Amenability of coarse spaces and \(\mathbb {K}\)algebras
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Abstract
In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra.
Keywords
Amenability Paradoxical decompositions Følner nets Coarse spaces Unital \(\mathbb {K}\)algebras Leavitt path algebras Translation algebrasMathematics Subject Classification
16P90 43A07 37A15 20F65 16S991 Introduction
Let us fix a field \(\mathbb {K}\). Elek introduced in [32] the notion of amenability for finitely generated unital algebras over \(\mathbb {K}\), and proved some essential results in the case where the algebra has no zerodivisors. The main definition he used also resembles Følner’s characterization, with subsets replaced by linear subspaces, and cardinalities replaced by dimensions. We generalize this notion to \(\mathbb {K}\)algebras of arbitrary dimensions and single out a more restrictive situation brought about by the additional requirement that the Følner net is exhaustive, which we term proper amenability.
Definition 1
Following Elek’s pioneering work, a number of authors have dealt with amenability for algebras from different perspectives, such as Bartholdi [15], CecheriniSilberstein and SaimetVaillant [25], and D’Adderio [26] (building on work of Gromov [39]). Special attention has been paid by Elek to the case of division algebras over a field, see [30, 33, 34]. In particular, the notion of amenability for division algebras plays an important role in the study of infinite dimensional representations of a finitedimensional algebra over a finite field undertaken in [30].
The fundamental result of Elek in [32] is the equivalence, for finitely generated unital \(\mathbb {K}\)algebras without zerodivisors, among three characterizations of algebraic amenability analogous to those in the cases of groups and metric spaces: algebraic amenability à la Følner as given in Definition 1, the nonexistence of paradoxical decompositions, and an analogue of von Neumann’s invariant means called invariant dimension measures. The definitions of the latter two notions enlist the involvement of linear bases of the algebra. We offer here generalizations of these notions (cf., Definitions 4.1 and 4.5) and of Elek’s theorem to encompass all \(\mathbb {K}\)algebras regardless of the size of the generating set or the existence of zerodivisors or a unit. Notably, invariant dimension measures in our definition exhibit delicate deviations from von Neumann’s invariant means on a group, owing to the fact that the lattice of subspaces of an algebra is not distributive, unlike the lattice of subsets of a group. For the sake of brevity, here we state the generalized theorem only for countably dimensional \(\mathbb {K}\)algebras.
Theorem 2
 (1)
\({\mathcal A}\) is algebraically amenable.
 (2)
There is a linear basis of \({\mathcal A}\) that cannot be paradoxically decomposed.
 (3)
There exists an invariant dimensionmeasure on \({\mathcal A}\) associated to some linear basis.
 (1)
Leavitt path algebras constructed from directed graphs (Definition 5.6): These algebras were introduced in [3] and [11] as generalizations of the classical algebras studied by Leavitt in [42, 43]. They also provide natural purely algebraic analogues of the widely studied graph \(C^*\)algebras (see e.g. [48]). The class of Leavitt path algebras has interesting connections with various branches of mathematics, such as representation theory, ring theory, group theory, and dynamical systems. We refer the reader to [2] for a recent survey on this topic.
 (2)
Translation algebras constructed from (locally finite) metric spaces (Definition 6.1):
These algebras were introduced by Roe as an intermediate step between coarse metric spaces and a class of \(C^*\)algebras now known as the (uniform) Roe \(C^*\)algebras, as part of his farreaching work on coarse geometry and the index theory for noncompact manifolds and metric spaces (cf., [49]). Their geometric nature enable them to serve as an important bridge between coarse geometry and the field of operator algebras, as well as a rich source of examples. We will further explore their connections to the theory of \(C^*\)algebras in relation to amenabilitytype properties in [7].
As corollaries of Theorem 2, we observe that properly infinite unital algebras are always nonamenable. Recall that a unital algebra \({\mathcal A}\) is said to be properly infinite if the unit is Murray–von Neumann equivalent to two mutually orthogonal idempotents. This condition itself expresses a form of paradoxicality, one that is generally strictly stronger than the notion of paradoxical decompositions used in Theorem 2. This Murray–von Neumann kind of paradoxical decomposition, along with some other forms of nonamenability, are discussed in [24, Section 4.5]. Indeed, there are division algebras which are nonamenable, and a division algebra cannot be properly infinite (cf. [33]). However, proper infiniteness and algebraic nonamenability coincide for the two main classes of examples we study.
Theorem 3
 (1)
a unital Leavitt path \(\mathbb {K}\)algebra of a finite graph, or
 (2)
a translation \(\mathbb {K}\)algebra (associated to a locally finite extended metric space),
In fact, in both cases, we pinpoint the necessary and sufficient properties of the underlying geometric data that give rise to the algebraic amenability of these algebras (cf., Theorems 5 and 5.10).
One novel aspect of our treatment is the careful distinction, in both the geometric setting and the algebraic setting, between the notion of amenability and the somewhat more restrictive notion of proper amenability, which, as described in Definition 1, asks for a Følner net that is exhaustive. In the group case as well as the case of ordinary metric spaces, these two concepts coincide (Corollary 2.19). However, subtle differences emerge once we engage extended metric spaces, that is, we allow the distance between two points to be infinite. A typical way for this to happen is for an infinite space to admit a finite coarse connected component (i.e., a finite cluster of points having finite distances among each other but infinite distances to the rest of the space), as this finite subset would immediately constitute a Følner net by itself, which is enough to witness amenability but not enough for proper amenability. In this sense, proper amenability ignores any Følner net that comes cheaply from an “isolated finite substructure”. It turns out such a typical way is, in fact, the only way to separate the two notions in this context (Corollary 2.20). In the algebraic setting, the distinction between the two concepts appears more pronounced, as they possess somewhat different permanence properties (cf., Proposition 3.6, Example 3.7 and Proposition 3.8). Nevertheless, we show that the disagreement between the two notions is always caused by the existence of a finitedimensional (onesided) ideal—again a prototypical “isolated finite substructure” in the relevant setting.
Theorem 4
(cf., Theorem 3.9) Let \({\mathcal A}\) be an infinite dimensional \(\mathbb {K}\)algebra over a field \(\mathbb {K}\) that is algebraically amenable but not properly algebraically amenable. Then \({\mathcal A}\) has a finitedimensional left ideal.
It follows from this theorem that algebraic amenability and proper algebraic amenability also agree for algebras without zerodivisors.^{1} The distinction between the two concepts eventually plays a role in the aforementioned generalization of Elek’s result in Theorem 2, even though the statement of the theorem does not mention proper algebraic amenability.
Although we only focus on the algebraic and the coarse geometric aspects of amenability in the present article, a major underlying motivation comes from their connections to the Følner property in the context of operator algebras. Such connections will be explored in [7], where we will investigate the close relationship between algebraic amenability and the existence of Følner nets of projections for operator algebras on a Hilbert space. We remark that Følner nets of projections are relevant in single operator theory [45], operator algebras (see, e.g., [9, 12, 18, 18]) as well as in applications to spectral approximation problems (see, e.g., [14, 20, 44] and references cited therein).
We conclude the article with some results connecting the two main objects of study in the paper—locally finite (extended) metric spaces and algebras over a field—through precisely the construction of the translation algebra of a locally finite (extended) metric space. With the help of the equivalent characterizations of amenability in both contexts, we obtain the satisfactory result that (proper) amenability of the metric space is equivalent to (proper) algebraic amenability of the corresponding translation algebra.
Theorem 5
(cf., Theorems 6.3 and 6.4) Let (X, d) be a locally finite extended metric space and let \(\mathbb {K}_\mathrm {u}(X)\) be its translation \(\mathbb {K}\)algebra of a field \(\mathbb {K}\). Then (X, d) is amenable (respectively, properly amenable) if and only if \(\mathbb {K}_\mathrm {u}(X)\) is algebraically amenable (respectively, properly algebraically amenable).
In the case where the field \(\mathbb {K}\) is the complex numbers \(\mathbb {C}\), suitable completions of the translation algebras, the socalled uniform Roe \(C^*\) algebras, will be considered in [7], where further equivalences involving the Følner property of these \(C^*\)algebras will be established.
Contents The paper is organized as follows. In Sect. 2, we begin by addressing the notion of amenability for locally finite extended metric spaces. We will recall in this context the relation to paradoxical decompositions and existence of invariant means in Theorem 2.11. Finally, we will completely clarify the relation between amenability and proper amenability for extended metric spaces in Sect. 2.1.
In Sect. 3, we analyze amenability issues in the context of algebras over a field \(\mathbb {K}\), and give a complete analysis of the difference between algebraic amenability and proper algebraic amenability (see Proposition 3.6 and Theorem 3.9). If the \(\mathbb {K}\)algebra has no zerodivisor, then algebraic amenability and proper algebraic amenability coincide (see Corollary 3.10).
Then we proceed in Sect. 4 to develop the relation between algebraic amenability, paradoxical decompositions and existence of dimension measures on the lattice of subspaces for general \(\mathbb {K}\)algebras (i.e., not necessarily countably dimensional). This extends previous results by Elek in [32] in the context of countably dimensional algebras without zerodivisors. In this general setting, and due to the fact that the lattice of subspaces of an algebra is not distributive, the notion of additivity and invariance of dimension measures are captured by inequalities instead of equalities (see Definition 4.5 for details). Finally, we give examples of how to produce algebras that are not algebraically amenable using the dimension measure.
In the last two sections, we apply our general theory to two vast classes of examples: the Leavitt path algebras and the translation algebras. In Sect. 5, we prove that algebraic nonamenability and proper infiniteness coincide for the class of all unital Leavitt path algebras (see Theorem 5.10). Using the construction of path algebras, we also give simple examples where left and right algebraic amenability differ from each other. In Sect. 6, we prove the same result for the class of translation algebras associated to locally finite extended metric spaces. In fact, we also establish equivalences between the algebraic amenability of the translation algebra and the amenability of the underlying metric space (see Theorem 6.3), and the analogous equivalence for proper amenability (see Theorem 6.4).
