@
Appl. Gen. Topol. 21, no. 1 (2020), 1-15

doi:10.4995/agt.2020.11488

c© AGT, UPV, 2020

Topological characterizations of amenability

and congeniality of bases

Sergio R. López-Permouth and Benjamin Stanley

Department of Mathematics, Ohio University, USA (lopez@ohio.edu,benqstanley@gmail.com)

Communicated by J. Galindo

Abstract

We provide topological interpretations of the recently introduced no-
tions of amenability and congeniality of bases of infinite dimensional
algebras. In order not to restrict our attention only to the countable
dimension case, the uniformity of the topologies involved is analyzed
and therefore the pertinent ideas about uniform topological spaces are
surveyed.
A basis B over an infinite dimensional F -algebra A is called amenable
if F B, the direct product indexed by B of copies of the field F , can
be made into an A-module in a natural way. (Mutual) congeniality is
a relation that serves to identify cases when different amenable bases
yield isomorphic A-modules.
(Not necessarily mutual) congeniality between amenable bases yields an
epimorphism of the modules they induce. We prove that this epimor-
phism is one-to-one only if the congeniality is mutual, thus establishing
a precise distinction between the two notions.

2010 MSC: 16K40; 16D99; 15B99.

Keywords: uniform topologies; linear vector spaces; amenable bases; con-
geniality of bases; Schauder bases; infinite-dimensional modules
and algebras.

1. Introduction and Preliminaries

The notions of amenability and congeniality for bases of an infinite dimen-
sional algebra were recently introduced in [1]. Since then, several papers have

Received 08 March 2019 – Accepted 28 November 2019

http://dx.doi.org/10.4995/agt.2020.11488


S. R. López-Permouth and B. Stanley

appeared exploring questions that arise naturally in that context (e.g. [2], [3],
[5].) The purpose of our paper is to give topological insights on the naturality
of those notions and to show some applications of these new perspectives. In
addition, as is the case also in [5], we extend the notions studied in [1] to bases
of any infinite dimensional module over an algebra of arbitrary dimension. We
start with a brief summary of background definitions and results, which are
straightforward adaptations of those in the literature. Unless otherwise stated,
F denotes a field, A denotes an F-algebra and T denotes a (left) A-module
(therefore, itself an F-vector space.)

Given two sets I and J, an I × J matrix over the field F is a function
f : I × J → F . As usual, I indexes rows of f and J indexes its columns. In
this sense, the i-th row of f is f|{i}×J and the j-th column of f is f|I×{j}.

We say that a matrix f is column finite if |{i ∈ I | f(i,j) 6= 0}| < ∞ for
all j ∈ J. Similarly, we say that f is row finite if |{j ∈ J | f(i,j) 6= 0}| < ∞
for all i ∈ I. If we have two matrices, f and g, such that f : I ×J → F and
g : J ×K → F then, when possible, we define the product fg of f and g, to
be the matrix

fg(i,k) =
∑
j∈J

f(i,j)g(j,k).

As infinite sums of non-zero elements are not defined, the product of two ma-
trices is also not necessarily defined. When the product does exist, the result
is a matrix with domain I ×K. Clearly, if either f is row finite or g is column
finite then the product fg exists.

We follow the usual definitions of algebras over fields and of unitary modules
over rings with unity. This paper contains results from the doctoral dissertation
[7].

1.1. Amenability and Congeniality. The F-vector space T is isomorphic to
F (B), the direct sum of copies of F indexed by B. Consequently, F (B) inherits
an A-module structure from T . A second vector space, FB, the collection of
all infinite linear combinations of elements of B with coefficients in F , clearly
contains F (B) ∼= T as a proper subspace. A natural question is for what bases
does FB have a (left) module structure over A that naturally extends that of
T . For the case T = A, the notion of an amenable basis was introduced in [1]
to answer that question. A basis B of T is amenable if for every r ∈A, c ∈B
there exist only finitely many d ∈ B such that (rd)c 6= 0. In other words, the
set rBc = {d ∈B | (rd)c 6= 0} is finite.

It is easy to see that B is amenable if for all r ∈A, the column-finite matrix
[`r]B, representing the F-linear map `r : T →T left multiplication by r given by
`r(a) = ra, is also row-finite. Basically, the row-finiteness requirement allows
us to multiply a vector v with infinite support on the left by [`r]B; each entry of
the product is the inner product of a vector with finite support and v, a finite
sum. In this paper we aim to explain in what sense this module structure is
naturally induced by the knowledge of the products of elements from the basis
B, as suggested in [1].

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 2



Topological characterizations of amenability and congeniality of bases

The expression FB = BT serves two purposes: it indicates that B is amenable
and gives a name to the A-module structure on FB. Given an infinite dimen-
sional module T over an algebra A and an amenable basis B for T , the resulting
module BT is said to be a basic module (or the natural module extension of T
by B.)

The proof of Theorem 2.6 of [1] may easily be adapted to get the following
result, which guarantees that amenable bases exist when the module considered
is countable dimensional over an at most countable dimensional algebra; it
is not know in general whether uncountable dimensional algebras must have
amenable bases.

Theorem 1.1. If dimA is either finite or countable and dimT is countable
then T has an amenable basis.

It is easy to see from [1] that some bases, but not necessarily all, are
amenable. In fact, the following theorem is a direct consequence of a theorem
in [5], which shows that, not only does such a module have a non amenable
basis under the hypothesis, but it actually has a contrarian basis (as defined in
[5].)

