The cb-norm approximation of generalized skew derivations by elementary operators

TL;DR

This paper characterizes generalized skew derivations in prime C*-algebras that are limits of elementary operators, showing they are of a specific form and that certain endomorphisms must be inner automorphisms.

Contribution

It provides a structural description of cb-norm limits of elementary operators for generalized skew derivations in prime C*-algebras, and characterizes when endomorphisms are inner automorphisms.

Findings

Generalized skew derivations in prime C*-algebras are of the form $d(x)=bx+axc$ for some $a,b,c$ in the multiplier algebra.

Endomorphisms in the cb-norm closure of elementary operators are necessarily inner automorphisms.

The results do not extend to certain non-prime C*-algebras like $C(X, M_2)$.

Abstract

Let $A$ be a ring and $\sigma: A \to A$ a ring endomorphism. A generalized skew (or $\sigma$-)derivation of $A$ is an additive map $d: A \to A$ for which there exists a map $\delta:A \to A$ such that $d(xy)=\delta(x)y+\sigma(x)d(y)$ for all $x,y \in A$. If $A$ is a prime $C^*$-algebra and $\sigma$ is surjective, we determine the structure of generalized $\sigma$-derivations of $A$ that belong to the cb-norm closure of elementary operators $\mathcal{E}\ell(A)$ on $A$; all such maps are of the form $d(x)=bx+axc$ for suitable elements $a,b,c$ of the multiplier algebra $M(A)$. As a consequence, if an epimorphism $\sigma: A \to A$ lies in the cb-norm closure of $\mathcal{E}\ell(A)$, then $\sigma$ must be an inner automorphism. We also show that these results cannot be extended even to relatively well-behaved non-prime $C^*$-algebras like $C(X,\mathbb{M}_2 )$.

Equations

$$d(xy)=\delta (x)y+\sigma (x)d(y)$$

$$\delta (xy)=\delta (x)y+\sigma (x)\delta (y)$$

$$d((x+y)z)=\delta (x+y)z+\sigma (x+y)d(z)=\delta (x+y)z+\sigma (x)d(z)+\sigma (y)d(z)$$

$$d(xz+yz)=d(xz)+d(yz)=\delta (x)z+\sigma (x)d(z)+\delta (y)z+\sigma (y)d(z).$$

$$d(xyz)=\delta (xy)z+\sigma (xy)d(z)=\delta (xy)z+\sigma (x)\sigma (y)d(z),$$

$$d(xyz)=\delta (x)yz+\sigma (x)d(yz)=\delta (x)yz+\sigma (x)\delta (y)z+\sigma (x)\sigma (y)d(z)$$

$$\rho (xy)=\sigma (x)\rho (y)$$

$$d=\delta +\rho ,$$

$$\delta (x)=ax-\sigma (x)a,$$

$$\rho (x)=\sigma (x)a,$$

$${\Vert \sum _i{M}_{a_i,b_i}\Vert }_{cb}\le {\Vert \sum _i{a}_i\otimes b_i\Vert }_h,$$

$$\Vert t{\Vert }_h=\inf \left\{{\Vert \sum _i{a}_i{a}_i^{\ast }\Vert }^{\frac{1}{2}}{\Vert \sum _i{b}_i^{\ast }{b}_i\Vert }^{\frac{1}{2}}:t=\sum _i{a}_i\otimes b_i\right\}.$$

$$(M(A)\otimes M(A),\Vert \cdot {\Vert }_h)\to (\mathcal{E}\ell (A),\Vert \cdot {\Vert }_{cb})\text{mbox}givenby\sum _i{a}_i\otimes b_i\mapsto \sum _i{M}_{a_i,b_i}.$$

$$\sigma =\mathrm{Ad}(a),\delta (x)=bx-\sigma (x)b\text{mbox}and\rho (x)=\sigma (x)b^{\prime },$$

$$d(x)=bx+axc,$$

$$ax=\sigma (x)a\forall x\in A.$$

$$q^{\ast }q+\sum _{i=1}^n{q}_i^{\ast }{q}_i=\lambda 1.$$

$$\Vert q{\Vert }^2=\inf \{\lambda \in {\mathbb{R}}_{+}:q^{\ast }q\le \lambda 1\},$$

$$AaA=\left\{\sigma (x)ay:x,y\in A\right\}=\left\{axy:x,y\in A\right\}\subseteq aA.$$

$$\mathcal{L}(ax)=\sigma (x)\text{mbox}and\mathcal{R}(ax)=x.$$

$$\mathcal{L}(\sigma (x)ay)=\mathcal{L}(axy)=\sigma (xy)=\sigma (x)\sigma (y)=\sigma (x)\mathcal{L}(ay)$$

$$\mathcal{L}(ax)ay=\sigma (x)ay=axy=ax\mathcal{R}(ay).$$

$$\sigma (x)=\mathcal{L}(ax)=axb=\sigma (x)ab\text{mbox}andx=\mathcal{R}(ax)=bax$$

$$\sum _{n=1}^{\infty }{a}_{n}x{b}_n=0$$

$$\rho (x)=\sigma (x)p=ax{a}^{-1}p$$

$$\sum _{n=1}^{\infty }(a_{n}x-\sigma (x)a_n)y{b}_n=0$$

$$a_{n}x=\sigma (x)a_n$$

$$\rho (x)=\sum _{n=1}^{\infty }{a}_{n}x{b}_n=\sigma (x)\left(\sum _{n=1}^{\infty }{a}_n{b}_n\right)=\sigma (x)p=ax{a}^{-1}p.$$

$$f(x)\otimes 1=\sum _{n=1}^{\infty }(a_{n}g(x)+h(x)a_n)\otimes b_n,$$

$$f(x)=bg(x)+h(x)b$$

Full text

This work was supported by the Croatian Science Foundation under the project IP-2016-06-1046.

