result of [13], it is convenient to consider a stronger property, called Property AEP*+*. The following is [13, Definition 4.9].

Definition 10.1**.**

A C*-algebra $A$ has Property AEP*+* if there exists a sequence $\left(\pi_{n}\right)$ of irreducible representations of $A$, a positive contraction $b_{0}$ of $A$, and a central sequence $\left(x_{n}\right)$ of pairwise orthogonal positive contractions of $A$, such that:

1. (1)

for every $\lambda\in\mathbb{C}$, the sequence $\left(\pi_{n}\left(\left(x_{n}-\lambda\right)b_{0}\right)\right)_{n\in\mathbb{N}}$ does *not *converge to [math]; 2. (2)

for every pair of distinct natural numbers $n$ and $m$, $x_{n}\in\mathrm{\mathrm{Ker}}\left(\pi_{m}\right)$.

The following perturbation lemma is obtained in [13, Lemma 4.10].

Lemma 10.2**.**

For every $\varepsilon>0$ there exists $\rho>0$ such that for every C-algebra $A$ and every pair of positive contractions $x,b$ of $A$ and $\mu\in\mathbb{C}$ such that*

[TABLE]

and

[TABLE]

one must have that

[TABLE]

for some $\lambda\in\mathbb{C}$.

With this lemma, we can show that Property AEP*+* is indeed stronger than Property APE.

Lemma 10.3**.**

Let $A$ be a separable C-algebra. If $A$ satisfies Property AEP+, then it satisfies Property APE.*

Proof.

We follow the outline of the proof of [13, Lemma 4.10]. Let $\left(\pi_{n}\right)$ be a sequence of irreducible representations of $A$, $b_{0}$ be a positive contraction of $A$ of norm $1$, and $\left(x_{n}\right)$ be a sequence of pairwise orthogonal positive contractions of $A$ that witness Property AEP*+* for $A$. Let $\boldsymbol{a}=\left(a_{n}\right)$ be a dense sequence in $\mathrm{\mathrm{Ball}}\left(A\right)$. After passing to a subsequence of $\left(x_{n}\right)$, we can assume that there exists an $\varepsilon>0$ such that for every $\mu\in\mathbb{C}$ and $i,n\in\mathbb{N}$ with $i\leq n$,

[TABLE]

and

[TABLE]

Thus, for $\delta>0$, $\mu\in\mathbb{C}$, $n\in\mathbb{N}$, and $t\in\left(\delta,1\right)$ we have

[TABLE]

We claim that $\boldsymbol{x}$ is an APE sequence for $\boldsymbol{a}$. Indeed, let $Y$ be an uncountable $\delta$-separated subset of $\left(0,1\right)^{\mathbb{N}}$ for some $\delta>0$. Suppose by contradiction that $\alpha_{\boldsymbol{s}}$ and $\alpha_{\boldsymbol{t}}$ are unitarily equivalent for all $\boldsymbol{s}$ and $\boldsymbol{t}$ in $Y$. Thus, there exists $z_{\boldsymbol{t},\boldsymbol{s}}$ in the center of $A^{\ast\ast}$ such that

[TABLE]

and hence

[TABLE]

Fix $\boldsymbol{s}\in Y$ and let $\boldsymbol{t}$ and $\boldsymbol{t}^{\prime}$ be distinct elements of $Y$. Then there exists $m\in\mathbb{N}$ such that

[TABLE]

Since $\pi_{m}$ is an irreducible representation of $A$, there exists $\mu_{\boldsymbol{t},\boldsymbol{t}^{\prime}}\in\mathbb{C}$ such that

[TABLE]

is a scalar multiple of the identity.