Notations Given sets \(X_1,X_2\) we write their cardinality by \(X_i\), \(i=1,2\) and their disjoint union by \(X_1\sqcup X_2\). We put \(\mathbb {N}_0=\{0,1,2,\ldots \}=\mathbb {N}\sqcup \{0\}\).
2 Amenable metric spaces
In this section we will study locally finite metric spaces from a large scale geometric point of view. There are many interesting examples, of which the most prominent is the case of a finitely generated discrete group endowed with the word length metric. More generally, one can always equip any (countable) discrete group with a right (or left)invariant proper metric and obtain a metric space. The dependence on the rightinvariant proper metric is a rather mild one, if one is only interested in the “largescale” behavior of the metric space. More precisely, different rightinvariant proper metrics on the same group induce metric spaces that are coarsely equivalent, see, e.g., Section 1.4 in [46]. Many important properties of groups are “largescale” in nature. Examples include amenability, exactness, Gromov hyperbolicity, etc. In this section, we will focus on the first property in this list. Amenability has been well studied in coarse geometry (see, e.g., [46] or [21, Section 5.5]), so we will only emphasize the aspects which are important for establishing parallelism with the algebraic amenability for \(\mathbb {K}\)algebras that we are going to investigate in the next sections. For the sake of simplicity, we will focus on locally finite metric spaces, i.e., those where any bounded set has finite cardinality.^{2}

Rboundary: \(\partial _R A:=\{x\in X: d(x,A)\le R\ \text {and}\ d(x,X{\setminus } A)\le R \}\);

outer R boundary: \(\partial ^+_R A := \{x\in X{\setminus } A:d(x,A)\le R\}\);

inner R boundary: \(\partial ^_R A := \{x\in A:d(x,X{\setminus } A)\le R\}\).
Definition 2.1
 (i)Let \(R>0\) and \(\varepsilon \ge 0\). A finite nonempty set \(F\subset X\) is called an \((R,\varepsilon )\)Følner set if it satisfiesWe denote by Open image in new window the collection of \((R,\varepsilon )\)Følner sets.$$\begin{aligned} \frac{\partial _R F}{F}\le \varepsilon . \end{aligned}$$
 (ii)
The metric space (X, d) is called amenable if for every \(R>0\) and \(\varepsilon > 0\) there exists Open image in new window .
 (iii)
The metric space (X, d) is called properly amenable if for every \(R>0\), \(\varepsilon >0\) and finite subset \(A\subset X\) there exists a Open image in new window with \(A\subset F\).
Remark 2.2
 (i)Amenability of (X, d) is equivalent to the existence of a net \(\{F_i\}_{i\in I}\) of finite nonempty subsets such that$$\begin{aligned} \lim _{i} \frac{\partial _R F_i}{F_i} = 0, \quad \mathrm {for~all}\quad R > 0. \end{aligned}$$
 (ii)
Proper amenability of (X, d) requires, in addition, that this net \(\{F_i\}_{i\in I}\) satisfies \( X = \liminf _{i} F_i \), where \( \liminf _{i} F_i := \bigcup _{j \in I} \bigcap _{i \ge j} F_i \).
Example 2.3
For a finitely generated discrete group \(\Gamma \) equipped with the word length metric both notions are equivalent to Følner’s condition for the group (see e.g., [46, Proposition 3.1.7]).
Remark 2.4
Remark 2.5
The following lemma shows that the definition of proper amenability can be already characterized in terms of the cardinality of the Følner sets.
Lemma 2.6
Let (X, d) be an infinite locally finite metric space. Then X is properly amenable if and only if for every \(R>0\), \(\varepsilon >0\) and \(N\in \mathbb {N}\) there exists an Open image in new window such that \(F \ge N\).
Proof
As in the group case, the notion of amenability for metric spaces comes with an important dichotomy in relation to paradoxical decompositions. To formulate it, we first need to introduce an important tool in the study of coarse geometry.
Definition 2.7
The set of all partial translations of X is denoted as \(\mathrm {PT}(X)\).
Definition 2.8
A mean \(\mu \) on a locally finite metric space (X, d) is a normalized, finitely additive map on the set of all subsets of X, \(\mu :\mathcal {P}(X)\rightarrow [0,1]\). The measure \(\mu \) is called invariant under partial translations if \(\mu (A)=\mu (B)\) for all partial translations (A, B, t).
Definition 2.9
Let (X, d) be a locally finite metric space. A paradoxical decomposition of X is a (disjoint) partition \(X= X_+ \sqcup X_\) such that there exist two partial translations \(t_i:X\rightarrow X_i\) for \(i \in \{+,  \}\).
Remark 2.10
Applying a BernsteinSchrödertype argument, one may slightly weaken the condition of having a paradoxical decomposition: it suffices to assume that there are two disjoint (nonempty) subsets \(X'_+, X'_ \subset X\) such that there exist partial translations \(t'_i :X \rightarrow X'_i\) for \(i \in \{+,\}\). Here we do not require their union to be X, in contrast with Definition 2.9. Indeed, assume we can find \((X'_+, t'_+, X'_, t'_)\) as above. We may then write \(X = X'_+ \sqcup X'_ \sqcup \widetilde{X}\). Now we define \( \widehat{X} = \bigcup \nolimits _{k=0}^\infty (t'_+)^k (\widetilde{X})\), where \((t'_+)^0\) is viewed as the identity map. This is a disjoint union because \(\widetilde{X}\) is disjoint from the image of \(t'_+\). Note also that \(t'_+\) maps \(\widehat{X}\) and \(X{\setminus }\widehat{X}\) into themselves, respectively, and \(\widehat{X} = \widetilde{X} \sqcup t'_+( \widehat{X} ) \). By the injectivity of \(t'_+\), we have \(t'_+(X \setminus \widehat{X}) = X'_+ \setminus t'_+ (\widehat{X} ) = X'_+ \setminus \widehat{X}\). This allows us to construct a paradoxical decomposition \((X_+, t_+, X_2, t_2)\) in the sense of Definition 2.9 by setting \(X_+ = X'_+ \sqcup \widetilde{X}\) (which is equal to \((X'_+ \setminus \widehat{X}) \sqcup \widehat{X}\)), \(X_2 = X'_\), \(t_+ = \left( t'_+  _{X \setminus \widehat{X}} \right) \sqcup \mathrm {Id}_{\widehat{X}}\) and \(t_2 = t'_\).
The following result gives some standard characterizations of amenable metric spaces that will be used later (see, e.g., [23, Theorems 25 and 32]; we give an alternative proof of the implication (2)\(\Rightarrow \)(1) in the more general context of extended metric spaces; see in Theorem 2.17).
Theorem 2.11
 (1)
(X, d) is amenable.
 (2)
X admits no paradoxical decomposition.
 (3)
There exists a mean \(\mu \) on X which is invariant under partial translations.
Remark 2.12
Deuber, Simonovits and Sós in [29] considered the exponential growth rate^{3} on locally finite metric spaces and they showed that this growth condition characterizes paradoxicality completely. It can be regarded as a Tarskialternativetype theorem for locally finite metric spaces and it also served as an inspiration for the proof of the Tarski alternative (see [23, Theorem 32]).
It is interesting to note that the notions of paradoxicality and invariant means have been recently introduced and studied for arbitrary Boolean inverse monoids in [41].
2.1 Amenability versus proper amenability for extended metric spaces
In many ways, the amenability for metric spaces generalizes the corresponding notion for groups, with certain properties paralleling those of the latter. However, caution should be taken when one tries to understand amenability for metric spaces from its similarity with groups. For example, amenability for metric spaces does not pass to subsets in general. As an example consider the free group \(\mathbb {F}_n\), \(n\ge 2\), with a ray attached to it. In this sense there is also a parallelism with the notion of Følner sequence in the context of operator algebras as considered in [9, Section 4].
Remark 2.13
Definitions 2.1, 2.7, 2.8 and 2.9 generalize directly to extended metric spaces. So does the BernsteinSchrödertype argument in Remark 2.10.
Remark 2.14
Proposition 2.15
Let (X, d) be a locally finite extended metric space. Then X is amenable if at least one of its coarse connected components is amenable. The converse is true in the case where there are only a finite number of coarse connected components.
Proof
The first statement is trivial. For the second, assume that \(X = \bigsqcup _{i=1}^N X_i\) is a union of finitely many coarse connected components \(X_i\), and that all the coarse connected components are nonamenable. We have to show that X is nonamenable. Since all coarse connected components \(X_i\) are nonamenable, it follows from Theorem 2.11 that each component \(X_i\) has a paradoxical decomposition. Since there is only a finite number of components, these paradoxical decompositions can be assembled to a paradoxical decomposition of X, hence X is nonamenable, as desired.\(\square \)
The second part of Proposition 2.15 cannot be generalized to extended metric spaces with an infinite number of coarse connected components, as the following example shows.
Example 2.16
We construct a locally finite extended metric space (X, d), with an infinite number of coarse connected components, such that neither of the connected components of X is amenable, but X is properly amenable. Let Y be the Cayley graph of the free nonAbelian group \(\mathbb F _2\) of rank two. For each \(n\in \mathbb {N}\), let \(Y_n\) be the graph obtained by attaching n new vertices \(v_1,\ldots , v_n\) and n new edges \(e_1,\ldots , e_n\) to Y, in such a way that \(e_i\) connects \(v_i\) with \(v_{i+1}\) for \(i=1,\ldots ,n1\), and \(e_n\) connects \(v_n\) with e, being e the neutral element of \(\mathbb F_2\) (seen as a vertex of Y). Note that \(Y_n\) is the graph obtained by attaching a trunk of length n to Y. Let \(X_n\) be the metric space associated to the connected graph \(Y_n\), and observe that all the metric spaces \(X_n\) are nonamenable. Let X be the extended metric space having the metric spaces \(X_n\) as coarse connected components. Then clearly X is properly amenable, because we can use the long trunks to localize the Følner sets of X of arbitrary large cardinality.
