Topologies, ring norms and algebra norms on some algebras of continuous functions.

Topologies, ring norms and algebra norms on some algebras of continuous functions. Javier Gómez-Pérez Javier Gómez-Pérez, Departamento de Matemáticas, Universidad de León, 24071 León, Spain. Corresponding author. Warren Wm. McGovern Warren Wm. McGovern, Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403. Abstract Let C(X) be the algebra of all real-valued continuous functions on a completely regular space X, and C (X) the subalgebra of bounded functions. Any intermediate algebra A between C (X) and C(X) is endowed with the m A -topology τ ma, defined in a similar way to the m-topology τ m on C(X). The relative topology that A inherits as a subspace of (C(X), τ m ) is finer than the m A -topology and, although (A, τ ma ) is generally a topological ring, it is not always a topological algebra. Some properties of (A, τ ma ) are studied, and the characterization of those intermediate algebras that are normed topological rings provides the tool for distinguish between two kinds of intermediate algebras which share the property of contain a countable cofinal subset. Key words: Rings of continuous functions, intermediate algebras, topological rings, normed rings 1991 MSC: Primary; 13J25, 54C35. Secondary; 54C40, 46E25 Email addresses: demjgp@unileon.es (Javier Gómez-Pérez), warrenb@bgnet.bgsu.edu (Warren Wm. McGovern). 1 First author partially supported by the Spanish D.G.I.C.Y.T grant BFM2002-04125-C02-02 and Junta de Castilla y León grant LE 66/03. Preprint submitted to Elsevier Preprint 4 May 2004

Introduction Let C(X) be the algebra of all real-valued continuous functions on a nonempty completely regular space X, and C (X) the subalgebra of bounded functions. This paper deals with subalgebras of C(X) containing C (X). We shall refer to them as intermediate algebras on X. They are sublattices of C(X) and so they are Φ-algebras in the sense of Henriksen-Johnson [10]. They have been studied from different points of view by several authors (see [5]). It has been shown in [5, 3.1] that any intermediate algebra on X is the ring of fractions of C (X) with respect to a saturated multiplicatively closed subset of C (X). As rings of fractions of C (X) the intermediate algebras inherit some algebraic properties from C (X). Hewitt describes in [11] some different topologies on C(X). Some of them as the uniform, the compact-open or the point-open topologies are very well known. The m-topology considered in that paper is truly less known than the others. Some authors refer to the m-topology as the fine topology, the Whitney topology or the Morse topology, and can also be described on spaces C(X, Y ) where Y is a metric space (see [13] and [14] for more information). Following Hewitt, for a given intermediate algebra A one can define four topologies, namely the u A -, k A -, p A - and m A -topologies. The u A -, k A -, p A - topologies on A are precisely the topologies that A inherits as a subspace of C(X) with the u-, k- and p-topologies. The m A -topology has a different behavior than the others; in fact the m A -topology is coarser than the topology induced by the m-topology on C(X). This paper deals with the m A -topology on A. The m A -topology on A is a ring topology, i.e. addition and multiplication in A are continuous under the m A -topology, but it is not an algebra topology when A contains an unbounded function. We shall study some topological properties of an intermediate algebra A endowed with the m A -topology. The family of intermediate algebras which we call countably generated intermediate algebras contains a strictly smaller family consisting of the singly generated ones. A special property, related with the order on the countably generated intermediate algebras is that they are exactly those that contain a countable cofinal subset, but there is no way to know if an intermediate algebra that contain a countable cofinal subset is singly generated or if it is not. By studying the ring normability of the intermediate algebras, we shall make the distinction between them, as we show that an intermediate algebra A endowed with the m A -topology is a normed ring if and only if it is singly generated. The last section of the paper deals with the non-existence of algebra norms 2