1. Introduction

A well-known consequence of Skolem-Noether theorem (see e.g. [8, Theorem 4.46]) is that a finite-dimensional central simple algebra $A$ over a field $\mathbb{F}$ admits only inner derivations and inner automorphisms. This fact can be also proved by observing that all $\mathbb{F}$-linear maps $\phi:A\to A$ are elementary operators, i.e. they can be written as finite sums of two-sided multiplications $x\mapsto axb$, with $a,b\in A$ (see [8, Lemma 1.25, Theorem 1.30]). Hence, one can ask the following general question:

Problem 1.1**.**

Under which conditions on a semiprime ring (or an algebra) $R$ are all derivations and/or automorphisms of $R$ that are also elementary operators necessarily inner?

In order to investigate Problem 1.1 it is sometimes convenient to consider maps $d:R\to R$ that comprise both (generalized) derivations and automorphisms. One particularly interesting class of such maps $d$ is the following (see e.g. [19, 20]):

Definition 1.2**.**

Let $R$ be a ring and let $\sigma:R\to R$ be a ring endomorphism. An additive map $d:R\to R$ is called a generalized $\sigma$-derivation (or a generalized skew derivation) if there exists a map $\delta:R\to R$ such that

[TABLE]

for all $x,y\in R$.

In case when $R$ is semiprime and $\sigma$ is surjective, in [10] we considered the problem of determining the structure of generalized $\sigma$-derivations $d:R\to R$ that are also elementary operators. In order to give a description of such maps, we used standard techniques of the theory of rings of quotients. As a consequence of the proof of [10, Theorem 1.2], which is the main result of that paper, we showed that if $R$ is semiprime and centrally closed, then a derivation $\delta:R\to R$ (resp. a ring epimorphism $\sigma:R\to R$) is an elementary operator if and only if $\delta$ (resp. $\sigma$) is an inner derivation (resp. inner automorphism), thus giving an affirmative answer to Problem 1.1 for this class of rings.

It is also interesting to consider Problem 1.1 in the setting of $C^{*}$-algebras. Moreover, when working with $C^{*}$-algebras, it is well-known that their derivations, automorphisms and elementary operators are completely bounded. This motivates us to consider the following analytic variation of Problem 1.1:

Problem 1.3**.**

Under which conditions on a $C^{*}$-algebra $A$ are all derivations and/or automorphisms of $A$ that admit a cb-norm approximation by elementary operators necessarily inner?

In our previous work we considered Problem 1.3 only for derivations. More precisely, we showed that all such derivations of $A$ are inner in a case when $A$ is prime [12, Theorem 4.3] or central [12, Theorem 5.6]. This result was further extended in [14, Theorem 1.5] for unital $C^{*}$-algebras whose every Glimm ideal is prime. The latter result in particular applies to derivations of local multiplier algebras (see e.g. [2]), since their Glimm ideals are prime [2, Corollary 3.5.10]. Hopefully, this result might be useful in order to give an answer to Pedersen’s problem from 1978, which asks whether all derivations of local multiplier algebras are inner [24].

Motivated by these results, in this paper we consider the problem of determining the structure of generalized $\sigma$-derivations of $C^{*}$-algebras $A$, with $\sigma$ surjective, that can be approximated by elementary operators in the cb-norm.

The main result of this paper is Theorem 3.1, where we fully describe the structure of such maps when $A$ is a prime $C^{*}$-algebra. In particular, if $A$ is prime, we show that if an epimorphism $\sigma:A\to A$ admits a cb-norm approximation by elementary operators, then $\sigma$ must be an inner automorphism of $A$ (Corollary 3.8). The proof of Theorem 3.1 is given through several steps in Section 3.

In Section 4 we consider the possible generalization of Theorem 3.1 and its consequences for $C^{*}$-algebras that are not necessarily prime. However, this generalization will not be possible even for some well-behaved $C^{*}$-algebras, like homogeneous $C^{*}$-algebras, even thought they have only inner derivations (see [31, Theorem 1] for unital case and [13, Proposition 3.2] for general case). In fact, we show that for each $n\geq 2$ there is a compact manifold $X_{n}$ such that the $C^{*}$-algebra $C(X_{n},\mathbb{M}_{n})$ (where $\mathbb{M}_{n}$ is the algebra of $n\times n$ complex matrices) admits outer automorphisms that are simultaneously elementary operators (Proposition 4.2). We also give an example of a unital separable $C^{*}$-algebra $A$ that admits both outer derivations and outer automorphisms that are also elementary operators (Proposition 4.6).

2. Preliminaries

Let $R$ be a ring. As usual, by $Z(R)$ we denote its centre. By an ideal of $R$ we always mean a two-sided ideal. An ideal $I$ of $R$ is said to be essential if $I$ has a non-zero intersection with every other non-zero ideal of $R$. If $R$ is unital, by $R^{\times}$ we denote the set of all invertible elements of $R$.

Recall that a ring $R$ is said to be semiprime if for $x\in R$, $xRx=\{0\}$ implies $x=0$. If in addition for $x,y\in R$, $xRy=\{0\}$ implies $x=0$ or $y=0$, $R$ is said to be prime.

If $R$ is a semiprime ring by $M(R)$ we denote its multiplier ring (see [2, Section 1.1]). Note that $M(R)$ is also a semiprime ring and that $R$ is prime if and only if $M(R)$ is prime [2, Lemma 1.1.7]. For each $a\in M(R)^{\times}$ we denote by $\mathrm{Ad}(a)$ the automorphism $x\mapsto axa^{-1}$. We call such automorphisms inner (when $R$ is unital this coincides with the standard notion of the inner automorphism).