We also remark that Theorem 2.11 given in [23] stays true in the case of extended metric space.
Theorem 2.17
 (1)
(X, d) is amenable.
 (2)
X admits no paradoxical decomposition.
 (3)
There exists a mean \(\mu \) on X which is invariant under partial translations.
Proof
The proofs of the implications (1) \(\Rightarrow \) (3) and (3) \(\Rightarrow \) (2) are standard and apply equally well to the extended metric space situation (see, e.g., [23, §26 and part III]).
The implication (2) \(\Rightarrow \) (1) is more interesting. Hereby we present a direct proof for the sake of completeness, adapting ideas from Kerr and Li in [40, Theorem 3.4, (vi) \(\Rightarrow \) (v)] to the setting of extended metric spaces (see also [40]). This proof should also serve as a motivation for the proof of Proposition 4.4 in the context of algebraic amenability.
The next proposition is the key to our results on the relationship between amenability and proper amenability for extended metric spaces.
Proposition 2.18
Let (X, d) be a nonempty locally finite extended metric space, and assume that all the coarse connected components of X are infinite. Then X is amenable if and only if X is properly amenable.
Proof
For \(k\in I_0\cup J_0\), set \(F_{0,k} = F_0\cap X_k\) and \(F_k= F\cap X_k\). (Note that some \(F_{0,k}\) or some \(F_k\) might be empty.)
 (a)If \(\partial _R(F) \ne \varnothing \), thenand so, \(N_0= F_0< F\). Hence Open image in new window with \(F>N_0\), which is a contradiction to the maximality of \(N_0\).$$\begin{aligned} \frac{1}{F} \le \frac{\partial _R(F)}{F} \le \varepsilon <\frac{1}{F_0} \end{aligned}$$
 (b)If \(\partial _R (F) = \varnothing \) we have two possibilities, for each \(j\in J_0\):Assume that condition (ii) holds for some \(j_0\in J_0\). Then \(\widetilde{F}:= F_0\sqcup F_{j_0}\) satisfies
 (i)
If \(F_j\cap F_{0, j}\not =\varnothing \), then \(F_{0,j} \subset F_j \) by using our assumption that \(\partial _R(F)=\varnothing \).
 (ii)
\(F_j \cap F_{0,j}=\varnothing \).
where the equality follows from the fact that \(\partial _R (F) = \varnothing \). Thus \(\widetilde{F}\) is a \((R_0,\varepsilon _0)\)Følner set with \( \widetilde{F} >N_0\) and we have a contradiction. If case (i) occurs for all \(j\in J_0\), then \(J_0\subset I_0\) and \(F_{0,j}\subset F_j\) for all \(j\in J_0\). Writing \(\widetilde{F} = F_0 \cup F\), we have that \(\widetilde{F} >F_0= N_0\), because \(F\not \subset F_0\). Setting \(I_0'':= I_0{\setminus } J_0\), we get, using that \(\partial _{R_0}F_j= \varnothing \) for all \(j\in J_0\),$$\begin{aligned} \frac{\partial _{R_0} (\widetilde{F})}{\widetilde{F}} \le \frac{\partial _{R_0} (F_0)+\partial _{R_0} (F_{j_0})}{F_0+F_{j_0}} = \frac{\partial _{R_0} (F_0)}{F_0+F_{j_0}} < \frac{\partial _{R_0} (F_0)}{F_0}\le \varepsilon _0 , \end{aligned}$$so that \(\widetilde{F}\) is a \((R_0,\varepsilon _0)\)Følner set of cardinality strictly larger than \(N_0\), which is again a contradiction.$$\begin{aligned} \frac{\partial _{R_0}\widetilde{F}}{\widetilde{F}}&= \frac{\sum _{j\in J_0} \partial _{R_0}F_j + \sum _{i\in I_0''} \partial _{R_0}F_{0,i}}{\widetilde{F}}\\&= \frac{ \sum _{i\in I_0''} \partial _{R_0}F_{0,i}}{\widetilde{F}} \le \frac{\partial _{R_0}F_0}{F_0}\le \varepsilon _0, \end{aligned}$$  (i)
As an immediate consequence of Proposition 2.18, we obtain the following result.
Corollary 2.19
Let (X, d) be a locally finite metric space. Then (X, d) is amenable if and only if (X, d) is properly amenable.
We can now obtain the characterization of the amenable but not properly amenable extended metric spaces. This should be compared to Theorem 3.9 in the algebraic setting.
Corollary 2.20
Let (X, d) be a locally finite extended metric space with infinite cardinality. Then X is amenable but not properly amenable if and only if \(X= Y_1\sqcup Y_2\), where \(Y_1\) is a finite nonempty subset of X, \(Y_2\) is nonamenable and \(d(x,y)= \infty \) for \(x\in Y_1\) and \(y\in Y_2\).
Proof
Suppose now that X is amenable but not properly amenable. We first show that there are only a finite number of finite components. Indeed, if \(X_1,X_2,\ldots , \) is an infinite sequence of finite coarse connected components, then \(\bigsqcup _{i=1}^n X_i\) are Følner (R, 0)subsets of unbounded cardinality in X, and so X is properly amenable by Remark 2.14, giving a contradiction. Hence there is only a finite number of finite coarse connected components \(X_1,\ldots , X_N\). Let \(Y_1= \bigsqcup _{i=1}^N X_i\), and let \(Y_2=X{\setminus } Y_1\). Then all the coarse connected components of \(Y_2\) are infinite. If \(Y_2\) is amenable, then it is also properly amenable by Proposition 2.18, and so X is also properly amenable, contradicting our hypothesis. Hence \(Y_2\) is nonamenable. Since X is amenable by hypothesis, we conclude that \(Y_1\ne \varnothing \). This concludes the proof. \(\square \)
3 Algebraic amenability
In this section we will analyze from different points of view a version of amenability for \(\mathbb {K}\)algebras, where \(\mathbb {K}\) is a field. Our definition will follow existing notions in the literature (see Section 1.11 in [38] and [25, 32]), but we aim to generalize previous definitions and results in a systematical fashion. To simplify terminology, we will often not mention \(\mathbb {K}\) explicitly. For instance, we may call \(\mathbb {K}\)algebras just algebras, and \(\mathbb {K}\)dimensions just dimensions.
Definition 3.1
 (i)Let \(\mathcal {F}\subset {\mathcal A}\) be a finite subset and \(\varepsilon \ge 0\). Then a nonzero finitedimensional linear subspace \(W \subset {\mathcal A}\) is called a left \((\mathcal {F}, \varepsilon )\) Følner subspace if it satisfiesThe collection of \((\mathcal {F}, \varepsilon )\)Følner subspaces of \({\mathcal A}\) is denoted by Open image in new window .$$\begin{aligned} \frac{\dim (a W +W)}{\dim (W)}\le 1+\varepsilon , \quad \mathrm {for~all}\quad a\in \mathcal {F}. \end{aligned}$$(3.1)
 (ii)
\({\mathcal A}\) is left algebraically amenable if for any \(\varepsilon >0\) and any finite set \(\mathcal {F}\subset {\mathcal A}\), there exists a left \((\mathcal {F}, \varepsilon )\)Følner subspace.
 (iii)
\({\mathcal A}\) is properly left algebraically amenable if for any \(\varepsilon >0\) and any finite set \(\mathcal {F}\subset {\mathcal A}\), there exists a left \((\mathcal {F}, \varepsilon )\)Følner subspace W such that \(\mathcal {F} \subset W\).
We may also define right Følner subspaces, right algebraic amenability and proper right algebraic amenability by replacing \({\mathcal A}\) with \({\mathcal A}^\mathrm {op}\) in the above definitions. Since the two situations are completely symmetric, we will stick with the left versions of the definitions. For simplicity we are going to drop the term “left” for the rest of this section. Any algebra satisfying \(\dim ({\mathcal A})<\infty \) is obviously properly algebraically amenable by taking \(W={\mathcal A}\).
Remark 3.2
 (i)Algebraic amenability of \({\mathcal A}\) is equivalent to the existence of a net \(\{W_i\}_{i\in I}\) of finitedimensional linear subspaces such that$$\begin{aligned} \lim _{i} \frac{\dim (a W_i +W_i)}{\dim (W_i)} = 1, \quad \mathrm {for~all}\quad a\in {\mathcal A}. \end{aligned}$$
 (ii)
Proper algebraic amenability of \({\mathcal A}\) requires, in addition, that this net \(\{W_i\}_{i\in I}\) satisfies \( {\mathcal A}= \liminf _{i} W_i \), where \( \liminf _{i} W_i := \bigcup _{j \in I} \bigcap _{i \ge j} W_i \).
Remark 3.3
 (i)
The notion given by Elek in Definition 1.1 of [32] in fact corresponds to proper algebraic amenability, as will become evident in the next proposition (see also Definition 3.1 in [25]). Nevertheless, since the main results in Elek’s paper restrict to the case of algebras with no zero divisors, amenability and algebraic amenability are equivalent (see Corollary 3.10 below).
 (ii)
In Definition 4.3 of [15], Bartholdi uses the name exhaustively amenable instead of properly amenable.
Notice that although the definition works for \(\mathbb K\)algebras of arbitrary dimensions, the property of algebraic amenability is in essence a property for countably dimensional algebras, as seen in the next proposition.
Proposition 3.4
A \(\mathbb K\)algebra \({\mathcal A}\) is (properly) algebraically amenable if and only if any countable subset in \({\mathcal A}\) is contained in a countably dimensional \(\mathbb K\)subalgebra that is (properly) algebraically amenable.
Proof

We let \({\mathcal B}_0\) be the subalgebra generated by \(\mathcal C\).