Theorem 1.2. If a module T has an amenable basis B with an element b ∈B
that is not an eigenvector for any left multiplication map lr with r ∈ A\ F ,
then T also has a non-amenable basis.

Following [1], given two bases B and C for a module T , we say that B is
congenial to C if [I]CB, the matrix representation of the identity map I : T →T
with respect to the bases cB and C, is row-finite. In general, this relation is not
symmetric; when B is congenial to C and C is congenial to B then we say that
B and C are mutually congenial. If B is congenial to C and C is not congenial
to B, we say that B is properly congenial to C and the matrix [I]CB is a proper
congeniality matrix. In other words, a proper congeniality matrix is a row and
column finite matrix whose inverse is column but not row finite.

It was proven in [1] that, given two mutually congenial bases B and C, one
of them is amenable if and only if the other one is and they induce isomorphic
natural module extensions. On the other hand, if B is properly congenial to
C, the possible amenability of either basis is largely independent from that of
the other one. It is know, however, that if they are both amenable then the
map from BT into CT given by left multiplication by the proper congeniality
matrix [I]CB is a module homomorphism and, perhaps surprisingly, it is onto.
This is one of the main results from [1]. The question about the significance of
the potential injectivity of multiplication by [I]CB was left open.

The final result in this paper, Theorem 3.0.2, shows that a proper conge-
niality matrix [I]CB never induces a one-to-one left multiplication map and con-
sequently draws a line separating mutual and proper congeniality. Our proof
of this fact relies on the topological machinery built throughout the paper.

Proper congeniality has turned out to be one of the most interesting notions
in this line of work; it has inspired the notions of simple and projective bases

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 3



S. R. López-Permouth and B. Stanley

and these types of bases have, in turned, fueled many ongoing projects (see,
for example, [1], [2] and [3].)

The unit vectors in F (B) are a basis B for F (B) which have the interesting
property that they can be, in some sense, a surrogate basis for FB. Indeed, any
element in FB can be viewed as an infinite linear combination of the elements
of B and the only way to represent 0 ∈ FB is using all zero coefficients. So, the
elements of B satisfy a stronger sort of linear independence. This is reminiscent
of the classical Schauder bases where a collection of elements in a topological
vector space play the role of a basis because their infinite linear combinations
converge to the element that they represent. Theorem 1.2.6 basically states
that for any basis B there indeed exists a topology on FB that makes B ⊂ FB
into a Schauder basis.

1.2. Nets and convergence. For convenience, we will briefly introduce some
topological concepts here. However, most necessary topology notions, such as
subspace topologies and discrete topologies (when all subsets are open), may
be found in any standard reference (c.g. [8]).

The product topology on the cartesian product of a family of topological
spaces is the smallest topology making all projection maps continuous. In
other words:

Definition 1.3 (Product topologies and projection maps).

(1) For a collection of topological spaces {(Xi,τi)}i∈I,
∏
i∈I Xi = {f :

I →
⋃
i∈I Xi | f(i) ∈ Xi} is a topological space under the topology

generated by the base B = {U ⊂
∏
i∈I Xi | U =

∏
i∈I Ui where each

Ui ∈ τi and Ui 6= Xi only finitely often}.
(2) The map πj :

∏
i∈I Xi → Xj, given by πj(f) = f(j) ∈ Xj, is called the

j-th projection map.

Not all topologies may be described in terms of the behavior of sequences, a
more powerful notion, called nets, is needed. Likewise, although the sequence
of images under a continuous map of the elements of a convergent sequence
converge to the image of the limit of that sequence, this property alone does
not in general characterize continuous functions. In order to obtain a charac-
terization of continuous functions, nets are once again needed. We will start
by describing the prerequisite term of directed sets.

Definition 1.4 (Directed sets). A directed set is a non-empty set A with a
binary relation (a direction on A) ≤ that satisfies the following:

D.1 a ≤ a for all a ∈ A
D.2 if a ≤ b and b ≤ c then a ≤ c
D.3 if a,b ∈ A then there exists c ∈ A such that a ≤ c and b ≤ c.

Example 1.5. Let X be a topological space and let Bx be a local base (also
called a neighborhood base) at a point x ∈ X. For U,V ∈ Bx define U ≤ V if
and only if V ⊂ U, then ≤ is a direction on Bx.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 4



Topological characterizations of amenability and congeniality of bases

Definition 1.6 (Nets and their convergence).

(1) A net is a function f : A → X where A is a directed set and X is
a topological space. We write {xα}α∈A to denote the net given by
f(α) = xα.

(2) For U ⊂ X we say that a net {xα}α∈A is residually (eventually) in U
if there exists β ∈ A such that α ≥ β =⇒ xα ∈ U.

(3) We say that {xα}α∈A converges to x ∈ X if and only if for all open
neighborhoods U of x, {xα}α∈A is eventually in U.

The significance of nets is highlighted by the following well-known result.

Proposition 1.7. For arbitrary topologies,

(1) a subset U ⊂ X is open if and only if when a net {xα}α∈A converges
to some x ∈ U then {xα}α∈A is residually in U, and

(2) a subset F ⊂ X is closed if and only if every convergent net {xα}α∈A ⊂
F converges to some x ∈ F .

(3) a function f : X → Y is continuous if and only if for every net xα → x
we have that f(xα) → f(x).

Theorem 1.8. When F is equipped with the discrete topology and FB with the
corresponding product topology, then F (B) is dense in FB.