Remark 2.1**.**

If $R$ is a semiprime ring and $d:R\to R$ a generalized $\sigma$-derivation (Definition 1.2), then a map $\delta$ is obviously uniquely determined by $d$. Moreover, $\delta$ is a $\sigma$-derivation, that is $\delta$ is an additive map that satisfies

[TABLE]

for all $x,y\in R$. Indeed, using (1.1) and additivity of $d$ and $\sigma$, for all $x,y,z\in R$ we have

[TABLE]

and

[TABLE]

Subtracting these two equations we get the additivity of $\delta$. Similarly, subtracting the next two equations

[TABLE]

[TABLE]

shows (2.1). Further, it is now easy to verify that the map $\rho:=d-\delta$ is a left $R$-module $\sigma$-homomorphism, that is $\rho:R\to R$ is an additive map that satisfies

[TABLE]

for all $x,y\in R$. Therefore, every generalized $\sigma$-derivation can be uniquely decomposed as

[TABLE]

where $\delta$ is a $\sigma$-derivation and $\rho$ is a left $R$-module $\sigma$-homomorphism. In particular, generalized $\sigma$-derivations simultaneously generalize $\sigma$-derivations of $R$ (we get them for $d=\delta$) and left $R$-module $\sigma$-homomorphisms of $R$ (we get them for $\delta=0$).

Example 2.2**.**

The simplest examples of $\sigma$-derivations $\delta:R\to R$ are inner $\sigma$-derivations, i.e. those of the form

[TABLE]

where $a$ is some element of $M(R)$. Further, any map of the form

[TABLE]

where $a\in M(R)$, is a left $R$-module $\sigma$-homomorphism. If $R$ is unital, then all left $R$-module $\sigma$-homomorphism of $R$ are of this form ($M(R)=R$ in this case).

Throughout this paper $A$ will be a $C^{*}$-algebra. Then $M(A)$ has a structure of a $C^{*}$-algebra and is called the multiplier algebra of $A$. It is well-known (and easily checked) that $A$, as a ring, is semiprime. As usual, by $\mathop{\mathrm{CB}}\nolimits(A)$ we denote the set of all completely bounded maps $\phi:A\to A$ (see e.g. [23]). For $S\subseteq\mathop{\mathrm{CB}}\nolimits(A)$ we denote by $\overline{\overline{S}}_{cb}$ the cb-norm closure of $S$.

The most prominent class of completely bounded maps on $A$ are elementary operators, i.e. those that can be expressed as finite sums of two-sided multiplications $M_{a,b}:x\mapsto axb$, where $a$ and $b$ are elements of $M(A)$. We denote the set of all elementary operators on $A$ by $\mathcal{E}\ell(A)$. It is well-known that elementary operators on $C^{*}$-algebras are completely bounded. In fact, we have the following estimate for their cb-norm:

[TABLE]

where $\|\cdot\|_{h}$ is the Haagerup tensor norm on the algebraic tensor product $M(A)\otimes M(A)$, i.e.

[TABLE]

By inequality (2.3) the mapping

[TABLE]

is a well-defined contraction. Its continuous extension to the completed Haagerup tensor product $M(A)\otimes_{h}M(A)$ is known as a canonical contraction from $M(A)\otimes_{h}M(A)$ to $\mathop{\mathrm{CB}}\nolimits(A)$ and is denoted by $\Theta_{A}$. We have the following result (see [2, Proposition 5.4.11]):

Theorem 2.3** (Mathieu).**

$\Theta_{A}$* is isometric if and only if $A$ is a prime $C^{*}$-algebra.*

If $A$ is not a prime note that $\Theta_{A}$ is not even injective. Indeed, in this case there are non-zero elements $a,b\in A$ such that $aAb=\{0\}$. Then $a\otimes b$ defines a non-zero tensor in $M(A)\otimes M(A)$ but $M_{a,b}=0$. For a unital but not necessarily prime $C^{*}$-algebra one can construct a central Haagerup tensor product $A\otimes_{Z,h}A$ and consider the induced contraction $\Theta_{A}^{Z}:A\otimes_{Z,h}A\to\mathop{\mathrm{CB}}\nolimits(A)$ (the questions when $\Theta_{A}^{Z}$ is isometric or injective were treated in [30, 5, 4]).

3. Results

We begin this section by stating the main result of this paper:

Theorem 3.1**.**

Let $A$ be a prime $C^{*}$-algebra and let $d:A\to A$ be a generalized $\sigma$-derivation, with $\sigma$ surjective. The following conditions are equivalent:

• (i)

$d\in\overline{\overline{\mathcal{E}\ell(A)}}_{cb}$.

• (ii)

Either $d$ is a left multiplication implemented by some element of $M(A)$ or $\sigma$ is an inner automorphism of $A$. In the latter case, if $d=\delta+\rho$ is a decomposition as in Remark 2.1, then $\delta$ is an inner $\sigma$-derivation and $\rho$ is a right multiplication of $\sigma$ by some element of $M(A)$.

Remark 3.2**.**

• (i)

Note that any left multiplication $d=M_{l,1}$, where $l\in M(A)$, is a generalized $\sigma$-derivation with respect to any ring endomorphism $\sigma:A\to A$. Indeed, for any such $\sigma$, let $\delta(x)=lx-\sigma(x)l$ and $\rho(x)=\sigma(x)l$. Then obviously $d=\delta+\rho$, so in this case we cannot say anything about the epimorphism $\sigma$ and the corresponding maps $\delta$ and $\rho$.

• (ii)

If $d$ is not a left multiplication, then following the second case of part (ii) of Theorem 3.1 we have

[TABLE]

for some $a\in M(A)^{\times}$ and $b,b^{\prime}\in M(A)$, $b\neq b^{\prime}$. In particular, $d$ is of the form

[TABLE]

where $c:=a^{-1}(b^{\prime}-b)$.

Remark 3.3**.**

In the sequel of this section we assume that $A$ is an infinite-dimensional prime $C^{*}$-algebra, since otherwise [22, Theorem 6.3.8] and the primeness of $A$ would imply that $A$ is isomorphic to the matrix algebra $\mathbb{M}_{n}$ for some non-negative integer $n$. Then every linear map $\phi:A\to A$ is an elementary operator (see e.g. [8, Lemma 1.25]), so Theorem 3.1 is just a simple consequence of [10, Theorem 1.2] (the maximal right ring of quotients $Q_{mr}(A)$ in this case coincides with $A$).