Suppose \({\mathcal B}_i\) has been defined. Let \(\{e_k\}_{k=1}^\infty \) be a basis of \({\mathcal B}_i\). By the (proper) algebraic amenability of \({\mathcal A}\), for each positive integer k, we may find a finite dimensional linear subspace \(W_k \subset {\mathcal A}\) that is \((\{e_1, \ldots , e_k\}, \frac{1}{k})\)Følner (and contains \(\{e_1, \ldots , e_k\}\) in the case of proper algebraic amenability). We define \({\mathcal B}_{i+1}\) to be the subalgebra generated by the countably dimensional linear subspace \({\mathcal B}_i + W_1 + W_2 + \ldots \).
Conversely, in order to check (proper) algebraic amenability of \({\mathcal A}\), we fix \(\varepsilon >0\) and an arbitrary finite subset \({\mathcal F}\subset {\mathcal A}\). By assumption, \({\mathcal F}\) is contained in a countably dimensional subalgebra that is (properly) algebraically amenable, which is enough to produce the desired \(({\mathcal F}, \varepsilon )\)Følner subspace. \(\square \)
Just as in the case of metric spaces in Sect. 2, we are interested in the distinctions and relations between amenability and proper amenability. For example, when \({\mathcal A}\) is finite dimensional, then the two notions clearly coincide. The general situation bears strong similarity to the case of metric spaces. To begin with, we present a few more ways to characterize proper algebraic amenability (for infinite dimensional algebras). The first half of the following proposition should be considered as the algebraic counterpart of what we already showed in Lemma 2.6 in the context of metric spaces.
Proposition 3.5
 (1)
\({\mathcal A}\) is properly algebraically amenable.
 (2)For any \(\varepsilon >0\), \(N \in \mathbb {N}\) and any finite set \(\mathcal {F}\subset {\mathcal A}\) there exists an \((\mathcal {F}, \varepsilon )\)Følner subspace W such that$$\begin{aligned} \dim (W) \ge N. \end{aligned}$$
 (3)
For any \(\varepsilon >0\) and any finite set \(\mathcal {F}\subset {\mathcal A}\) there exists an \((\mathcal {F}, \varepsilon )\)Følner subspace that contains \( \mathbb {1}_{{\mathcal A}} \).
Proof
The implication (1) \(\Rightarrow \) (2) is immediate from the definition, since \({\mathcal F}\subset W\) implies \(\dim (W) \ge \dim ( \mathrm {span}(\mathcal {F}))\), while the latter may be made arbitrarily large since \({\mathcal A}\) is infinite dimensional.
Now assume \({\mathcal A}\) is unital. The implication (1) \(\Rightarrow \) (3) is trivial from the definition, while (3) \(\Rightarrow \) (2) is also easy in view of Remark 3.2, after observing that \( \mathbb {1}_{{\mathcal A}} \in W \) implies \(\dim ( \mathrm {span}( \mathcal {F} W +W)) \ge \dim ( \mathrm {span}(\mathcal {F}))\). This shows that (3) is equivalent to (1) and (2). \(\square \)
A notable difference between algebraic amenability and proper algebraic amenability lies in their behaviors under unitization. Recall that for a (possibly unital) \(\mathbb {K}\)algebra, the unitization of \({\mathcal A}\), denoted by \(\widetilde{{\mathcal A}}\), is defined to be the unital algebra linearly isomorphic to \({\mathcal A}\oplus \mathbb {K}\), with the product defined by \((a, \lambda ) (b, \mu ) = (ab + \mu a + \lambda b, \lambda \mu )\) for any \((a, \lambda ), (b, \mu ) \in {\mathcal A}\oplus \mathbb {K}\). The element (0, 1) now serves as the unit \(\mathbb {1}_{\widetilde{{\mathcal A}}}\). Observe that when \({\mathcal A}\) already has a unit, then \(\widetilde{{\mathcal A}} \cong {\mathcal A}\times \mathbb {K}\) as an algebra.
Proposition 3.6
 (1)
\(\widetilde{{\mathcal A}}\) is algebraically amenable if \({{\mathcal A}}\) is algebraically amenable.
 (2)
\(\widetilde{{\mathcal A}}\) is properly algebraically amenable if and only if \({\mathcal A}\) is properly algebraically amenable.
Proof
Let \(\pi {:}\, {\mathcal A}\oplus \mathbb {K} \rightarrow {\mathcal A}\) be the projection onto the first coordinate and \(\iota :{\mathcal A}\rightarrow {\mathcal A}\oplus \mathbb {K}\) be the embedding onto \({\mathcal A}\times \{0\}\). We also assume that \({\mathcal A}\) is infinite dimensional, as otherwise there is nothing to prove.
To prove (1), we assume \({\mathcal A}\) is algebraically amenable. Then for any \(\varepsilon > 0\) and any finite subset \({\mathcal F}\subset \widetilde{{\mathcal A}}\), we pick an \((\pi ({\mathcal F}), \varepsilon )\)Følner subspace W in \({\mathcal A}\). Then \(\iota (W) \subset \widetilde{{\mathcal A}}\) is \(({\mathcal F}, \varepsilon )\)Følner because for any \((a,\lambda ) \in {\mathcal F}\), \((a,\lambda ) \cdot \iota (W) + \iota (W) = \iota (aW + W)\). Thus \(\widetilde{{\mathcal A}}\) is algebraically amenable.
As for (2), we first observe that the “if” part is proved similarly as above, except for that we also use the fact that \(\dim (\iota (W)) = \dim (W)\) and apply Proposition 3.5.
The following example exhibits the difference between algebraic amenability and proper algebraic amenability, and also demonstrate that the converse of (1) in Proposition 3.6 is false (see also Theorem 3.2 in [45] for an operator theoretic counterpart).
Example 3.7
Let \({\mathcal A}\) be a \(\mathbb {K}\)algebra with a nonzero left ideal I of finite \(\mathbb K\)dimension. Then \({\mathcal A}\) is always algebraically amenable, since I is an \(({\mathcal A}, \varepsilon =0)\)Følner subspace. Therefore an easy way to construct an amenable \(\mathbb {K}\)algebra that is not properly amenable is to take a direct sum of a finite dimensional algebra and a nonalgebraicallyamenable algebra (e.g., the group algebra of a nonamenable group; see Example 3.12). In particular, if \({\mathcal A}\) is a nonamenable unital algebra, then \(\widetilde{{\mathcal A}} \cong {\mathcal A}\oplus \mathbb {K}\) is algebraically amenable but not properly algebraically amenable. Moreover, this is the only way in which a unitization \(\widetilde{{\mathcal A}}\) can be algebraically amenable but not properly algebraically amenable, as we will show in Corollary 3.11.
The next result refers to twosided ideals.
Proposition 3.8
Let \({\mathcal A}\) be a \(\mathbb {K}\)algebra with a nonzero twosided ideal I of finite \(\mathbb K\)dimension. Then, \({\mathcal A}\) is properly algebraically amenable if and only if the quotient algebra \({\mathcal A}/I\) is.
Proof
Let \(\pi :{\mathcal A}\rightarrow {\mathcal A}/ I\) is the natural projection, then for any \(\varepsilon >0\) and any finite set \(\mathcal {F}\subset {\mathcal A}\), \(V \mapsto \pi ^{1}(V)\) defines a map from Open image in new window to Open image in new window with \(\mathrm {dim} (\pi ^{1}(V) ) \ge \mathrm {dim} (V)\).
Next we show that the only situation where algebraic amenability and proper algebraic amenability differ is when the \(\mathbb K\)algebra contains a nonzero left ideal of finite \(\mathbb K\)dimension, as demonstrated by the following theorem. This situation is similar to what is known for Hilbert space operators (cf., [45, Theorem 4.1]).
Theorem 3.9
Proof
Corollary 3.10
Let \({\mathcal A}\) be a \(\mathbb K\)algebra without zerodivisor, then \({\mathcal A}\) is algebraically amenable if and only if it is properly algebraically amenable.
Proof
Corollary 3.11
Suppose that \({\mathcal A}\) is a nonalgebraically amenable algebra such that its unitization \(\widetilde{{\mathcal A}}\) is algebraically amenable. Then \({\mathcal A}\) is a unital algebra.
Proof
Example 3.12
([15, Corollary 4.5]) The group algebra \(\mathbb {K}G\) is algebraically amenable if and only if it is properly algebraically amenable if and only if G is amenable.
4 Paradoxical decompositions and invariant dimension measures of \(\mathbb {K}\)algebras
Elek showed that, analogous to the situation for groups, there is a dichotomy between algebraic amenability and a certain kind of paradoxical decomposition defined for algebras (cf., [32, Theorem 2]). However, in his paper, the conditions of countable dimensionality and the nonexistence of zerodivisors are required.
We remark here that these conditions can be removed if one replaces Elek’s definition [corresponding to proper algebraic amenability as in Definition 3.1 (ii)] with algebraic amenability as in Definition 3.1 (i). By Theorem 3.9 the assumption of no zerodivisors happens to have the effect that the properness for algebraic amenability comes for free. We will state and prove this general version of Elek’s theorem below.
The following definition of paradoxicality is equivalent to the one given by Elek in [32]. We prefer this formulation because it is formally closer to the usual condition for actions of groups, (cf., [54, Definition 1.1]).
Definition 4.1
If such a paradoxical decomposition exists, we say \(\{ e_i \}_{i \in I}\) is paradoxically decomposed by \(\mathcal {S}\).
Note that, in particular, \(g_i_{A_i}\) and \(h_j_{B_j}\) are injective, where \(A_i\) is the linear span of \(L_i\) and \(B_j\) is the linear span of \(R_j\).