Proof. Let U(b1, . . . ,bn|k1, . . . ,kn) = {f ∈ FB | f(bi) = ki, 1 ≤ i ≤ n}, note
that sets of this form are open in the product topology as {k} is open in
F for all k ∈ F. Furthermore, the set {U(b1, . . . ,bn | g(b1), . . . ,g(bn)) | g ∈
F (B), n ∈ N, b1, . . . ,bn ∈B} is a basis for the product topology on FB. Clearly
U(b1, . . . ,bn | g(b1), . . . ,g(bn)) contains g|{b1,...,bn} ∈ F

(B). Thus, every open

set in FB has non-trivial intersection with F (B). Now, let f ∈ FB and let U
be open around it. We have seen that U intersects F (B) so that f is a cluster
point of F (B) and so, f ∈ cl(F (B)) so that F (B) is dense. �

1.3. Uniform Topological Spaces. The condition of Uniform continuity is
strictly stronger than continuity. One could say that uniform continuity is the
condition of being ‘continuous the same way’ everywhere. The main property of
uniformly continuous functions that we will make use of here is that uniformly
continuous functions preserve Cauchy sequences. That is, if {xn}n∈N is Cauchy
and f : X → Y is uniformly continuous then {f(xn)}n∈N is Cauchy in Y .

We start by recalling the familiar definition, in the context of metric spaces,
of a uniformly continuous function.

Definition 1.9. Let X,Y be metric spaces, whose metrics are dX,dY respec-
tively. We say a function f : X → Y is uniformly continuous if for all � > 0 there
exists δ > 0 such that when dX(x,x

′) < δ we have that dY (f(x),f(x
′)) < �.

In this paper we deal with potentially uncountable products, which may fail
to be metrizable. Thus, we introduce Uniform Spaces, which will allow us to
talk about uniformly continuous functions in the absence of a metric. We first
develop some notation; the reader may find a more thorough treatment in [8].

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 5



S. R. López-Permouth and B. Stanley

Definition 1.10. If X is a set, we denote by ∆ the diagonal {(x,x) | x ∈ X}
in X × X. If U,V ⊂ X × X, then U ◦ V is the set {(x,y) | for some z ∈
X, (x,z) ∈ V and (z,y) ∈ U}. Notice that U and V are just relations on X
and ◦ is a natural extension of the notion of composition of functions.

The definition we will use comes from the notion that x and y are close
together in a metric space if and only if (x,y) is close to the diagonal in X×X.
Definition 1.11. A diagonal uniformity on a set X is a collection D(X), or
just D, of subsets of X ×X, called entourages, which satisfy:

(D.a) D ∈D =⇒ ∆ ⊂ D
(D.b) D1,D2 ∈D =⇒ D1 ∩D2 ∈D
(D.c) D ∈D =⇒ E ◦E ⊂ D for some E ∈D
(D.d) D ∈D =⇒ E−1 ⊂ D for some E ∈D where E−1 = {(y,x) | (x,y) ∈

E}
(D.e) D ∈D,D ⊂ E =⇒ E ∈D

Proposition 1.12. If D is a diagonal uniformity and D ∈D then D−1 ∈D.
Proof. Let D ∈D then, by (D.d) we have that there is some E ∈D such that
E−1 ⊂ D. Therefore, E ⊂ D−1 so by (D.e) we have that D−1 ∈D. �

Each uniformity gives rise to a topology. To see this, we define neighbor-
hoods of points x ∈ X for a uniform space.
Definition 1.13. For x ∈ X and D ∈ D let D[x] = {y ∈ X | (x,y) ∈ D}.
This is extended to subsets A ⊂ X as follows: D[A] =

⋃
x∈A D[x].

Definition 1.14. A collection E ⊂ D is a base for the uniformity on D if for
all D ∈D there is some E ∈E such that E ⊂ D. That is, D can be recovered
from E by applying (D.e).

That these sets, D[x], are neighborhoods is justified by the following theorem
which can be found [8].

Theorem 1.15. For each x ∈ X, the collection Ux = {D[x] | D ∈D} forms a
neighborhood base at x, making X a topological space.

The way that X is made a topological space is as follows. We say that O is
D-open if for all x ∈ O there is some U ∈ D such that U[x] ⊂ O. That is, O
is open if and only if is a neighborhood of all its points. This topology will be
called the uniform topology on X generated by D.
Example 1.16. Let D� = {(x,y) | d(x,y) < �}⊂ X ×X where X is a metric
space with metric d : X ×X → R. The set D = {D� | � > 0} is a uniformity
on X. In fact, the uniform topology generated by D coincides with the metric
topology on X.

Example 1.17. Let X be a set and let D = {D ⊂ X × X | ∆ ⊂ D}. The
uniform topology generated by D is the discrete topology on X. Indeed, let
A ⊂ X then ∆[x] = {x} ⊂ A for all x ∈ A. This uniformity is called the
discrete uniformity on X.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 6



Topological characterizations of amenability and congeniality of bases

Now we generalize the notion of a uniformly continuous function to all uni-
form spaces.

Definition 1.18. Let X and Y be sets with diagonal uniformities D and E
respectively. A function f : X → Y is uniformly continuous if and only if for
each E ∈E, there is some D ∈D such that (x,y) ∈ D =⇒ (f(x),f(y)) ∈ E.