For the proof of Theorem 3.1 we will need some auxiliary results. We start with the following:

Lemma 3.4**.**

Let $A$ be a prime $C^{*}$-algebra. Suppose that $\sigma:A\to A$ is a ring epimorphism for which there exists a non-zero element $a\in M(A)$ such that

[TABLE]

Then $a$ is invertible in $M(A)$, so that $\sigma=\mathop{\mathrm{Ad}}(a)$ is an inner automorphism of $A$.

Before proving Lemma 3.4 recall from [2] (see also [6]) that an essentially defined double centralizer on a semiprime ring $R$ is a triple $(\mathcal{L},\mathcal{R},I)$, where $I$ is an essential ideal of $R$, $\mathcal{L}:I\to R$ is a left $R$-module homomorphism, $\mathcal{R}:I\to R$ is a right $R$-module homomorphism such that $\mathcal{L}(x)y=x\mathcal{R}(y)$ for all $x\in I$. One can form the symmetric ring of quotients $Q_{s}(R)$ which is characterized (up to isomorphism) by the following properties:

• (i)

$R$ is a subring of $Q_{s}(R)$;

• (ii)

for any $q\in Q_{s}(R)$ there is an essential ideal $I$ of $R$ such that $qI+Iq\subseteq R$;

• (iii)

if $0\neq q\in Q_{s}(R)$ and $I$ is an essential ideal of $R$, then $qI\neq 0$ and $Iq\neq 0$;

• (iv)

for any essentially defined double centralizer $(\mathcal{L},\mathcal{R},I)$ on $R$ there exists $q\in Q_{s}(R)$ such that $\mathcal{L}(x)=qx$ and $\mathcal{R}(x)=xq$ for all $x\in I$.

In a case when $R=A$ is a $C^{*}$-algebra, $Q_{s}(A)$ has a natural structure as a unital complex $*$-algebra, whose involution is positive definite. An element $q\in Q_{s}(A)$ is called bounded if there is $\lambda\in\mathbb{R}_{+}$ such that $q^{*}q\leq\lambda 1$, in a sense that there is a finite number of elements $q_{1},\ldots,q_{n}\in Q_{s}(A)$ such that

[TABLE]

The set $Q_{b}(A)$ of all bounded elements of $Q_{s}(A)$ has a pre-$C^{*}$-algebra structure with respect to the norm

[TABLE]

which clearly extends the norm of $A$. One can easily check that an element $q\in Q_{s}(A)$ is bounded if and only if it can be represented by a bounded (continuous) essentially defined double centralizer (see [2, p. 57]). We call $Q_{b}(A)$ the bounded symmetric algebra of quotients of $A$ and its completion $M_{\mathrm{loc}}(A)$ the local multiplier algebra of $A$. Note that $M_{\mathrm{loc}}(A)$ has a structure of a $C^{*}$-algebra as a completion of a pre-$C^{*}$-algebra.

Proof of Lemma 3.4.

First note that a non-zero element $a\in M(A)$ that satisfies (3.1) cannot be a zero-divisor. Indeed, if there exists $x\in M(A)$ such that $ax=0$ then for each $y\in A$ we have $ayx=\sigma(y)ax=0,$ so that $aAx=\{0\}$. Since $A$ is prime and since $A$ is an essential ideal of $M(A)$, $a\neq 0$ implies $x=0$. Similarly, if $xa=0$ then for all $y\in A$ we have $x\sigma(y)a=xay=0$. Since $\sigma$ is surjective, this is equivalent to $xAa=\{0\}$, so the primeness of $A$ again implies $x=0$.

We now show that $a$ is invertible in $M(A)$. Since $M(A)$ is a unital $C^{*}$-subalgebra of $M_{\mathrm{loc}}(A)$, we have $M(A)^{\times}=M(A)\cap M_{\mathrm{loc}}(A)^{\times}$, so it suffices to show that $a$ is invertible in $M_{\mathrm{loc}}(A)$. In order to do this, first note that $aA$ is a non-zero ideal of $A$, hence essential, since $A$ is prime. Indeed, since $\sigma$ is surjective, we have $A=\sigma(A)$, hence

[TABLE]

In particular, for $\alpha\in\mathbb{C}$ and $x\in A$ we have $(\sigma(\alpha x)-\alpha\sigma(x))aA=\{0\}$, which implies $\sigma(\alpha x)=\alpha\sigma(x)$. Therefore $\sigma$ is a linear map, hence an algebra epimorphism.

We define maps $\mathcal{L},\mathcal{R}:aA\to A$ by

[TABLE]

That the maps $\mathcal{L}$ and $\mathcal{R}$ are well-defined follows from the fact that $a$ is not a (left) zero-divisor. Clearly, $\mathcal{R}$ is a right $A$-module homomorphism. Next, for $x,y\in A$ we have

[TABLE]

and

[TABLE]

This shows that $(\mathcal{L},\mathcal{R},aI)$ is an essentially defined double centralizer on $A$. Since $A$ is prime, by [2, Corollary 2.2.15] all essentially defined double centralizers are automatically continuous. In particular, there exists an element $b\in Q_{b}(A)\subseteq M_{\mathrm{loc}}(A)$ such that

[TABLE]

for all $x\in A$. Since $\sigma(A)=A$, this is equivalent to $A(1-ab)=\{0\}$ and $(1-ba)A=\{0\}$. Hence, $a$ is invertible in $M_{\mathrm{loc}}(A)$ and $a^{-1}=b$. This completes the proof. ∎

Recall from [29, Definition 3.2] that a sequence $(a_{n})$ in an infinite-dimensional $C^{*}$-algebra $B$ such that the series $\sum_{n=1}^{\infty}a_{n}^{*}a_{n}$ is norm convergent is said to be strongly independent if for every sequence $(\alpha_{n})\in\ell^{2}$, equality $\sum_{n=1}^{\infty}\alpha_{n}a_{n}=0$ implies $\alpha_{n}=0$ for all $n\in\mathbb{N}$.