Let $\rho>0$ be obtained from $\delta\varepsilon/2$ as in Lemma 10.2. Then by the choice of $\rho$ one has that for every $t\in\left(\delta,1\right)$, $n\in\mathbb{N}$, and $\mu\in\mathbb{C}$,

[TABLE]

Thus, we have

[TABLE]

As this holds for every distinct $\boldsymbol{t}$ and $\boldsymbol{t}^{\prime}$ in $Y$, it contradicts the separability of $A$, concluding the proof. ∎

The following lemmas establishing Property AEP*+* for certain C*-algebra are [13, Lemma 4.12] and [13, Lemma 4.13]. Their proofs are contained in the proofs of Lemma 3.6 and Lemma 3.7 of [1] and the implication $(i)\Rightarrow(ii)$ at page 139 in [6].

Lemma 10.4**.**

Let $A$ be a separable C-algebra with $T_{1}$ primitive spectrum $\mathrm{Prim}\left(A\right)$. Let $\left(\xi_{n}\right)$ be a sequence of separated points in $\mathrm{Prim}\left(A\right)$, let $F$ be the (closed) set of limit points of $\left(\xi_{n}\right)$, and let $I$ be the closed ideal of $A$ corresponding to $F$. If either the quotient $A/I$ does not have continuous trace, or the multiplier algebra of $A/I$ has nontrivial center, then $A$ has Property AEP+.*

Lemma 10.5**.**

*Let $A$ be a separable liminary C-algebra with $\mathrm{Prim}\left(A\right)$ homeomorphic to $\omega+1$, i.e., the one-point compactification of a countable discrete space. Then $A$ has Property AEP+.

11. Proof of the main theorem

The following can is the main result of this paper. It is a refinement of the Akemann–Elliott–Pedersen Theorem characterizing C*-algebras with outer derivations; see Theorem 1.1.

Theorem 11.1**.**

Let $A$ be a separable C-algebra. The following assertions are equivalent:*

1. (1)

$A$* has an outer derivation;* 2. (2)

$A$* has an outer derivable automorphism;* 3. (3)

$A$* has Property APE;* 4. (4)

*the relation of outer equivalence of (contractive) -derivations of $A$ is not $c_{0}$-orthogonal; 5. (5)

the relation of unitary equivalence of derivable automorphism of $A$ is not $c_{0}$-orthogonal; 6. (6)

*the relation of $E_{0}^{\mathbb{N}}$ of tail equivalence of countably many binary sequences is Borel-reducible to the relation of outer equivaelnce of (contractive) -derivations of $A$; 7. (7)

the relation $E_{0}^{\mathbb{N}}$ is Borel-reducible to the relation of unitary equivalence of derivable automorphisms of $A$.

Proof.

(1)$\Rightarrow$(3) We proceed as in the proof of [13, Theorem 4.5]. If $\mathrm{Prim}\left(A\right)$ is not $T_{1}$, then the conclusion follows from Corollary 9.2. If $A$ has a liminal quotient $B$ with $\mathrm{Prim}\left(B\right)$ homeomorphic to $\omega+1$, then $A$ has Property APE by Lemma 10.5 and Lemma 9.1. Otherwise, by [6, Lemma 16], $\mathrm{Prim}\left(A\right)$ is $T_{1}$ but not Hausdorff. By [5], separated points are dense in $\mathrm{Prim}\left(A\right)$. Thus there exists a sequence $\boldsymbol{\xi}=\left(\xi_{n}\right)$ of separated points such that the (closed) set

[TABLE]

contains at least two points. Let $I$ be the closed ideal corresponding to $F$. Then we have that $A/I$ is nonsimple.

If $A/I$ has continuous trace, then by [1, Theorem 3.9 and Lemma 3.1], $M(A/I)$ has nontrivial center, whence $A/I$ has Property APE by Lemma 10.4. If $A/I$ does not have continuous trace, then $A/I$ has Property APE by Lemma 10.4 again. It follows that $A$ has Property APE by Lemma 9.1. This concludes the proof of the implication (1)$\Rightarrow$(3).

All the other implications are either trivial or follow from Lemma 7.3. ∎