# Onα-Cogenerated Commutative Unital C*-Algebras

Authors: Ehsan Momtahan
Journal: Algebra
 (2013)

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## Abstract

Gelfand-Naimark's theorem states that every commutativeC*-algebra is isomorphic to a complex valued algebra of continuous functions over a suitable compact space. We observe that for a completely regular space X, βX is dense-α-separable if and only if C(X) is α-cogenerated if and only if every family of maximal ideals of C(X) with zero intersection has a subfamily with cardinal number less than α and zero intersection. This gives a simple characterization of α-cogenerated commutative unital C*-algebras via their maximal ideals.


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## Body

## 1. Introduction

In this paper, byR we always mean a commutative ring with identity. Let F denote the reals or the complexes. For a completely regular (topological space) X, let C(X) stand for the F-algebra of continuous maps X→F. The reader is referred to [1] for undefined terms and notations. By βX, we mean the Stone-Čech compactification of X. We denote the ring of all bounded continuous functions by C*(X). It is well known that for every completely regular space X, we have C*(X)≅C(βX) (see [1, 7.1]). This note is a continuation of [2], in which we showed that for a compact space X, the following are equivalent: X is dense-separable if and only if C(X) is ℵ1-cogenerated if and only if C(X) is separable. Here, we will drop the compactness condition of the space X and improve our main result in [2]. Furthermore we generalize our results to any regular cardinal α.Letα be a regular cardinal. A set A is said to be an α-set if |A|<α. Following Motamedi in [3], we call a ring Rα-cogenerated if for any set {Ai∣i∈I} of ideals of R with ⋂i∈I‍Ai=(0) there exists an α-subset I0 of I such that ⋂i∈I0‍Ai=(0) and α is the least regular cardinal with this property. Any left or right Artinian ring is ℵ0-cogenerated. Any ring with countably many distinct ideals is α-cogenerated, where α is one of ℵ0 or ℵ1. In [2], it has been observed that C[0,1], C(𝕂), where 𝕂 is the Cantor perfect set and C(βℚ) are ℵ1-cogenerated. We call a ring Rα-separable if it has the following property: if {Mi}i∈I is a family of maximal ideals with ⋂i∈I‍Mi=(0), then there exists an α-subset I0 of I such that ⋂i∈I0‍Mi=(0). In this note ℵ1-separable rings are also called separable. Every α-cogenerated ring is α-separable. However, the converse is not true. In [2], we give an example of a separable ring which is not ℵ1-cogenerated.The density of a spaceX is defined as the smallest cardinal number of the form |A|, where A is a dense subset of X; this cardinal number is denoted by d(X) (see [4]). A space X is called dense-α-separable if every dense subset A of X has a dense-α-subset B, which implies that d(A) and hence d(X) are less than α. Dense-separable (or in our terminologies dense-ℵ1-separable) spaces are of great interest. Dense-separable spaces were introduced and studied by Levy and McDowell in [5]. It is evident that every dense-separable space is separable and every second countable space is dense-separable. It is well known that ℝl the Sorgenfrey line satisfies all the countability axioms but the second (see [6], page 195, example 3). Since every dense subset of ℝ is also dense in ℝl, the Sorgenfrey line is dense-separable. In [5], it has been shown that βℚ and βℚ∖ℚ are dense-separable.