As we primarily deal with product spaces in this document it will be useful to
have a way to define a uniformity on a product of uniform spaces. In particular,
we will do so in such a way that the uniform topology on the product coincides
with the product of the uniform topologies.

Definition 1.19. If Xα is a set for each α ∈ A and X =
∏
Xα, the αth bipro-

jection is the map Pα : X×X → Xα×Xα defined by Pα(x,y) = (πα(x),πα(y))
where πα is the αth projection. That is, πα(x) = x(α) ∈ Xα for each x ∈ X.

Theorem 1.20. If Dα is a diagonal uniformity on Xα, for each α ∈ A, then
the sets P−1α1 (Dα1 )∩·· ·∩P

−1
αn

(Dαn ), where Dαn ∈Dαn for i = 1, . . . ,n, form a
base for a uniformity D on X =

∏
Xα whose associated topology is the product

topology on X.

1.4. Complete Uniform Spaces. Let us deal next with completeness for
uniform spaces; a notion which, when dealing with metric spaces, relies on
Cauchy sequences. Consequently, we we will lean on the more general concept
of Cauchy nets.

Definition 1.21. Let X be a uniform space with diagonal uniformity D and
let {xλ}λ∈Λ ⊂ X be a net. We say that {xλ}λ∈Λ is a Cauchy Net if for all
D ∈D there exists λD ∈ Λ such that γ,δ > λD =⇒ (xγ,xδ) ∈ D.

Much like the usual case with uniformly continuous functions and Cauchy
sequences we have that uniformly continuous functions preserve Cauchy Nets.
That is, if f : X → Y where X,Y are uniform spaces and {xλ} ⊂ X is a
Cauchy net then {f(xλ)}⊂ Y is a Cauchy net.

Definition 1.22. A uniform space X is complete if every Cauchy net converges
to some element of X.

Theorem 1.23. If {xλ}λ∈Λ ⊂ X is a Cauchy net and f : X → Y is uniformly
continuous then {f(xλ)}λ∈Λ is also a Cauchy net.

Proof. By definition, for each entourage V in X we have that there is some
λV ∈ Λ such that λ,λ′ > λV implies that (xλ,xλ′) ∈ V .
Let V be an entourage in Y . As f is uniformly continuous there is some
entourage U in X such that (x,y) ∈ U implies that (f(x),f(y)) ∈ V . Let
{xλ}λ∈Λ be a Cauchy net. Let λX,U be such that λ,λ′ ≥ λX,U ⇒ (xλ,xλ′) ∈ U.
Then, (f(xλ),f(xλ′)) ∈ V so that {f(xλ)}λ∈Λ is a Cauchy net. �

Uniformly continuous functions between complete uniform spaces can be
continuously extended to the whole space; furthermore, that extension is also
uniformly continuous.

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S. R. López-Permouth and B. Stanley

Theorem 1.24. If X,Y are complete uniform spaces and A ⊂ X is dense
then every f : A → Y that is uniformly continuous has a unique uniformly
continuous extension f̂ : X → Y .

The above result, an easy consequence of Theorem 8.3.10 in [4], will help
us determine when natural module extensions exist.

1.5. Topological Vector Spaces. It turns out that F (B) and FB are topo-
logical vector spaces so we write down some useful properties of these spaces.

Definition 1.25. A topological F−vector space E is an F-vector space E
that is endowed with a topology such that the mappings (x,y) → x + y and
(λ,x) → λx from E × E to E and F × E to E respectively are continuous.
(E ×E and F ×E having the product topologies.)

Example 1.26. Let F be a field and endow F with the discrete topology.
Then, for non-empty A, X =

∏
α∈A F is a topological vector space with the

product topology.

Proof. We proceed by showing that addition and scalar multiplication are con-
tinuous. Let + : X ×X → X be such that +(x,y) = x + y. We aim to show
that this map is continuous by showing that πj◦+ : X×X → Fj is continuous
for each j ∈ A. Note, of course, that each Fj is just a copy of F . Note that
πj ◦ +(x,y) = πj(x + y) = x(j) + y(j). As singletons make up a base for the
topology on F we look at the inverse image of a singleton under this map.
(πj ◦ +)−1(z) = ∪x∈Fπ−1j (x) × π

−1
j (z − x). As π

−1
j (x) × π

−1
j (z − x) is open

in X ×X for each x ∈ F we have that (πj ◦ +)−1(z) is the arbitrary union of
open sets and so is open itself. Thus πj ◦ + is continuous for each j ∈ A so
that + : X ×X → X is continuous.

Now we show that · : F × X → X defined by ·(λ,f) = λf is continuous.
Let j ∈ A and note that πj ◦ ·(λ,f) = λf(j). Again, we examine the inverse
image of singletons, (πj ◦ ·)−1(z) = ∪λ∈F{λ}×π−1j

(z
λ

)
each of which is open

in F ×X so that the inverse image is an arbitrary union of open sets. Thus, ·
is continuous as well. �

As we saw in definition 1.19 we have that the space defined in example 1.26 is
indeed a uniform space. With that in mind, we show that, for linear operators,
continuity and uniform continuity are the same. To do this, we rely on some
more elementary facts.

Proposition 1.27. Let U be all the basic open sets around 0 as defined by the
product topology. Let DU = {(x,y) | x−y ∈ U} and D = {DU | U ∈U}. Then,
D is a base for the product uniformity, E, on X =

∏
α∈A F . That is D ⊂ E

and for every entourage E ∈E there is some entourage D ∈D with D ⊂ E.