Remark 4.2
 (i)
The slight formal inhomogeneity with \(L_0\) and \(R_0\) can be fixed by adding the unit \(\mathbb {1}_{\mathcal A}\) into \(\mathcal {S}\), when \({\mathcal A}\) is unital. This way, we may write \(L_0\) as \(\mathbb {1}_{\mathcal A}L_0\), and \(R_0\) as \(\mathbb {1}_{\mathcal A}R_0\). When \({\mathcal A}\) is not unital, we can still fix it by considering \(\mathcal {S}\) as a subset of \(\widetilde{{\mathcal A}}\) and adding \(\mathbb {1}_{\widetilde{{\mathcal A}}}\) into it.
 (ii)Following [32, Definition 1.2], we may also present a variant of the above definition involving only one partition. Namely, we define a onepartition paradoxical decomposition of \(\{ e_i \}_{i \in I}\) by \(\mathcal {S}\) so that it consists of a partition \(\{ e_i \} _{i \in I} = T_1 \sqcup \ldots \sqcup T_k \) and elements \(g_1,\ldots ,g_k, h_1,\ldots , h_k \in \mathcal {S}\) with the property thatis a disjoint union and linearly independent family in \({\mathcal A}\). Though this is seemingly a more restrictive notion, the existence of this onepartition version is equivalent to that of a general paradoxical decomposition, provided that \(\mathcal {S}\) contains the unit (of \({\mathcal A}\) or \(\widetilde{{\mathcal A}}\)). Indeed, starting from a general paradoxical decomposition$$\begin{aligned} g_1T_1 \cup \ldots \cup g_k T_k \cup h_1T_1 \cup \ldots \cup h_kT_k \end{aligned}$$we may define a onepartition paradoxical decomposition by setting \(T_{ij}: = L_i\cap R_j\), \(g_{ij}:= g_i\), and \(h_{ij}:= h_j\) for \(i=0,\ldots , n\) and \(j=0,\ldots , m\), with the understanding that \(g_0 = h_0 = \mathbb {1}_{\mathcal A}\) or \(\mathbb {1}_{\widetilde{{\mathcal A}}}\).$$\begin{aligned} \big ((L_0, \ldots , L_n), (R_0, \ldots , R_m), (g_1,\ldots ,g_n), (h_1,\ldots , h_m) \big ), \end{aligned}$$
 (iii)
The relation to Elek’s definition in [32] is thus as follows: a unital countably dimensional algebra is paradoxical in the sense of [32, Definition 1.2] if and only if for any (countable) basis \(\{ e_i \}_{i \in I}\) of \({\mathcal A}\), there is a paradoxical decomposition of \(\{ e_i \}_{i \in I}\) by \({\mathcal A}\).
The following lemma generalizes [32, Lemma 2.2].
Lemma 4.3
Proof
The following is a key proposition of this section. It generalizes Proposition 2.2 in [32] to arbitrary \(\mathbb K\)algebras which may have zerodivisors, have no unit, or have uncountable dimensions. To prove this, we adapt ideas from [40, Theorem 3.4, (vi) \(\Rightarrow \) (v)] (see also [40]) in the context of groups and metric spaces to the algebraic setting.
Proposition 4.4
Assume that \({\mathcal A}\) is a \(\mathbb K\)algebra which is not algebraically amenable. Then there exists a finite subset \({\mathcal F}\subset {\mathcal A}\) such that for any basis \(\{e_i\}_{i \in I}\) of \({\mathcal A}\), there is a paradoxical decomposition of \(\{e_i\}_{i \in I}\) by \({\mathcal F}\).
Proof
Our goal is to “trim down” the above constant setvalued function to a singletonvalued function in \(\Omega \). For this purpose, we use the natural partial order on \(\Omega \) given by pointwise inclusion: \(\omega \le \omega '\) if \(\omega (i,j) \subset \omega '(i,j)\) for any \((i,j) \in I \times \{0,1\}\). Since any descending chain in \(\Omega \) has a nonempty lower bound given by pointwise intersection, by Zorn’s Lemma, we can find a minimal element \(\omega _0 \in \Omega \).
Now we define a suitable notion of invariant dimensionmeasure for \(\mathbb {K}\)algebras, an analogue of invariant mean for amenable groups. Note that the lack of distributivity in the lattice of subspaces of a vector space makes it necessary to give up some of the properties one would expect for this concept.
Definition 4.5
 (i)
\( \mu ({\mathcal A}) =1\).
 (ii)
If A, B are linear subspaces in \({\mathcal A}\) with \(A\,\cap \, B = \{0 \}\), then \(\mu (A\,\oplus \, B) \ge \mu (A)\,+\, \mu (B)\).
 (iii)
For every partition \(L_1\sqcup L_2\sqcup \ldots \sqcup L_m\) of \(\{e_i \}_{i \in I}\), we have \(\sum _{k=1}^m \mu (\mathrm {span}(L_k)) =1\).
 (iv)
For any \(s\in \mathcal {S}\) and any linear subspace \(A \subset {\mathcal A}\) such that \(s_A\) is injective, we have \(\mu (sA) \ge \mu (A)\).
Note that if \(\mu \) is a dimensionmeasure on \({\mathcal A}\) and \(A\subseteq B\) are subspaces of \({\mathcal A}\), then, by property (ii), it follows that \(\mu (A) \le \mu (B)\).
We can now state the following generalization of [32, Theorem 1].
Theorem 4.6
 (1)
\({\mathcal A}\) is algebraically amenable.
 (2)
For any finite subset \({\mathcal F}\subset {\mathcal A}\), there is a basis of \({\mathcal A}\) that cannot be paradoxically decomposed by \({\mathcal F}\).
 (3)
For any countably dimensional linear subspace \(W \subset {\mathcal A}\), there is a basis of \({\mathcal A}\) that cannot be paradoxically decomposed by W.
 (4)
For any countably dimensional linear subspace \(W \subset {\mathcal A}\), there exists a Winvariant dimensionmeasure on \({\mathcal A}\) (associated to some basis).
Proof
The implication (2) \(\Rightarrow \) (1) follows from Proposition 4.4. The implication (3) \(\Rightarrow \) (2) is immediate by setting \(W = \mathrm {span}({\mathcal F})\).
 Case 1:\({\mathcal A}\) is properly algebraically amenable. By Proposition 3.4, there is a countably dimensional subalgebra \({\mathcal B}\subset {\mathcal A}\) that is properly algebraically amenable and contains W. Let \(\{ W_i \}_{i=1}^{\infty }\) be an increasing sequence of finitedimensional subspaces of \({\mathcal A}\) such that \({\mathcal B}= \cup _{i=1}^{\infty } W_i\), and such thatfor all \(a\in {\mathcal B}\). Let \(\omega \) be a free ultrafilter on \(\mathbb N\), and let \(\{e_i \}_{i=1}^{\infty }\) be a basis for \({\mathcal B}\) obtained by successively enlarging basis of the spaces \(W_i\) (cf. [32, Proposition 2.1]). We then enlarge \(\{e_i \}_{i=1}^{\infty }\) to a basis \(\{e_i \}_{i \in I}\) of \({\mathcal A}\), where \(\mathbb {N} \subset I\). For a linear subspace A of \({\mathcal A}\), set$$\begin{aligned} \lim _{i\rightarrow \infty } \frac{\dim (aW_i+W_i)}{\dim (W_i)} = 1 \end{aligned}$$Obviously, we have \(\mu ({\mathcal A}) =1 \) and \(0 \le \mu (A)\le 1\) for every subspace A. Moreover, properties (ii) and (iii) in Definition 4.5 clearly hold, so we only need to check (iv). To prove (iv) we first show that for any \(a\in W\) and any linear subspace A we have$$\begin{aligned} \mu (A) = \lim _{\omega } \frac{\dim (A\cap W_i)}{\dim (W_i)}. \end{aligned}$$Write \(T_i = (W_i+aW_i)\cap A\). Then \(T_i\cap W_i = A\cap W_i \), so that \(T_i = (W_i\cap A)\oplus T_i'\) with \(T_i'\cap W_i = \{ 0\}\). Hence$$\begin{aligned} \mu (A) = \lim _{\omega }\frac{\dim ( (W_i+aW_i)\cap A ) }{\dim (W_i)}. \end{aligned}$$(4.1)Since \(\dim (T_i') /\dim (W_i)\rightarrow 0\), we obtain the result. We now show (iv). Let \(a \in W\) be such that \(a_A\) is injective. Then we have$$\begin{aligned} \frac{\dim (T_i)}{\dim (W_i)} = \frac{\dim (W_i\cap A)}{\dim (W_i)} + \frac{\dim (T_i')}{\dim (W_i)}. \end{aligned}$$where in the second equality we have used that aA is injective.$$\begin{aligned} \mu (aA)&= \lim _{\omega } \frac{\dim ((W_i+aW_i)\cap aA)}{\dim (W_i)} \ge \lim _{\omega } \frac{\dim (aW_i\cap aA)}{\dim (W_i)}\\&\ge \lim _{\omega } \frac{\dim (a(W_i\cap A))}{\dim (W_i)} = \lim _{\omega } \frac{\dim (W_i \cap A)}{\dim (W_i)}= \mu (A) , \end{aligned}$$
 Case 2:\({\mathcal A}\) is algebraically amenable but not properly algebraically amenable. By Theorem 3.9, we only need to build a dimensionmeasure in the case where \({\mathcal A}\) has a nonzero finitedimensional left ideal I. This is easily taken care of by definingfor each linear subspace \(A \subset {\mathcal A}\).$$\begin{aligned} \mu (A) = \frac{\dim (I\cap A)}{\dim (I)} \end{aligned}$$
For countably dimensional (or equivalently, countably generated) \(\mathbb {K}\)algebras, the statement of the previous theorem can be somewhat simplified:
Corollary 4.7
 (1)
\({\mathcal A}\) is algebraically amenable.
 (2)
There is a basis of \({\mathcal A}\) that cannot be paradoxically decomposed by \({\mathcal A}\).