The next fact can be deduced from [7, Proposition 1.5.6], [29, Lemma 4.1] and [1, Lemma 2.3].

Remark 3.5**.**

Let $B$ be an infinite-dimensional $C^{*}$-algebra.

• (i)

Every tensor $t\in B\otimes_{h}B$ has a representation as a convergent series $t=\sum_{n=1}^{\infty}a_{n}\otimes b_{n},$ where $(a_{n})$ and $(b_{n})$ are sequences in $B$ such that the series $\sum_{n=1}^{\infty}a_{n}a_{n}^{*}$ and $\sum_{n=1}^{\infty}b_{n}^{*}b_{n}$ are norm convergent. Moreover, the sequence $(b_{n})$ can be chosen to be strongly independent.

• (ii)

If $t=\sum_{n=1}^{\infty}a_{n}\otimes b_{n}$ is a representation of $t$ as above, with $(b_{n})$ strongly independent, then $t=0$ if and only if $a_{n}=0$ for all $n\in\mathbb{N}$.

Corollary 3.6**.**

Let $A$ be a prime $C^{*}$-algebra and suppose that $(a_{n})$, $(b_{n})$ are sequences in $M(A)$ such that the series $\sum_{n=1}^{\infty}a_{n}a_{n}^{*}$ and $\sum_{n=1}^{\infty}b_{n}^{*}b_{n}$ are norm convergent, with $(b_{n})$ strongly independent. If

[TABLE]

for all $x\in A$, then $a_{n}=0$ for all $n\in\mathbb{N}$.

Proof.

If $t:=\sum_{n=1}^{\infty}a_{n}\otimes b_{n}\in M(A)\otimes_{h}M(A)$, then (3.2) is equivalent to $\Theta_{A}(t)=0$. Since $A$ is prime, by Theorem 2.3 $\Theta_{A}$ is isometric (hence injective), so $t=0$. The claim now follows from part (ii) of Remark 3.5. ∎

Proposition 3.7**.**

Let $A$ be a prime $C^{*}$-algebra and let $\sigma:A\to A$ be a ring epimorphism. If $\rho:A\to A$ is a non-zero left $A$-module $\sigma$-homomorphism, then the following conditions are equivalent:

• (i)

$\rho\in\overline{\overline{\mathcal{E}\ell(A)}}_{cb}$.

• (ii)

There are elements $a,p\in M(A)$, with $a$ invertible and $p\neq 0$, such that $\sigma=\mathop{\mathrm{Ad}}(a)$ and

[TABLE]

for all $x\in A$.

Proof.

Since $A$ is prime, by Theorem 2.3 the canonical contraction $\Theta_{A}:M(A)\otimes_{h}M(A)\to\mathop{\mathrm{CB}}\nolimits(A)$ is isometric. In particular, the image of $\Theta_{A}$ is closed in the cb-norm so $\overline{\overline{\mathcal{E}\ell(A)}}_{cb}$ coincides with the image of $\Theta_{A}$. Hence, there is a tensor $t\in M(A)\otimes_{h}M(A)$ such that $\rho=\Theta_{A}(t)$. By Remark 3.5, we can write $t=\sum_{n=1}^{\infty}a_{n}\otimes b_{n}$, where $(a_{n})$ and $(b_{n})$ are sequences in $M(A)$ such that the series $\sum_{n=1}^{\infty}a_{n}a_{n}^{*}$ and $\sum_{n=1}^{\infty}b_{n}^{*}b_{n}$ are norm convergent, with $(b_{n})$ strongly independent. Then (2.2) implies

[TABLE]

for all $x,y\in A$. By Corollary 3.6 we have

[TABLE]

for all $n\in\mathbb{N}$. Since $\rho$ is non-zero, there is $n_{0}\in\mathbb{N}$ such that $a_{n_{0}}\neq 0$. By Lemma 3.4 $a:=a_{n_{0}}$ is invertible in $M(A)$. Hence $\sigma=\mathop{\mathrm{Ad}}(a)$ is an inner automorphism of $A$. Finally, if $p:=\sum_{n=1}^{\infty}a_{n}b_{n}\in M(A)$, using (3.3) we get

[TABLE]

∎

As a direct consequence of Proposition 3.7 we get:

Corollary 3.8**.**

If $A$ is a prime $C^{*}$-algebra then every ring epimorphism $\sigma:A\to A$ that lies in $\overline{\overline{\mathcal{E}\ell(A)}}_{cb}$ must be an inner automorphism of $A$.

The next fact can be deduced from the proof of [12, Theorem 4.3]. For completeness, we include a proof.

Lemma 3.9**.**

Let $B$ be a unital infinite-dimensional $C^{*}$-algebra and let $f,g,h:B\to B$ be any functions with $f\neq 0$. Suppose that for all $x\in B$ we have the following equality of tensors in $B\otimes_{h}B$

[TABLE]

where $(a_{n})$ and $(b_{n})$ are sequences in $B$ such that the series $\sum_{n=1}^{\infty}a_{n}a_{n}^{*}$ and $\sum_{n=1}^{\infty}b_{n}^{*}b_{n}$ are norm convergent, with $(b_{n})$ strongly independent. Then there is a non-zero element $b\in B$ such that

[TABLE]

for all $x\in B$.

Proof.