Proof. Let E be a base entourage from E, that is, E = P−1α1 (Eα1 ) ∩ ·· · ∩
P−1αn (Eαn ), where Eαn ∈Eαn for i = 1, . . . ,n where Eα is the discrete uniformity
on Fα. Take U = {x ∈ X | x(αi) = 0 for all i = 1, . . . ,n}. Then, DU =

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 8



Topological characterizations of amenability and congeniality of bases

{(x,y) | x(αi) = y(αi) for i = 1, . . . ,n} thus DU ⊂ E as (x(α),y(α)) ∈ Eα for
all Eα ∈Eα. Finally, note that DU = P−1α1 (∆α1 ) ∩·· ·∩P

−1
αn

(∆αn ). �

Proposition 1.28. Let X be as in example 1.26 and let f : X → X. Then, f
is uniformly continuous if and only if for all open sets U around 0 there exists
an open set V around 0 such that (x,y) ∈ DV =⇒ (f(x),f(y)) ∈ DU where
DU = {(x,y) | x−y ∈ U}.

Proof. This follows from the definition of uniformly continuous and the fact
that sets of the form DU give a base for the uniformity on X. �

Corollary 1.29. Let X be as in example 1.26 and let f : X → X. Then, f
is uniformly continuous if and only if for all open sets U around 0 there exists
an open set V around 0 such that x−y ∈ V =⇒ f(x) −f(y) ∈ U.

Proposition 1.30. Let X be as in example 1.26 and let f : X → X be a linear
map. Then, f is continuous on all of X if f is continuous at 0.

Proof. Let x ∈ X and let U be open around f(x). Let U′ = U − f(x) and
note that U′ is open around 0. By continuity at 0 there exists V ′ ⊂ X around
0 such that f(V ′) ⊂ U′. Let V = V ′ + x. V is open around x and f(V ) =
f(V ′) + f(x) ⊂ U′ + f(x) = U. Thus, for linear maps, continuity at 0 is
sufficient to show continuity in general. �

Theorem 1.31. Let X be as in example 1.26 and let f : X → X be a linear
map. Then, f is continuous if and only if f is uniformly continuous.

Proof. That uniform continuity implies continuity is trivial. Let U ⊂ X be an
open neighborhood of 0. Let V be such that f(V ) ⊂ U which is guaranteed
by continuity. Now, suppose x − y ∈ V so that f(x − y) ∈ U. However, by
linearity of f, f(x−y) = f(x) −f(y) so that x−y ∈ V =⇒ f(x) −f(y) ∈ U
so that f is uniformly continuous. �

While some of the above results hold for topological vector spaces in general,
we opted for leaving our discussion self-contained and focused only on the
specific case considered here.

2. Topological characterization of amenable bases and the
natural module extensions they induce.

It would certainly be nice if, in general, we could define rt for r ∈ A and
t ∈ FB to be a limit as can be done when the power series are viewed as a
module over polynomials with the standard basis. However, if T has a higher
dimension than ω this will not work. In particular, elements of FB with un-
countable support can not even be found as a limit of elements of T when B
is uncountable. That is, the limit of any sequence of finite linear combina-
tions can be at most a countable linear combination, or in the language of our
function spaces, a function f : B → F with at most countable support.

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S. R. López-Permouth and B. Stanley

2.1. Module Multiples Defined By Approximations. For r ∈ A, `r :
T → T denotes the map given by `r(t) = rt, γB : F (B) → T denotes the
bijection such that γB(f) =

∑
b∈B f(b)b, and ft = [t]B = γ

−1
B (t) ∈ F

B for t ∈T
(the coordinates of t with respect to B.) We think of [t]B as a matrix with one
column.

Also, for r ∈A, the [`r]B represent the map `r with respect to B in the usual
way, namely, the rows and columns are labeled by B and, for b ∈ B, the b-th
column of [`r]B is given by the equation [`r]B|B×{b} = rfb := [rb]B. With this
notation, one gets, as is customary in the theory of matrix representations of
linear transformations, the identity granting that, for t ∈ T and r ∈ A, the
product [`r]B[t]B is equal to [rt]B = γ

−1
B (rt) = rγ

−1
B (t) = rft.

In other words, multiplication on the left of f ∈ F (B) by [`r]B is a matrix
representation of left multiplication by r with respect to the basis B.

Consequently, for t ∈T the product [`r]B[t]B is equal to (γ−1B ◦ `r ◦γB)(ft).
We may then think of left multiplication by [`r]B as a function from F

(B)

to F (B). As [`r]B already has a function description we avoid using parenthesis
when we mean the matrix product and we will write [`r]B· : F (B) → F (B).
Clearly, this means that [`r]B· = γ−1B ◦ `r ◦γB

Handling the uncountable dimension situation requires a little finesse beyond
the standard approach with sequences. If f ∈ FB has uncountable support,
there is no sequence in F (B) that converges to f. We must therefore rely on
nets. Indeed, recall that a local base at f ∈ FB, Bf is given by sets of the
form U(b1, . . . ,bn | f(b1), . . . ,f(bn)) and that Bf is a directed set under the
direction in Example 1.5. That is, U ≤ V if and only if V ⊆ U.

Now we construct a net whose directed set is (Bf,≤). For U = U(b1, . . . ,bn
| f(b1), . . . ,f(bn)) ∈ Bf let fU = f|{b1,...,bn} ∪ 0|B\{b1,...,bn}.