 (3)
There exists an \({\mathcal A}\)invariant dimensionmeasure on \({\mathcal A}\) (associated to some basis).
Proof
This is immediate after we set \(W = {\mathcal A}\) in the statement of Theorem 4.6.\(\square \)
Remark 4.8
Recall the usual Murray–von Neumann equivalence \(\sim \) and comparison \(\gtrsim \) for idempotents of an algebra, defined as follows: for idempotents e, f in \({\mathcal A}\), write \(e\sim f\) if there are \(x,y\in {\mathcal A}\) such that \(e=xy\) and \(f=yx\); write \(e \gtrsim f\) if there are \(x,y\in {\mathcal A}\) such that \(xy \in e{\mathcal A}e\) and \(f=yx\). These relations naturally extends to the infinite matrix algebra \(M_\infty ({\mathcal A}) := \bigcup _{n=1}^\infty M_n({\mathcal A})\) where the \(M_n({\mathcal A})\) embeds into \(M_{n+1}({\mathcal A})\) blockdiagonally as \(M_n({\mathcal A}) \oplus 0\).
An idempotent e in an algebra \({\mathcal A}\) is said to be properly infinite if there are orthogonal idempotents \(e_1, e_2\) in \(e{\mathcal A}e\) such that \(e_1 \sim e\sim e_2\). Equivalently, e is properly infinite if \(e \gtrsim e \oplus e\). A (nonzero) unital algebra \({\mathcal A}\) is said to be properly infinite in case \(\mathbb {1}\) is a properly infinite idempotent.
As an application of the dichotomy shown in Theorem 4.6, we present a method of producing nonalgebraically amenable \(\mathbb {K}\)algebras:
Corollary 4.9
A properly infinite unital \(\mathbb {K}\)algebra is not algebraically amenable.
Proof
5 Leavitt algebras and Leavitt path algberas
In this section we study the amenability of Leavitt algebras and Leavitt path algebras (see below for the specific definitions). Classical Leavitt algebras were invented by Leavitt ([42, 43]) to provide universal examples of algebras without the invariant basis number property. As such, they cannot be algebraically amenable, by a result of Elek [32, Corollary 3.1(1)]. Leavitt path algebras provide a wide generalization of classical Leavitt algebras, in much the same way as graph \(C^*\)algebras generalize Cuntz algebras (see e.g. [48] for an introduction to the theory of graph \(C^*\)algebras).
5.1 Leavitt algebras
Definition 5.1
 (i)
Let n, m be integers such that \(1\le m< n\). Then the Leavitt algebra \(L(m,n)= L_\mathbb {K}(m,n)\) is the algebra generated by elements \(X_{ij}\) and \(Y_{ji}\), for \(i=1,\ldots ,m\) and \(j=1,\ldots ,n\), such that \(XY = \mathbb {1}_m\) and \(YX= \mathbb {1}_n\), where X denotes the \(m\times n\) matrix \((X_{ij})\) and Y denotes the \(n\times m\) matrix \((Y_{ji})\).
 (ii)
The algebra \(L_{\infty }= L_{\mathbb {K}, \infty }\) is the unital algebra generated by \(x_1,y_1,x_2, y_2, \ldots \) subject to the relations \(y_jx_i = \delta _{i,j} \mathbb {1}\).
The algebras L(m, n) are simple if and only if \(m=1\) [43, Theorems 2 and 3]. The algebra \(L_{\infty }\) is simple [10, Theorem 4.3].
The following is wellknown (cf. [2] or [42]):
Proposition 5.2
 (1)
\({\mathcal A}\) does not satisfy the IBN property if and only if there is a unital homomorphism \(L(m,n) \rightarrow {\mathcal A}\) for some \(1\le m < n\).
 (2)
\({\mathcal A}\) is properly infinite if and only if there is a unital embedding \(L_{\infty }\rightarrow {\mathcal A}\).
Proof
(1) By definition, if an algebra \({\mathcal A}\) does not have the IBN property, then there are m, n with \(1\le m <n\) such that \({\mathcal A}^m \cong {\mathcal A}^n\), and this isomorphism of free modules will be implemented by matrices \(X'\in M_{m\times n}({\mathcal A})\) and \(Y' \in M_{n\times m} ({\mathcal A})\) such that \(X'Y'=I_m\) and \(Y'X'= I_n\). We thus obtain a unital homomorphism \(L(m,n) \rightarrow {\mathcal A}\). The converse is trivial.
(2) If \({\mathcal A}\) is properly infinite, we may inductively find an infinite sequence \(e_1, e_2,\ldots \) of mutually orthogonal idempotents such that \(e_i\sim 1\) for all i. This enables us to define a homomorphism \(L_{\infty } \rightarrow {\mathcal A}\) which is injective because \(L_{\infty }\) is simple. The converse is obvious.\(\square \)
Note that \(L_{\infty }\) is properly infinite but does have the IBN property.
Proposition 5.3
If \({\mathcal A}\) is a unital algebraically amenable algebra, then \({\mathcal A}\) has the IBN property.
Proof
Suppose that \({\mathcal A}\) does not have the IBN property. Then there are integers m, n with \(1\le m <n\) and there is a unital homomorphism \(L(m,n) \rightarrow {\mathcal A}\). Now \(M_n({\mathcal A})\cong M_m({\mathcal A})\) is properly infinite, so that by Corollary 4.9, \(M_n({\mathcal A})\) is not algebraically amenable. If \({\mathcal A}\) were amenable then \(M_n({\mathcal A}) \cong {\mathcal A}\otimes M_n(\mathbb {K})\) would be amenable too ([25, Proposition 4.3(2)]). Therefore \({\mathcal A}\) is not algebraically amenable, showing the result. \(\square \)
Corollary 5.4
A unital \(\mathbb {K}\)algebra \({\mathcal A}\) that unitally contains the Leavitt algebra L(m, n) for some \(1\le m < n\) is not algebraically amenable. \(\square \)
5.2 Leavitt path algebras
In general, a nonalgebraically amenable algebra need not be properly infinite, as the noncommutative free algebra shows. We now show that, within a certain class of algebras, the class of Leavitt path algebras, both properties are indeed equivalent. Note that this class of algebras includes the algebras L(1, n) and \(L_{\infty }\) as distinguished members. (The algebras L(m, n), with \(1<m<n\) are not included in the class of Leavitt path algebras, but they are Moritaequivalent to Leavitt path algebras associated to separated graphs [8].) We refer the reader to [2] and the references therein for more information about Leavitt path algebras.
We recall some definitions needed here.
Definition 5.5
A (directed) graph \(E=(E^{0},E^{1},r,s)\) consists of two sets \(E^{0}\) and \(E^{1}\) together with range and source maps \(r,s:E^{1}\rightarrow E^{0}\). The elements of \(E^{0}\) are called vertices and the elements of \(E^{1}\) edges.
A vertex v is called a sink if it emits no edges, that is, \(s^{1}(v)=\varnothing \), the empty set. The vertex v is called a finite emitter if \(s^{1}(v)\) is finite; otherwise it is an infinite emitter. A finite emitter which is not a sink is also called a regular vertex. For each \(e\in E^{1}\), we call \(e^{*}\) a ghost edge. We let \(r(e^{*})\) denote s(e), and we let \(s(e^{*})\) denote r(e).
The Leavitt path algebras are built on top of these directed graphs.
Definition 5.6
 (1)
\(s(e)e=e=er(e)\) for all \(e\in E^{1}\).
 (2)
\(r(e)e^{*}=e^{*}=e^{*}s(e)\) for all \(e\in E^{1}\).
 (3)
(The “CK1 relations”) For all \(e,f\in E^{1}\), \(e^{*}e=r(e)\) and \(e^{*}f=0\) if \(e\ne f\).
 (4)(The “CK2 relations”) For every regular vertex \(v\in E^{0}\),$$\begin{aligned} v=\sum _{e\in E^{1},s(e)=v}ee^{*}. \end{aligned}$$
In a sense, the definition of a Leavitt path algebra treats the graph as a dynamical system: its multiplication is based on the ways one can traverse the vertices of the graph via the edges. This naturally brings into the picture notions such as paths and cycles.
Definition 5.7
A (finite) path \(\mu \) of length \(n>0\) is a finite sequence of edges \(\mu =e_{1}e_{2}\cdot \cdot \cdot e_{n}\) with \(r(e_{i})=s(e_{i+1})\) for all \(i=1,\cdot \cdot \cdot ,n1\). In this case, \(\mu ^{*}=e_{n}^{*}\cdot \cdot \cdot e_{2}^{*}e_{1}^{*}\) is the corresponding ghost path. The set of all vertices on the path \(\mu \) is denoted by \(\mu ^{0}\). Any vertex v is considered a path of length 0.
A nontrivial path \(\mu \) \(=e_{1}\dots e_{n}\) in E is closed if \(r(e_{n} )=s(e_{1})\), in which case \(\mu \) is said to be based at the vertex \(s(e_{1})\). By cyclically permuting the edges of a closed path \(\mu =e_{1}\dots e_{n}\), we obtain a closed path \(e_{k}\dots e_{n}e_{1}\dots e_{k1}\) based at the vertex \(s(e_{k})\) for any \(k = 1, \ldots , n\). A closed path \(\mu \) as above is called simple provided it does not pass through its base more than once, i.e., \(s(e_{i})\ne s(e_{1})\) for all \(i=2,...,n\).
The closed path \(\mu \) is called a cycle based at v if \(s(e_1)=v\) and it does not pass through any of its vertices twice, that is, if \(s(e_{i})\ne s(e_{j})\) whenever \(i\ne j\). A nontrivial cyclic permutation of a cycle based at a vertex v is then a cycle based at a different vertex. Cyclic permutation thus induces an equivalence relation on the set of all cycles based at vertices. An equivalence class of it is called a cycle. Note that it is meaningful to talk about the set of vertices of a cycle, which we denote by \(c^0\). A cycle c is called an exclusive cycle if it is disjoint with every other cycle; equivalently, no vertex v on c is the base of a different cycle other than the cyclic permutation of c based at v.