Choose $x_{0}\in B$ such that $f(x_{0})\neq 0$ and let $\varphi\in B^{*}$ be an arbitrary bounded linear functional such that $\varphi(f(x_{0}))\neq 0$. If for $x=x_{0}$ we act on the equality (3.4) with the right slice map $R_{\varphi}:B\otimes_{h}B\to B$, $R_{\varphi}:a\otimes b\mapsto\varphi(a)b$ (see e.g. [29, Section 4]), we obtain

[TABLE]

For $n\in\mathbb{N}$ let

[TABLE]

Note that $(\alpha_{n})\in\ell^{2}$, since all bounded linear functionals on $C^{*}$-algebras are completely bounded (see e.g. [23, Proposition 3.8]) and the series $\sum_{n=1}^{\infty}(a_{n}g(x_{0})+h(x_{0})a_{n})(a_{n}g(x_{0})+h(x_{0})a_{n})^{*}$ is norm convergent. Then (3.6) can be rewritten as $\sum_{n=1}^{\infty}\alpha_{n}b_{n}=1$, so by (3.4) we have

[TABLE]

for all $x\in B$. Consequently, since $(b_{n})$ is strongly independent, Remark 3.5 (ii) implies that

[TABLE]

for all $n\in\mathbb{N}$ and $x\in B$. Since $\sum_{n=1}^{\infty}\alpha_{n}b_{n}=1$, there is some $n_{0}\in\mathbb{N}$ such that $\alpha_{n_{0}}\neq 0$. If $b:=(1/\alpha_{n_{0}})a_{n_{0}}$, then the above equation is obviously equivalent to (3.5). Also, $b\neq 0$ since $f\neq 0$. ∎

Proof of Theorem 3.1.

(ii) $\Longrightarrow$ (i). This is trivial (see also Remark 3.2).

(i) $\Longrightarrow$ (ii). Assume that $d\in\overline{\overline{\mathcal{E}\ell(A)}}_{cb}$ and that $d\neq M_{l,1}$ for all $l\in M(A)$. In particular $d\neq 0$. Using the same arguments from the beginning of the proof of Proposition 3.7, we see that there is a tensor $t\in M(A)\otimes_{h}M(A)$ such that $d=\Theta_{A}(t)$. By Remark 3.5, we can write $t=\sum_{n=1}^{\infty}a_{n}\otimes b_{n}$, where $(a_{n})$ and $(b_{n})$ are sequences of $M(A)$ such that the series $\sum_{n=1}^{\infty}a_{n}a_{n}^{*}$ and $\sum_{n=1}^{\infty}b_{n}^{*}b_{n}$ are norm convergent, with $(b_{n})$ strongly independent. Using (1.1) for all $x,y\in A$ we get

[TABLE]

or equivalently

[TABLE]

Again, since $\Theta_{A}$ is isometric (hence injective), (3.7) is equivalent to the equality

[TABLE]

of tensors in $M(A)\otimes_{h}M(A)$ for all $x\in A$. If $\delta=0$, then $d$ must be a non-zero left $A$-module $\sigma$-homomorphism of $A$ (see Remark 2.1) so the claim follows directly from Proposition 3.7. If $\delta\neq 0$, Lemma 3.9 implies that there is a non-zero element $b\in M(A)$ such that

[TABLE]

for all $x\in A$. If we decompose $d=\delta+\rho$ as in Remark 2.1, the map $\rho^{\prime}:A\to A$ defined by

[TABLE]

is obviously a left $A$-module $\sigma$-homomorphism of $A$ that lies in $\overline{\overline{\mathcal{E}\ell(A)}}_{cb}$ (since $d$ does). Since, by assumption, $d$ is not a left multiplication, $\rho^{\prime}$ is non-zero. Hence, by Proposition 3.7 there are elements $a,p\in M(A)$ with $a$ invertible and $p\neq 0$ such that $\sigma=\mathop{\mathrm{Ad}}(a)$ and $\rho^{\prime}(x)=\sigma(x)p$ for all $x\in A$. In particular, if we put $b^{\prime}:=b+p$, we get $\rho(x)=\sigma(x)b^{\prime}$, which completes the proof. ∎

4. Counterexamples and further remarks

In [12, 14] we considered derivations of unital $C^{*}$-algebras $A$ that lie in $\overline{\overline{\mathcal{E}\ell(A)}}_{cb}$. We showed that all such derivations are inner in a case when $A$ is prime [12, Theorem 4.3] or central [12, Theorem 5.6], or more generally, when $A$ is a unital $C^{*}$-algebra whose every Glimm ideal is prime [14, Theorem 1.5].

In light of this, it is natural to ask if one can extend Corollary 3.8 in its original form (and consequently Theorem 3.1) for similar classes of $C^{*}$-algebras. However, this will not be possible, even for relatively well-behaved $C^{*}$-algebras like homogeneous $C^{*}$-algebras. In fact, we will now show that for all $n\geq 2$, a $C^{*}$-algebra $A_{n}=C(PU(n),\mathbb{M}_{n})$, where $PU(n)=U(n)/\mathbb{S}^{1}$ is the projective unitary group, admits outer automorphisms which are simultaneously elementary operators on $A_{n}$ (Proposition 4.2).

In order to show this, first suppose that $A$ is a general separable $n$-homogeneous $C^{*}$-algebra (i.e. all irreducible representations of $A$ have the same finite dimension $n$). Then by [17, Theorem 4.2] the primitive spectrum $X:=\mathrm{Prim}(A)$ is a (locally compact) Hausdorff space and by a well-known theorem of Fell [11, Theorem 3.2] and Tomiyama-Takesaki [32, Theorem 5] there is a locally trivial bundle $\mathcal{E}$ over $X$ with fibre $\mathbb{M}_{n}$ and structure group $\mathrm{Aut}^{*}(\mathbb{M}_{n})\cong PU(n)$ such that $A$ is isomorphic to the $C^{*}$-algebra $\Gamma_{0}(\mathcal{E})$ of continuous sections of $\mathcal{E}$ that vanish at infinity. Moreover, any two such algebras $A_{i}=\Gamma_{0}(\mathcal{E}_{i})$ with primitive spectra $X_{i}$ ($i=1,2$) are isomorphic if and only if there is a homeomorphism $f:X_{1}\to X_{2}$ such that $\mathcal{E}_{1}\cong f^{*}(\mathcal{E}_{2})$ (the pullback bundle) as bundles over $X_{1}$ (see [32, Theorem 6]). Thus, we may identify $A$ with $\Gamma_{0}(\mathcal{E})$. Further, by Dauns-Hofmann theorem [27, Theorem A.34] we can identify $Z(A)$ with $C_{0}(X)$ and $Z(M(A))$ with $C_{b}(X)$.