Definition 2.1. The net constructed above will be called the canonical net
for f.

Claim 2.2. The canonical net for f, {fU | U ∈ Bf}, converges to f.

Proof. Let U ∈ Bf be a basic open neighborhood of f. Let V ∈ Bf be such
that U ≤ V so that V ⊂ U. Therefore, U = U(b1, . . . ,bn | f(b1), . . . ,f(bn))
and V = U(b1, . . . ,bn,bn+1, . . . ,bn+k | f(b1), . . . ,f(bn),f(bn+1), . . . ,f(bn+k)).
Thus, fV |{b1,...,bn} ≡ fU|{b1,...,bn} so that fV ∈ U. That is, the net is eventually
in U for any basic neighborhood of f. �

These canonical nets are integral in our description of naturality. Note, the
canonical net for f consists of better and better approximations of f. Therefore,
if we can define scalar multiplication naturally, we would like [`r]BfU , U ∈ Bf
to consist of better and better approximations to [`r]Bf.

Definition 2.3. Let r ∈A and f ∈ FB. We say that [Lr]B· : FB → FB is the
natural extension of [`r]· if {[`r]BfU}→ [Lr]Bf for all f ∈ FB where {fU}U∈Bf
is the canonical net described above.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 10



Topological characterizations of amenability and congeniality of bases

Here we must justify the use of the word ‘the’ in our above definition. That
is, we must verify that this extension is indeed unique. This is handled by
a well-known topological fact. The arbitrary product of Hausdorff spaces is
again a Hausdorff space. As limits are unique in Hausdorff spaces we know
that {[`r]BfU}U∈Bf can converge to at most one element in F

B, thus, if one
natural extension exists, it is the only one.

With it in mind that [`r]Bf = rf for r ∈ A and f ∈ F (B) we see that F (B)
is a module and {[`r]B · | r ∈A} is a collection of matrices that describe scalar
multiplication in F (B). In fact, the definition of γB shows us that this module
coincides with T .

Definition 2.4. We say that AF
B is the natural extension of the left A-

module structure on F (B) if {[Lr]B · | r ∈ A} is a set of maps that describe
scalar multiplication by r and each [Lr]B· is the natural extension of [`r]B·.

Note, if any [`r]B· fails to have a natural extension then AFB can not be a
natural extension of F (B) with the aforementioned module structure. Note, for
any f ∈ FB the scaled canonical net, {rfU}U∈Bf is a net in F

(B). Whether it
converges for every f is the determining factor for scaling by r being a valid
operation on FB. To help us understand when this happens, we introduce the
notion of a uniformity on our topoligical spaces as well as what it means for
a net to be Cauchy. Much like in the case with Cauchy sequences in metric
spaces, a Cauchy net is one such that if we desire two elements of the net to
be close to each other, we need only traverse far enough into the net. The
notion of closeness will be handled by the uniformity as we lack a metric in
general. Under this new structure we will utilize the notion of a uniformly
continuous function. In particular, uniformly continuous functions map Cauchy
nets to Cauchy nets. Therefore, if multiplication by r is found to be uniformly
continuous, then we shall be able to define rf as the limU∈Bf rfU as this limit
will therefore exist. The details are the subject of the following subsection.

2.2. Uniform Continuity and Amenability. We will see that if [`r]B is a
row finite matrix, then it is uniformly continuous and, therefore, has a uniformly
continuous extension as guaranteed by theorem 1.24.

Theorem 2.5. If [`r]B· : F (B) → F (B) is row finite then it is uniformly con-
tinuous.

Proof. As [`r]B is row finite we know that for each b ∈ B there are only
finitely many b′ ∈ B such that [`r]B(b,b′) 6= 0. As it is F-linear, we show
that it is continuous at 0 to establish that it is uniformly continuous. Let
U(b1, . . . ,bn | 0, . . . , 0) be an open set around 0. That is, U = {f ∈ FB | f(bi) =
0, i = 1, . . . ,n}. Let BU = {b ∈B | [`r]B(bi,b) 6= 0 for any i = 1, . . . ,n}. Note,
BU is finite as each row of [`r]B is finite. Let V = U(BU | 0) = {f ∈ FB |f(b) = 0
for all b ∈BU}.

Now suppose that f1,f2 ∈ F (B) such that f := f1 − f2 ∈ V. Then, for
i ∈ {1, . . . ,n}, [`r]Bf(bi) =

∑
b′∈B[`r]B(bi,b

′)f(b′), yet, [`r]B(bi,b
′) = 0 for all

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 11



S. R. López-Permouth and B. Stanley

b′ ∈B\BU so that [`r]Bf(bi) =
∑
b′∈BU [`r]B(bi,b

′)f(b′) = 0 as f(b′) = 0 for all

b′ ∈BU so that [`r]Bf ∈U. Thus, [`r]B· is uniformly continuous. �

Theorem 2.6. If [`r]B is row finite then, [Lr]B· : FB → FB defined by

[Lr]Bf(b) =
∑
b′∈B

[`r]B(b,b
′)f(b′)

is the uniformly continuous extension of [`r]B·.