## 2. Dense-α-Separable Spaces

Since it is easy to observe thatC(X) is ℵ0-cogenerated if and only if X is a finite space, and in this case C(X) is a finite direct product of F, we may suppose that in our discussion X is an infinite space. Before proving our results, we need two easy (but useful) lemmas. Comparing these two lemmas with [2, Lemmas 1 and 2], one may observe that these two will improve and generalize [2, Lemmas 1 and 2]. In fact they give us enough space to use the full power of Mcknight's Theorem.Lemma 1.
LetX be a completely regular space; then the following hold.(1)
SupposeY is a subset of X; then Y is a dense subset of X if and only if f∈C(X); f|Y=0 implies that f≡0.(2)
SupposeY is a subset of X; then Y is a dense subset of X if and only if f∈C(X); Y⊂intxZ(f) implies that f≡0.Proof.
Part 1.(⇐): let x∈X and Uxan open set containing x. We must show that Ux∩Y≠∅, and suppose on the contrary that Ux∩Y=∅; then Y⊆X∖Ux; by complete regularity of X, there exists a function f∈C(X) such that f(x)=1 and f(Y)=0, and this is a contradiction to our hypothesis.
(⇒): since ℝ is a T1-space and {0} is closed in ℝ, hence f-1{0} is closed in X. Since Y⊆f-1{0}, we conclude that X=Y¯⊆f-1{0}. Hence f≡0.
Part 2. It is enough to show the necessary part: suppose thatY¯≠X; hence there exists z∈X∖Y¯ and a function f:X→[0,1] such that f(z)=1 and f(Y)={0}. In as much as Y and z are contained in disjoint zero sets, there exists a function g such that Y¯⊆intZ(g) and g(z)≠0 (see [1, 1.15]). This is a contradiction.Lemma 2.
ForA⊆βX, OA=(0) if and only if MA=(0).Proof.
Suppose thatOA=(0); then by the previous lemma A¯=βX. Now suppose that f∈MA. There is a positive unit u in C(X) such that f-=uf and -1≤f-≤1 (see [1, 1E.1]). Since A⊆clβXZ(f)⊆Z(f-β) and A¯=βX, we have f-β=0. Hence f-≡0 and this implies that f≡0.In the next theorem, which is the main result of this note, we have generalized [2, Theorem 3] by removing the compactness hypothesis and also replacing ℵ0 with an arbitrary (regular) cardinal. Since βX, by its very definition, is compact and βX=X whenever X is compact, the earlier form of our result is just a special case of the new one. On the other hand since βX always exists, we can (always, i.e., for an arbitrary completely regular space X) judge when the ring C(X) is α-cogenerating and also α-separating by looking at βX.Remark 3.
Before stating our main theorem, we need some useful facts. LetX be a completely space and A⊆βX. Suppose that OA={f∈C(X)∣A⊆intβXclβXZ(f)} and MA={f∈C(X)∣A⊆clβXZ(f)}. Lemma 1 (Lemma 2, resp.) shows if MA=(0) (OA=(0), resp.), then A is dense in X and vice versa. Dietrich Jr. in [7] has shown that for every ideal I of C(X), there exists A⊆βX such that OA⊆I⊆MA. By Mcknight Theorem [7, Theorem 1.3], the set A is ⋂f∈I‍clβXZ(f). Dietrich Jr. has also proved that [7, Lemma 1.6] ⋂‍Z[OA]=⋂‍Z[MA]=A and if OA⊆MB, then B¯⊆A¯.Theorem 4.
LetX be an infinite completely regular space. The following are equivalent: (1)
βX is dense-α-separable;(2)
C(X) is α-cogenerated;(3)
C(X) is α-separable.Proof.
(1)⇒(2): let ⋂j∈J‍Ij=(0). For each j∈J, there exists Aj⊆βX such that OAj⊆Ij⊆MAj. By the previous observations from [7] we have
(1)O∪Aj=⋂j∈J‍OAj⊆⋂j∈J‍Ij⊆⋂j∈J‍MAj=M∪Aj,
but ⋂j∈J‍Ij=(0); therefore O∪j∈JAj=⋂j∈J‍OAj=(0), and hence by Lemma 1, ⋃j∈J‍Aj is dense in βX. Since βX is a dense-α-separable space, there exists an α-subset B of ⋃j∈J‍Aj, such that B¯=X. Hence there exists an α-set J0⊂J such that ⋃j∈J0‍Aj is dense in βX; that is, O⋃j∈J0‍Aj=(0). Now by Lemma 2, ⋂j∈J0‍MAj=(0), and this latter observation in its turn shows that ⋂j∈J0‍Ij=(0).
(2)⇒(3): it is evident.
(3)⇒(1): let D be a dense subset of βX. Then by Lemma 1, ⋂t∈D‍Ot=OD=(0). Now by Lemma 2, MD=(0). Since C(X) is α-separable, there is an α-subset A of D such that MA=(0)=OA, and again by Lemma 1, this shows that A is dense in βX and hence dense in D.Observe that whenX is finite, C(X) is artinian and hence ℵ0-cogenerated. If C(X) is separable, then X is dense-separable. A ring R is called von-Neumann regular if for every a∈R, there exists b∈R such that aba=a. I. Kaplansky has shown that every ideal in a commutative von-Neumann regular ring can be written as the intersection of some family of maximal ideals. Hence, a commutative von-Neumann regular ring is α-separable if and only if it is α-cogenerated. This implies that for any p-space X, C(X) is α-separable if and only if C(X) is α-cogenerated. A ring is called right V-ring (after Villamayor) if every right simple R-module is injective. It is well known that over right V-rings, every submodule of a right R-module M can be written as the intersection of a family of maximal submodules of M. Hence over a right V-ring, a right R-module is α-cogenerated if and only if it is α-separable. Let {Ji}i∈I be a family of ideals of R; we have ⋂i∈I‍Mn(Ji)=Mn(⋂i∈I‍Ji). Hence as far as one is concerned with two sided ideals of Mn(R), one obtains that Mn(R) is α-separable (α-cogenerated, resp.) when R is α-separable (α-cogenerated, resp.).Corollary 5.
The following are equivalent:(1)
βX is dense-α-separable;(2)
C(βX) is α-cogenerated;(3)
C(βX) is α-separable;(4)
C(X) is α-cogenerated;(5)
C(X) is α-separable;(6)
C*(X) is α-cogenerated;(7)
C*(X) is α-separable.Proof.
It is well known thatC(βX)≅C*(X), and by Theorem 4 the verification is immediate.Corollary 6.
IfX is either (1) separable metric or (2) separable and ordered, then C(X) is ℵ1-cogenerated.Proof.
By [5, Corollary 3.2.], for these two cases βX is dense-separable. Now by Theorem 3 the proof is thorough.WhenC(X) is separable, then X is also separable. However, the converse is not true. In [5, example 5.3], a separable compact space Y has been introduced which is not dense-separable; for the space Y, C(Y) is not separable. Otherwise Y=βY should be dense-separable which is not the case. Based on these observations we have the following.Example 7.
There exists a separable spaceY, such that C(Y) is not separable.However, the converse is true when we have a much stronger property as we observe in the next proposition.Proposition 8.
LetR be a commutative ring. Then R is α-separable if and only if X=Max(R) is dense-α-separable.Proof.
LetX be dense-α-separable and 𝒜={Mi}i∈I a family of maximal ideals with zero intersection. This family then will be dense in X. By dense-α-separability of X, 𝒜 has an α-subset with zero intersection. Let R be an α-separable ring and 𝒜 a dense subspace of X. By definition 𝒜 has a zero intersection and by α-separability of R, it has an α-subspace which is dense in X.According to Gelfand-Naimark's theorem every commutativeC*-algebra with identity is isomorphic to C(X,ℂ), where X is a suitable compact Hausdorff space. Based on Theorem 3 and Gelfand-Naimark's theorem, we have the following.Corollary 9.
LetA be a commutative C*-algebra with identity and α an arbitrary regular cardinal. Then the following are equivalent: (1)
A is α-separable;(2)
A is α-cogenerated.



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