Let us denote by $\mathrm{Aut}_{Z}^{*}(A)$ the set of all $Z(M(A))$-linear $*$-automorphisms of $A$ (i.e. $\sigma(x^{*})=\sigma(x)^{*}$ and $\sigma(zx)=z\sigma(x)$ for all $z\in Z(M(A))$ and $x\in A$) and by $\mathrm{InnAut}^{*}(A)$ the set of automorphisms $\sigma$ of $A$ that are of the form $\sigma=\mathrm{Ad}(u)$, for some unitary element $u\in M(A)$. Obviously $\mathrm{InnAut}^{*}(A)\subseteq\mathrm{Aut}_{Z}^{*}(A)$. It is a very interesting (and non-trivial) problem to describe when do we have $\mathrm{Aut}_{Z}^{*}(A)=\mathrm{InnAut}^{*}(A)$ (see e.g. [28, 18, 25, 26] for some results regarding this question). In particular, we have the following consequence of [25, Theorem 2.1] (see also [25, 2.19]):

Theorem 4.1** (Phillips-Raeburn).**

If $A=\Gamma_{0}(\mathcal{E})$ is a separable $n$-homogeneous $C^{*}$-algebra with primitive spectrum $X$, we have the exact sequence

[TABLE]

of abelian groups, where $\check{H}^{2}(X;\mathbb{Z})$ is the second integral Čech cohomology group of $X$. Further, the image of $\eta$ is contained in the torsion subgroup of $\check{H}^{2}(X;\mathbb{Z})$.

Proposition 4.2**.**

For $n\geq 2$ let $A_{n}=C(PU(n),\mathbb{M}_{n})$. Then every derivation of $A_{n}$ is inner, but $A_{n}$ admits an outer automorphism that is also an elementary operator.

Remark 4.3**.**

A $C^{*}$-algebra $A$ is said to be quasicentral if any element $a\in A$ can be decomposed as $a=zb$ for some $b\in A$ and $z\in Z(A)$ (see [9, 3, 12] for other characterizations of such algebras). If every closed ideal of $A$ is quasicentral, note that any $Z(A)$-linear map $\phi:A\to A$ preserves all closed ideals of $A$ (i.e. $\phi(I)\subseteq I$ for any such ideal $I$). Indeed, since any $a\in I$ can be decomposed as $a=zb$, with $z\in Z(I)$ and $b\in I$ and since $Z(I)\subseteq Z(A)$, we have $\phi(a)=\phi(zb)=z\phi(b)\in I$.

This observation in particular applies to $n$-homogeneous $C^{*}$-algebras $A\cong\Gamma_{0}(\mathcal{E})$. Indeed, using the fact that the bundle $\mathcal{E}$ is locally trivial one can easily check that $n$-homogeneous $C^{*}$-algebras are quasicentral. Also, every closed ideal of an $n$-homogeneous $C^{*}$-algebra is also an $n$-homogeneous $C^{*}$-algebra, hence quasicentral. Further, if $A$ is unital (and $n$-homogeneous), every bounded $Z(A)$-linear map $\phi:A\to A$ is an elementary operator on $A$. This follows directly from the above observation and Magajna’s theorem [21, Theorem 1.1]. Therefore, for every unital $n$-homogeneous $C^{*}$-algebra $A$ we have $\mathrm{Aut}^{*}_{Z}(A)\subseteq\mathcal{E}\ell(A)$.

Proof of Proposition 4.2.

That all derivations of $A_{n}$ are inner follows from [31, Theorem 1]. On the other hand, by [16, Section IV] $A_{n}$ admits an automorphism $\sigma\in\mathrm{Aut}^{*}_{Z}(A_{n})\setminus\mathrm{InnAut}^{*}(A_{n})$ (note that $\check{H}^{2}(PU(n);\mathbb{Z})=\mathbb{Z}_{n}$, so all elements of $\check{H}^{2}(PU(n);\mathbb{Z})$ are torsion elements). By Remark 4.3, $\sigma\in\mathcal{E}\ell(A_{n})$. Suppose that $\sigma=\mathrm{Ad}(a)$ for some $a\in A_{n}^{\times}$. Then, since $\sigma$ is $*$-preserving, for all $x\in A_{n}$ we have

[TABLE]

Hence $a^{*}a\in Z(A_{n})$, so $|a|=\sqrt{a^{*}a}\in Z(A_{n})$. Therefore, if $u:=|a|^{-1}a$, then $u$ is a unitary element of $A_{n}$ and $\sigma=\mathrm{Ad}(u)\in\mathrm{InnAut}^{*}(A_{n})$; a contradiction. ∎

On the other hand, if $A_{n}=C(PU(n),\mathbb{M}_{n})$ as before, every $\sigma\in\mathrm{Aut}^{*}_{Z}(A_{n})$ is implemented by some unitary element of $Q_{b}(A)$. This follows from the following fact:

Proposition 4.4**.**

Let $A=\Gamma_{0}(\mathcal{E})$ be a separable $n$-homogeneous $C^{*}$-algebra whose primitive spectrum $X$ is locally contractable. Then for every $\sigma\in\mathrm{Aut}^{*}_{Z}(A)$ there is a unitary element $u\in Q_{b}(A)$ such that $\sigma=\mathrm{Ad}(u)$.

Proof.