Proof. As [`r]B is row finite we know that for each b ∈B there are only finitely
many b′ ∈ B such that [`r]B(b,b′) 6= 0 so that [Lr]Bf(b) is defined. Therefore,
[Lr]B is indeed a function from F

B to FB. The proof from here is identical to
the above proof, except that f1,f2 are taken from F

B. �

Therefore, we have established that if B is amenable, then [`r]B is uni-
formly continuous, that it has a uniformly continuous extension, and that
this extension is the natural extension of [`r]B. Therefore, with the collec-
tion {[Lr]B · | r ∈ A} FB has a left A-module structure. Furthermore, this
structure is the natural extension of the A-module structure on F (B).

Lemma 2.7. Any injective net {bλ}λ∈Λ converges to 0.

Proof. Let U be open around 0 and let b1, . . . ,bn ∈B be such that if f(bi) = 0
for all 1 ≤ i ≤ n then f ∈ U. Choose λ ∈ Λ such that γ > λ implies that
bγ 6∈ {b1, . . . ,bn}. Then, γ > λ implies bγ ∈ U. �

Theorem 2.8. If [`r]B· : F (B) → F (B) is continuous, then [`r]B is row finite.

Proof. Suppose that [`r]B fails to be row finite so that there is some infinite
row. That is, there is some b ∈B such that [`r]B(b,b′) 6= 0 for infinitely many
b′. Let f : N →B be an injective map such that for all b′ ∈ f(N) we have that
[`r]B(b,b

′) 6= 0. Now, f describes an injective net so that lemma 2.7 applies.
By continuity, we see that [`r]Bf(n) → 0. Therefore, there is some N ∈ N such
that for n > N we have that [[`r]Bf(n)](b) = 0. This is a contradiction as
the b-th coordinate of [`r]Bf(n) is non-zero for all n ∈ N. Therefore we must
conclude that [`r]B is indeed row finite. �

Corollary 2.9. The basis B is amenable if and only if [`r]B : F (B) → F (B) is
uniformly continuous for all r ∈A.

Theorem 2.10. The left A-module F (B) has a natural extension if and only
if B is amenable.

3. Characterizations of Mutual and Proper Congeniality

It is known that if B is mutually congenial to C then BT and CT are iso-
morphic. In fact, they are ‘naturally ’isomorphic as the congeniality map is an
isomorphism between them. Furthermore, for T of countable dimension, it is
known that congeniality maps are onto. Here we show that if a congeniality
map is one-to-one as well as being onto and, as always, continuous, then it is,

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 12



Topological characterizations of amenability and congeniality of bases

in fact, a mutual congeniality map. We do so by establishing continuity of the
inverse map.

Theorem 3.1. Let C and B be countable and let g : C ×B → F be a row
and column finite matrix such that g· maps FB bijectively onto FC, then g−1
is continuous.

The proof of this lemma uses a lot of the same tools that are used in the
proof that congeniality maps are onto, which is provided in [1].

Proof. Enumerate B = {bi}i∈Z+ . Let V be the row-span of g. For n ∈ Z+,
define Vn = {v ∈ V : v(bj) = 0 if j > n}; one could informally write ‘Vn =
F{b1,...,bn} ∩ V ’. Then V1 ⊆ V2 ⊆ ··· ⊆ ∪∞n=1Vi = V . Also, for any infinite
matrix h and for n ∈ N, h>n or h≥n+1 denote the infinite matrix made up of
all rows of h but beginning with the (n + 1)-th.

Note that Vn is a subspace of V and dim Vn ≤ n. Recursively, choose, for all
n ∈ Z+, Cn, a basis for Vn, in such a way that Cn ⊂Cn+1. Then C =

⋃
n∈Z+ Cn

is a basis for V . Let f be a matrix such that fg = h is a matrix having, as
rows, the elements of C in such a way that the elements of Ci appear no later
than those of Cj when i < j. Clearly, f is a row-finite invertible matrix with
an inverse f−1 that is also row-finite.

Let {nk | k ≥ 1} and be strictly increasing sequences of positive integers and
let {mk | k ≥ 1} be a sequence of positive integers so that the rows of h may
be viewed as a sequence of mk × nk rectangular matrices Ck with mk ≤ nk.
Let `k =

∑k
j=1 mj. in the following manner:

Now we have

fg = h =


 C1 0 0C2 0

. . .


 ,

and h· : FB → FC → FE where E = {ei}i∈Z+ and ei = γC
(
f−1C×{ei}

)
. We begin

by examining the image of sets of the form U
p
j := U(bj | p) = {s ∈ F

B | s(bj) =
p} for some bj ∈ B and p ∈ F . Let bj ∈ B and p ∈ F be arbitrary. Let
k = min{k ∈ Z+ | j ≤ nk}. Let FBnp = {t ∈ F

{b1,...bnk} | t(bj) = p} and
for t ∈ FBnp let U(t) = {s ∈ FB | s(bj) = t(bj) for all 1 ≤ j ≤ nk} and let
t̂ = t ∪ 0|{bj | j>nk} ∈ F

B. Note that hs(bi) = ht̂(bi) for 1 ≤ i ≤ `k for all
s ∈ U(t). Thus, hU(t) ⊂ Û(t) = {v ∈ FE | v(ej) = ht̂(ej) for all 1 ≤ j ≤ `k}.
Now, for v ∈ Û(t) we show that hx = v has a solution x ∈ U(t).