The following lemma was shown in the rowfinite case in [13, Lemma 7.3]. We include the identical proof for completeness.
Lemma 5.8
Let E be an arbitrary graph and let \(\mathbb {K}\) be a field. If \(v\in E^0\) belongs to a nonexclusive cycle, then v is a properly infinite idempotent in \(L_\mathbb {K}(E)\).
Proof
Below we summarize some additional basic terminologies and properties for graphs and Leavitt path algebras. For this we follow the book in preparation [1].
Remark 5.9
 (1)
If there is a path from a vertex u to a vertex v, we write \(u\ge v\). This defines a preorder on \(E^0\). As we have shown above, \(u\ge v\) implies \(u \gtrsim v\) in \(L_{\mathbb {K}}(E)\). Since all vertices on a cycle are equivalent with regard to the preorder \(\ge \), it induces a preorder on the set of all cycles, so that for any cycles \(c_1\) and \(c_2\), we have \(c_1\ge c_2\) if and only if there is path from a vertex of \(c_1\) to a vertex of \(c_2\).
 (2)
Let C be the set of all cycles in E. Let \(C/{\sim } \) be the partially ordered set obtained by antisymmetrization of the preorder \(\le \) on C, so that \(c\sim c'\) if and only if \(c \le c'\) and \(c'\le c\). Note that the exclusive cycles are precisely those cycles c such that \([c]= \{ c \}\), and that \(C/{\sim } \) is a finite set if E has a finite number of vertices.
 (3)
The Leavitt path algebra \(L_\mathbb {K}(E)\) is unital if and only if \( E^0  < \infty \), in which case the unit is given by \(\sum _{v \in E^0} v\).
 (4)
Every finite path \(\mu = e_1 \cdots e_n\) induces the elements \(\mu = e_1 \cdots e_n\) and \(\mu ^{*}=e_{n}^{*}\cdots e_{1}^{*}\) in \(L_{\mathbb {K}}(E)\). By a simple induction, we see that the Leavitt path algebra \(L_\mathbb {K}(E)\) is linearly spanned by terms of the form \(\lambda \rho ^*\), where \(\lambda \) and \(\rho \) are paths such that \(r(\lambda )= r(\rho )\).
 (5)
The graph E is called acyclic if it contains no cycle, and finite if both \(E^0\) and \(E^1\) are finite sets. A finite acyclic graph clearly contains finitely many paths. Thus by (4), we see that \(L_{\mathbb {K}}(E)\) is finitedimensional. In fact, in this case, \(L_{\mathbb {K}}(E)\) is a finite direct sum of matrix algebras over \(\mathbb {K}\) (cf., [2, Theorem 3.1]).
 (6)A subset H of \(E^{0}\) is called hereditary if, whenever \(v\in H\) and \(w\in E^{0}\) satisfy \(v\ge w\), then \(w\in H\). A hereditary set is saturated if, for any regular vertex v, \(r(s^{1}(v))\subseteq H\) implies \(v\in H\). For \(X\subseteq E^0\), we denote by \(\overline{X}\) the hereditary saturated closure of X. To compute \(\overline{X}\), one can first compute the tree of X, \(T(X) := \{ w\in E^0 : w\le v \text { for some } v\in X \}\), which is the smallest hereditary subset of \(E^0\) containing X, and then, setting \(\Lambda _0(T(X)) := T(X)\), compute inductivelyfor \(n = 1, 2 , \ldots \), where \(E^0_{\mathrm {reg}}\) is the set of regular vertices. It is easy to see \(\overline{X} = \bigcup _{n=0}^{\infty } \Lambda _n(T(X))\).$$\begin{aligned} \Lambda _n(T(X)) := \{y\in E^0_{\mathrm {reg}} : r(s^{1}(y))\subseteq \Lambda _{n1}(T(X))\} \cup \Lambda _{n1}(T(X)) \end{aligned}$$
 (7)We shall use the following constructions from [52]. A breaking vertex of a hereditary saturated subset H is an infinite emitter \(w\in E^{0}{\setminus } H\) with the property that \(1\le s^{1}(w)\cap r^{1}(E^{0}{\setminus }H)<\infty \). The set of all breaking vertices of H is denoted by \(B_{H}\). For any \(v\in B_{H}\), we define an idempotent \(v^{H} \in L_{\mathbb {K}}(E)\) byGiven a hereditary saturated subset H and a subset \(S\subseteq B_{H}\), (H, S) is called an admissible pair. Given an admissible pair (H, S), I(H, S) denotes the ideal generated by \(H\cup \{v^{H}:v\in S\}\). Then we have an isomorphism \(L_{\mathbb {K}}(E)/I(H,S)\cong L_{\mathbb {K}} (E / (H,S))\). Here E / (H, S) is the quotient graph of E in which \((E / (H,S))^{0}=(E^{0}\backslash H)\cup \{v^{\prime }:v\in B_{H}\backslash S\}\) and \((E / (H,S))^{1}=\{e\in E^{1}:r(e)\notin H\}\cup \{e^{\prime }:e\in E^{1},r(e)\in B_{H}\backslash S\}\) and r, s are extended to \((E / (H,S))^{1}\) by setting \(s(e^{\prime })=s(e)\) and \(r(e^{\prime })=r(e)^{\prime }\). Thus when \(S=B_{H}\), we can identify the quotient graph \(E{\setminus }(H,B_{H})\) with the subgraph E / H of E, where \((E / H)^0= E^0{\setminus }H\) and \((E / H)^1= \{ e\in E^1 : r(e) \notin H \}\). It was shown in [52] that the graded ideals of \(L_{\mathbb {K}}(E)\) are precisely the ideals of the form I(H, S) for some admissible pair (H, S), though we will not make use of this.$$\begin{aligned} v^{H} := v\sum _{s(e)=v,r(e)\notin H}ee^{*}. \end{aligned}$$
 (8)A subgraph \(E'\) of E is called full if \((E')^1 = \{e \in E^{1}:s(e), r(e) \in (E')^0 \}\). For a subset \(X \subset E^0\), we define a full subgraph M(X) so thatIf \(X = \{v\}\) for some \(v \in E^0\), we also write \(M(v) = M(\{v\})\). Also define$$\begin{aligned} M(X)^0=\{w\in E^{0}:w\ge v \text { for some } v \in X \}. \end{aligned}$$which is hereditary by design. Note that any edge e is in a cycle if and only if \(r(e) \notin H(s(e))\) if and only if \(r(e) \in M(s(e))^0\). It follows that if v belongs to a cycle, then H(v) is a hereditary saturated subset of E. \(\square \)$$\begin{aligned} H(v)=E^{0}{\setminus }M(v)^0, \end{aligned}$$
Theorem 5.10
 The following are equivalent:
 (A1)
\(L_\mathbb {K}(E)\) is not algebraically amenable.
 (B1)
\(E^0\) is finite, \(E^0 {\setminus } H = \varnothing \), and every maximal cycle is nonexclusive.
 (C1)
\(L_\mathbb {K}(E)\) is unital and properly infinite
 (A1)
 The following are equivalent:
 (A2)
\(L_\mathbb {K}(E)\) is algebraically amenable but not properly algebraically amenable.
 (B2)
\(E^0\) is finite, E is not acyclic, \(E^0 {\setminus } H\) consists of a nonzero number of finite emitters, and every maximal cycle is nonexclusive.
 (C2)
\(L_\mathbb {K}(E) = L_\mathbb {K}(E') \oplus L_\mathbb {K}(E'') \) for some directed graphs \(E'\) and \(E''\) such that \(L_\mathbb {K}(E') \) has nonzero finite dimension and \(L_\mathbb {K}(E'') \) is not algebraically amenable.
 (A2)
 The conditionholds if and only if one or more of the following conditions hold:
 (A3)
\(L_\mathbb {K}(E)\) is properly algebraically amenable
 (B3a)
E is acyclic;
 (B3b)
\(E^0\) is infinite;
 (B3c)
\(E^0 {{\setminus }} H\) contains at least one infinite emitter;
 (B3d)
E has an exclusive maximal cycle.
 (A3)
Proof

(B1) \(\Rightarrow \) (C1) \(\Rightarrow \) (A1) \(\Rightarrow \) (B1),

(B2) \(\Rightarrow \) (C2) \(\Rightarrow \) (A2) \(\Rightarrow \) (B2), and

(B3) \(\Rightarrow \) (A3) \(\Rightarrow \) (B3).
(B3a)\(\vee \)(B3b)\(\vee \)(B3c)\(\vee \)(B3d) \(\Rightarrow \) (A3): We first observe that when (B3a) holds and (B3b) fails, i.e., when E is finite and acyclic, Remark 5.9 (5) tells us that \(L_\mathbb {K}(E)\) is finite dimensional and thus properly algebraically amenable.
Apart from this easy case, \(L_\mathbb {K}(E)\) is always infinitedimensional, so by Proposition 3.5, it suffices to show that, given any \(\varepsilon >0\), any \(N \in \mathbb {N}\), and any finite subset \(\mathcal F\) of \(L_\mathbb {K}(E)\), we can find an \((\mathcal {F}, \varepsilon )\)Følner subspace W in \(L_\mathbb {K}(E)\) with \(\dim (W) \ge N\). Since each element of \(L_\mathbb {K}(E)\) is a linear combination of terms of the form \(\lambda \rho ^*\), where \(\lambda \) and \(\rho \) are paths such that \(r(\lambda )= r(\rho )\), without loss of generality we can assume that \(\mathcal F\) consists of elements of this form, say \(\mathcal F = \{ \lambda _1\rho _1^*, \ldots , \lambda _r \rho _r^* \}\).