We first show that there is a dense open subset $U$ of $X$ such that $\check{H}^{2}(U;\mathbb{Z})=0$. This can be shown by using the similar arguments as in the proof of [15, Lemma 3.1]. Indeed, let $\mathfrak{F}$ be a collection of all families $\mathcal{V}$ consisting of mutually disjoint open contractable subsets of $X$. We use the standard set-theoretic inclusion for partial ordering. If $\mathfrak{C}$ is a chain in $\mathfrak{F}$, then obviously $\bigcup\mathfrak{C}$ is an upper bound of $\mathfrak{C}$ in $\mathfrak{F}$. Therefore, applying Zorn’s lemma, we obtain a maximal family $\mathcal{M}$ in $\mathfrak{F}$. Let $U$ be the union of all members of $\mathcal{M}$. Since $U$ is a disjoint union of contractable spaces, we have $\check{H}^{2}(U;\mathbb{Z})=0$. Since $X$ is locally contractable (and regular), the maximality of $\mathcal{M}$ implies that $U$ is a dense (evidently open) subset of $X$.

Now let $I:=\Gamma_{0}(\mathcal{E}|_{U})$, where $\mathcal{E}|_{U}$ is a restriction bundle of $\mathcal{E}$ to $U$. Since $U$ is dense in $X$, $I$ is an essential closed ideal of $A$. If $\sigma\in\mathrm{Aut}^{*}_{Z}(A)$, by Remark 4.3 we have $\sigma(I)\subseteq I$, so $\sigma|_{I}\in\mathrm{Aut}^{*}_{Z}(I)$. Since $\mathrm{Prim}(I)=U$ and $\check{H}^{2}(U;\mathbb{Z})=0$, by Theorem 4.1 there is a unitary element $u\in M(I)$ such that $\sigma(x)=uxu^{*}$ for all $x\in I$. Since $I$ is an essential ideal of $A$, we also have $\sigma(x)=uxu^{*}$ for all $x\in A$. If we define $\mathcal{L}_{u},\mathcal{R}_{u}:I\to A$ by $\mathcal{L}_{u}(x)=xu$ and $\mathcal{R}_{u}(x)=ux$, then obviously $(\mathcal{L}_{u},\mathcal{R}_{u},I)$ is a bounded essentially defined double centralizer of $A$, so $u\in Q_{b}(A)$ and $\sigma=\mathrm{Ad}(u)$. ∎

Problem 4.5**.**

Can we omit the assumption of local contractibility of the space $X$ in Proposition 4.4, that is if $A$ is a general separable $n$-homogeneous $C^{*}$-algebra, are all automorphisms $\sigma\in\mathrm{Aut}^{*}_{Z}(A)$ of the form $\sigma=\mathrm{Ad}(u)$ for some unitary $u\in Q_{b}(A)$?

In [12, Example 6.1] we gave an example of a unital separable $C^{*}$-algebra $A$ which admits outer derivations that are also elementary operators. We now show that the same $C^{*}$-algebra admits outer automorphisms that are also elementary operators:

Proposition 4.6**.**

Let $B:=C([1,\infty],\mathbb{M}_{2})$ be a $C^{*}$-algebra that consists of all continuous functions from the extended interval $[1,\infty]$ to the $C^{*}$-algebra $\mathbb{M}_{2}$. If $A$ is a $C^{*}$-subalgebra of $B$ that consists of all $a\in B$ such that

[TABLE]

for some convergent sequence $(\lambda_{n}(a))$ of complex numbers, then $A$ admits an outer automorphism which is also an elementary operator on $A$.

Proof.

Let $f:[1,\infty]\to\mathbb{C}$ be a continuous function such the series $\sum_{n=1}^{\infty}f(n)$ does not converge and such that the range of $f$ is a subset of the imaginary axis. We define an element $b\in B$ by

[TABLE]

It was observed in [12, Section 6] that $\delta:=\mathrm{ad}(b)$ defines an outer derivation of $A$ which lies in the operator norm closure of the space of all inner derivations of $A$. Since $b^{*}=-b$, $\delta$ is a $*$-derivation (i.e. $\delta(x^{*})=\delta(x)^{*}$ for all $x\in A$). Hence $\sigma:=\exp(\delta)$ defines a $*$-automorphism of $A$ (see e.g. [2, Section 4.3]). Note that $\sigma=\mathrm{Ad}(u)$, where

[TABLE]

is a unitary element of $B$. Since the exponential map is continuous, and since $\delta$ in fact lies in the operator norm closure of the space of all inner $*$-derivations of $A$, we conclude that $\sigma$ lies in the operator norm closure of the set of all inner $*$-automorphisms of $A$.

Claim 1. $\sigma$ is an outer automorphism of $A$.

On the contrary, suppose that there exists an invertible element $a\in A$ such that $\sigma=\mathrm{Ad}(a)$. Then $u^{*}ax=xu^{*}a$ for all $x\in A$. Since

[TABLE]

defines an closed essential ideal of both $A$ and $B$, we conclude that $u^{*}a$ is an invertible central element of $B$. Hence, there exists an invertible continuous function $\varphi\in C([1,\infty])$ such that

[TABLE]

Since $a\in A$, we conclude that $\varphi(n+1)=(\exp{f(n)})\varphi(n)$, and consequently

[TABLE]

for all $n\in\mathbb{N}$. Finally, since $\lim_{x\to\infty}\varphi(x)$ exists and since $\varphi(1)\neq 0$, (4.1) implies that the series $\sum_{n=1}^{\infty}f(n)$ converges, a contradiction.

Claim 2. $\sigma$ is an elementary operator on $A$.

As noted, $\sigma$ lies in the operator norm closure of the set of all inner $*$-automorphisms of $A$. In particular, $\sigma$ lies in the operator norm closure of the set of all elementary operators on $A$. But the latter set is closed in the operator norm by [12, Lemma 6.6]. Hence, $\sigma$ is an elementary operator. ∎

We end this paper with the following question:

Problem 4.7**.**

Is Corollary 3.8 true for all von Neumann algebras? In particular, if an automorphism $\sigma$ of a von Neumann algebra $A$ is also an elementary operator, is $\sigma$ necessarily an inner automorphism?