The existence of a solution for the system can be obtained as the limit of a
convergent sequence. We build the sequence as follows. Let dk = v|{ej | 1 ≤ j ≤ `k}
and for every y > k, dy = v|{ej | `y−1+1 ≤ j ≤ `y} and, for all y ≥ k, Dy =
v|{ej | 1 ≤ j ≤ `y}. Clearly, t is such that such that hkt = dk where

hy−1 =




C1 0 0
C2 0

. . .
Cy−1




c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 13



S. R. López-Permouth and B. Stanley

for y > k. For y > k + 1 Suppose a y − 1-th approximation Xy−1 has been
obtained; in other words, hy−1Xy−1 = Dy−1. We construct Xy, but first we
partition Cy as

(
Cy,1 | Cy,2

)
where Cy,1 has ny−1 columns. We note that

the rows of Cy,2 are linearly independent. To see this we assume that they
aren’t linearly independent and that a1r

′
1 + . . .amyr

′
my

= 0 where r′i is a row
of Cy,2 and not all ai are zero. Now consider the same linear combination,
except of the rows of Cy. By the linear independence of the rows of Cy we
know a1r1 + amyrmy 6= 0, however, its non-zero entries must exist in the first
ny−1 entries. Thus, a1r1 + amyrmy ∈ Vy−1 = span{

⋃y−1
j=1 Cj} so that Cy is a

linearly dependent collection. This is a contradiction, so the rows of Cy,2 are
indeed linearly independent.

Now we obtain a solution xy to the equation Cy,2xy = Dy−1−Cy,1Xy−1 and
setting Xy = Xy−1 ∪xy. The existence of xy can be assured because my, the
number of rows of Cy,2, is less than or equal to ny −ny−1, the number of its
columns, due to linear independence. We note that Xy is a vector of length ny,

let Sy = Xy∪0|{bj | j>ny} ∈ F
(B). Now, clearly {Sy}∞y=1 is a Cauchy sequence.

That is, if we want Sn − Sm to have its first non-zero entry after ` then we
choose j such that nj > ` and let n,m > nj. Let S = limy→∞Sk which is
guaranteed to exist by completeness.

Furthermore, (hSy)(ej) = (hyXy)(ej) = Dy(ej) for 1 ≤ j ≤ `y and hSy →
hS by the continuity of h. Let D̂y = Dy ∪ 0|{ej | j>`y} and note that D̂y is
a Cauchy sequence and clearly D̂y → ∪Dy = v. Additionally, hSy − Dy → 0
as (hSy)(ej) = (hyXy)(ej) = Dy(ej) for 1 ≤ j ≤ `y. Therefore, if we want
hSy(ez) − Dy(ez) = 0 we need only choose y > Y where 1 ≤ z ≤ `Y . Thus,
hSy → v so v = hS. Thus, Û(t) ⊂ hU(t) so that Û(t) = hU(t) which is open
in the product topology on FE.

Then, as U
p
j = ∪t∈FBnp U(t) we have that hUj = h

(
∪
t∈FBnp

U(t)
)

= ∪
t∈FBnp

hU(t)

so that hUj is open in F
E. Now, let U = U(bj1, . . . ,bjz | p1, . . . ,pz) be a basic

open set in FB. Then, U =
⋂z

1=1 U
pi
ji

so that hU = h(
⋂z

1=1 U
pi
ji

) ⊂
⋂z

1=1 hU
pi
ji

.
However, as g· and f· are bijections we know that fg· = h· is a bijection as
well so that hU = h(

⋂z
1=1 U

pi
ji

) =
⋂z

1=1 hU
pi
ji

so that hU is open in FE.

As f is row finite we know that f· : FC → FE is continuous. Therefore,
f−1hU is open in FC. However, f−1hU = gU so that g· maps open sets to
open sets. Therefore, g· is an open map. Now, by bijectivity we know that
(g·)−1 : FC → FB is well-defined. Furthermore, [(g·)−1]−1U is open for all
open U. Thus, (g·)−1 is continuous and therefore is uniformly continuous by
virtue of linearity. We also have that (g·)−1|F(C) = g−1 · |F(C) so that g−1· is
continuous and g−1· = (g·)−1. �

Theorem 3.2. Let B and C be amenable. Then [I]CB· bijectively maps
BT onto

CT if and only if B is mutually congenial to C.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 14



Topological characterizations of amenability and congeniality of bases

Proof. Let B and C be amenable such that [I]CB· bijectively maps
BT onto CT .

Then, [I]CB satisfies the conditions of theorem 3.1 so that ([I]
C
B·)
−1 = [I]BC · is

continuous.
The other direction is trivial. �

References

[1] L. M. Al-Essa, S. R. López-Permouth and N. M. Muthana, Modules over infinite-

dimensional algebras, Linear and Multilinear Algebra 66 (2018), 488–496.
[2] P. Aydŏgdu, S. R. López-Permouth and R. Muhammad, Infinite-dimensional algebras

without simple bases, Linear and Multilinear Algebra, to appear.

[3] J. Dı́az Boils, S. R. López-Permouth and R. Muhammad, Amenable and simple bases
of tensor products of infinite dimensional algebras, preprint.

[4] R. Engelking, General Topology, Sigma Series in Pure Mathematics, vol. 6 (1989).
[5] S. R. López-Permouth and B. Stanley, On the amenability profile of an infinite dimen-

sional module over an algebra, preprint.

[6] P. Nielsen, Row and column finite matrices, Proc. Amer. Math. Soc. 135, no. 9 (2007),
2689–2697.

[7] B. Stanley, Perspectives on amenability and congeniality of bases, Ph. Dissertation, Ohio

University, February 2019.
[8] S. Willard, General Topology, Dover Publications (1970).

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 15