First, we assume (B3b) holds, i.e., \(E^0\) is infinite. Then we can find a subset \(X \subset E^0\) with \(X  = N\) and \(X \cap \{ s(\rho _1) , \ldots , s(\rho _r) \} = \varnothing \). Put \(W = \mathrm {span}(X)\). It then follows that \(\lambda _j \rho _j^* W = 0 \) for \(j = 1, \ldots , r\). Hence W is an \((\mathcal {F}, 0)\)Følner subspace with \(\dim (W) \ge N\).
Therefore any of the conditions (B3a), (B3b), (B3c) and (B3d) implies that \(L_\mathbb {K}(E)\) is properly algebraically amenable. \(\square \)
We highlight the following trivial consequence of Theorem 5.10:
Corollary 5.11
Let E be a graph with finitely many vertices and let \(\mathbb {K}\) be a field. Then the (unital) Leavitt path algebra \(L_\mathbb {K}(E)\) is not algebraically amenable if and only if it is properly infinite.
Remark 5.12
It is wellknown ([32, Proposition 3.1]) that a finitely generated \(\mathbb {K}\)algebra of subexponential growth is amenable. On the other hand, it has been shown in [5] that, for a finite graph E, the Leavitt path algebra \(L_{\mathbb {K}}(E)\) either has exponential growth or has polynomially bounded growth. Moreover, by [5, Theorem 5 (1)], \(L_{\mathbb {K}}(E)\) has polynomially bounded growth if and only if every cycle of E is an exclusive cycle, and in this case a precise formula for the Gelfand–Kirillov dimension of \(L_{\mathbb {K}}(E)\) is obtained ([5, Theorem 5 (2)]). Comparing this with Theorem 5.10, we see that there are finite graphs such that \(L_{\mathbb {K}}(E)\) is algebraically amenable and has exponential growth (just consider the graph E of Example 5.15).
Since \(L_\mathbb {K}(E)\) admits an involution (see for instance [52]), left and right amenability is equivalent for these algebras. Moreover the above proof shows that we can “localize” amenability in certain parts of the graph (in analogy with the metric space situation, cf., Sect. 2.1). We provide a simple example that shows that the situation is quite different when we consider the usual path algebras.
Definition 5.13
Given an arbitrary graph E and a field \(\mathbb {K}\), the path \(\mathbb {K}\) algebra \(\mathbb {K} E\) is defined to be the \(\mathbb {K}\)algebra generated by a set \(\{v:v\in E^{0}\}\) of pairwise orthogonal idempotents together with a set of variables \(\{e:e\in E^{1}\}\) which satisfy \(s(e)e=e=er(e)\) for all \(e\in E^{1}\).
In other words, the path algebra is linearly spanned by all paths in E, with the multiplication given by concatenation of paths (or zero if two paths cannot be concatenated).
Example 5.14
To see that \({\mathcal A}\) is left properly algebraically amenable, we choose an arbitrarily large finitedimensional subspace W of \( x {\mathcal A}w\) and note that \({\mathcal A}W = {\mathcal A}(v W ) = \mathbb {K}v W = W\), i.e., W is an \(({\mathcal A}, 0)\)Følner subspace.
The next example is similar to the above. It shows that having a maximal exclusive cycle is not enough to guarantee the (right) amenability of path algebras (compare with Theorem 5.10).
Example 5.15
6 Translation algebras on coarse spaces
To conclude we will illustrate the close relation between amenability for metric spaces and algebraic amenability for \(\mathbb {K}\)algebras, in view of the natural bridge between the two settings—the construction of translation algebras (see, e.g., [49, Chapter 4]). Let us recall this construction.
Let (X, d) be a locally finite extended metric space as in Sect. 2 and \(\mathbb {K}\) an arbitrary field. We denote by \(\mathbb {K}[X]\) the \(\mathbb {K}\)linear space generated by the basis X, and by \(\mathrm {End}_{\mathbb {K}}(\mathbb {K}[X])\) the algebra of \(\mathbb {K}\)linear endomorphism of \(\mathbb {K}[X]\). For the sake of clarity, we denote by \(\delta _x\) the basis element of \(\mathbb {K}[X]\) corresponding to a point \(x \in X\). We also sometimes think of an element \(T \in \mathrm {End}_{\mathbb {K}}(\mathbb {K}[X])\) as a matrix indexed by X, and define \(T_{xy} \in \mathbb {K}\) as its entry at \((x,y ) \in X \times X\), so that \(T(\delta _y) = \sum _{x\in X} T_{xy} \delta _x \) for any \(y \in X\).
Definition 6.1
The translation \(\mathbb {K}\) algebra \(\mathbb {K}_\mathrm {u}(X)\) is the (unital) \(\mathbb {K}\)subalgebra of \(\mathrm {End}_{\mathbb {K}}(\mathbb {K}[X])\) generated by \(V_t\) for all the partial translations t on X.
Remark 6.2
Theorem 6.3
 (1)
(X, d) is amenable.
 (2)
\(\mathbb {K}_\mathrm {u}(X)\) is algebraically amenable.
 (3)
\(\mathbb {K}_\mathrm {u}(X)\) is not properly infinite.
 (4)
\(\mathbb {K}_\mathrm {u}(X)\) does not contain the Leavitt algebra \(L_{\mathbb {K}}(1,n)\) as a unital \(\mathbb {K}\)subalgebra.
Proof
(2) \(\Rightarrow \) (3): This implication follows from Corollary 4.9.
(3) \(\Rightarrow \) (4): Suppose that for some \(n\ge 2\) the Leavitt algebra L(1, n) unitally embeds into \(\mathbb {C}_\mathrm {u}(X)\). Then, any two distinct pairs of generators \(X_i,Y_i\), \(X_j,Y_j\), \(i\not =j\), of L(1, n) implement the proper infiniteness of \(\mathbb {K}_\mathrm {u}(X)\).
We also have an analogous result for proper amenability. We will use the following terminology. Given two algebras \(\mathcal A\) and \(\mathcal B\), we say that \(\mathcal A\) is a finitedimensional extension of \(\mathcal B\) in case there is a finitedimensional twosided ideal I of \(\mathcal A\) such that \(\mathcal A/ I \cong \mathcal B\).^{4}
Theorem 6.4
 (1)
(X, d) is properly amenable.
 (2)
\(\mathbb {K}_\mathrm {u}(X)\) is properly algebraically amenable.
 (3)
\(\mathbb {K}_\mathrm {u}(X)\) is not a finitedimensional extension of a properly infinite \(\mathbb {K}\)algebra.
Proof
(2) \(\Rightarrow \) (3): Suppose that \(\mathbb {K}_\mathrm {u}(X)\) is a finitedimensional extension of a properly infinite \(\mathbb {K}\)algebra, that is, there is a finitedimensional twosided ideal I of \(\mathbb {K}_\mathrm {u}(X)\) such that \(\mathbb {K}_\mathrm {u}(X) / I\) is properly infinite. By Corollary 4.9, \(\mathbb {K}_\mathrm {u}(X) / I\) is not algebraically amenable, and thus not properly algebraically amenable, either. By Proposition 3.8, it follows that \(\mathbb {K}_\mathrm {u}(X)\) is not properly algebraically amenable.
(3) \(\Rightarrow \) (1): Assume that \(\mathbb {K}_\mathrm {u}(X)\) is not a finitedimensional extension of a properly infinite \(\mathbb {K}\)algebra. In particular, itself is not properly infinite. Then Theorem 6.3 implies that (X, d) is amenable. Now suppose that (X, d) were not a properly amenable metric space. Corollary 2.20 shows that there would be a partition \(X= Y_1\sqcup Y_2\), where \(Y_1\) is a finite nonempty subset of X, \(Y_2\) is nonamenable and \(d(x,y)= \infty \) for \(x\in Y_1\) and \(y\in Y_2\). As in Remark 6.2, this would induce a direct sum decomposition \(\mathbb {K}_\mathrm {u}(X) \cong \mathbb {K}_\mathrm {u}(Y_1)\oplus \mathbb {K}_\mathrm {u}(Y_2)\), with \(\mathbb {K}_\mathrm {u}(Y_1)\) being finitedimensional. In particular, \(\mathbb {K}_\mathrm {u}(X)\) would be a finitedimensional extension of \(\mathbb {K}_\mathrm {u}(Y_2)\), the latter being properly infinite, again by Theorem 6.3. This would contradict our assumption. \(\square \)
Footnotes
 1.
In fact, Elek’s original definition in [32] corresponds formally to our definition of proper algebraic amenability, instead of algebraic amenability. For general algebras with possible zerodivisors, we prefer to assign the term “algebraic amenability” to the concept without the exhaustion requirement because of its central role in Theorem 2.
 2.
Recall that a metric space is locally finite if and only if it is discrete and proper, the latter meaning that any closed ball is compact (see, e.g., [21, Section 5.5]). We avoid this terminology because we use the term “proper” in a different sense in this article.
 3.
 4.
This is in agreement with the nonuniversal convention of calling the algebra \(\mathcal A\) above an extension of \(\mathcal B\) by I.
Notes
Acknowledgements
The secondnamed author is partially supported by Deutsche Forschungsgemeinschaft (SFB 878). The thirdnamed author thanks Wilhelm Winter for his kind invitation to the Mathematics Department of the University of Münster in April 2014 and March–June 2016. Financial support was provided by the DFG through SFB 878, as well as, by a DAADGrant during these visits. He would also like to thank the organizers of the Thematic Program on Abstract Harmonic Analysis, Banach and Operator Algebras at Fields Institute in Toronto in May 2014 for the stimulating atmosphere. The fourthnamed author are grateful to David Kerr for some very helpful suggestions. We are also grateful to Javier Rodríguez Chatruc for his comments on Section 6. Part of the researc h was conducted during visits and workshops at Universitat Autònoma de Barcelona, University of Copenhagen, University of Münster and Institut Mittag–Leffler. The authors owe many thanks and great appreciation to these institutes and hosts for their hospitality.
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