on intermediate algebras. Yood [18] showed that there is no Banach algebra norm on C(X), and later Pruss [15] showed that C(X) does not admit any algebra norm, even permitting C(X) to be incomplete. We show in this section that the same property is verified by any intermediate algebra different from C (X). 1 Preliminaries. Concerning rings of continuous functions we basically adhere to the notation and terminology in [8]. In the realm of topological rings, we use the notations in [16] and [17]. You can also have some more information about topological rings in [1]. With respect to general topology concepts the reader may consult [7]. A ring R is said to be a topological ring if a topology T is given on R in such a way that addition and multiplication in R are continuous on R R, and the map x x is continuous from R to R. The topology T is called a ring topology on R. If R is a ring and P, Q R, we shall write P + Q to denote the set {p + q : p P, q Q}, and P Q = {pq : p P, q Q}, but we write p + Q and pq when P = {p}. The following characterization of a ring topology on a ring R can be seen in [17]. 1.1 Proposition. Let V be a fundamental system of neighborhoods of 0 for a topology T on R. Then T is a ring topology if and only if T satisfies the following four conditions: (a) For each V V there exists U V such that U + U V. (b) For each V V there exists U V such that U V. (c) For each V V there exists U V such that UU V. (d) For each V V and each b R there exists U V such that bu V and Ub V. Moreover, this topology is Hausdorff if and only if {0} is the intersection of the neighborhoods of 0. 1.2 Definition. Let G be a group (we use additive notation). A map N : G R + {0} is a norm on G if for all x, y G verifies: (a) N(x) = 0 if and only if x = 0. (b) N(x + y) N(x) + N(y). (c) N( x) = N(x). 3

If G is a ring, the group norm N is a ring norm if it also verifies (d) N(xy) N(x)N(y). If in addition G is a K-algebra (K = R or K = C), N is an algebra norm if it is a ring norm and for every λ K and every x G (e) N(λx) = λ N(x). 1.3 Definition. A subset S of a commutative topological ring R is said to be bounded if for every neighborhood U of zero there is a neighborhood V of zero such that US V. If R is a K-algebra (K = R or K = C), a subset U of R is said to be absorbent if for any x R there exists a positive number r such that λf U for all λ r. Next we recall some definitions from [5], [8] and [11]. Throughout, X will denote a completely regular topological space, C(X) the ring (and algebra) of all real-valued continuous functions on X, C (X) the subring (and subalgebra) of C(X) of bounded continuous functions; A any subalgebra of C(X) containing C (X), which we shall call intermediate algebra on X, or simply intermediate algebra. It is shown in [5] that A is an intermediate algebra on X if and only if it is a ring of fractions of C (X) with respect to a saturated multiplicatively closed subset of C (X) whose elements are units of C(X) (see [5] for more details). We shall denote this saturated multiplicatively closed subset by S A. Note that every function in S A is a bounded function that is a unit of A, and conversely any bounded unit in A is in S A. The intermediate algebras on X are sublattices of C(X) and so they are Φ- algebras in the sense of Henriksen-Johnson. They are also characterized as the absolutely convex subalgebras of C(X) ([5, 3.3]), where by absolutely convex is meant that any continuous function g is in A whenever g f for some f A. 1.4. Topologies in rings of continuous functions. We discuss next some ways to topologize C(X). For a real number r we denote by r the constant function equals to r. The u-topology (or uniform topology) τ u on C(X) is defined by taking as a fundamental system of neighborhoods of f C(X) the sets B(f, ε) = {g C(X) : f g < ε}, where ε is a positive real number. As usual, we will refer to it as the uniform topology on C(X). The k-topology (or compact-open topology) τ k on C(X) is defined by taking as a fundamental system of neighborhoods of f the sets B(f, K, ε) = {g 4

C(X) : f(x) g(x) < ε for all x K}, where K is an arbitrary compact subset of X and ε is any positive real number. The p-topology (or point-open topology) τ p on C(X) is obtained by substituting arbitrary finite subsets of X for arbitrary compact subsets in the definition of the k-topology. The m-topology τ m on C(X) is defined by taking as a fundamental system of neighborhoods of f C(X) the sets B(f, u) = {g C(X) : f g < u}, where u is any positive unit in C(X). The following describes some properties of (C(X), τ m ). 1.5 Theorem. [11, Theorem 3] Under the m-topology the set C(X) is a commutative algebra with unit over R, in which operations of addition and multiplication are continuous, the set of elements with inverses is an open set, and the inverse is a continuous operation wherever defined. If there exists an unbounded function in C(X), then C(X) is not metrizable under the m-topology, and, indeed, fails at every point to satisfy the first axiom of countability. If every function in C(X) is bounded, then the m-topology and the u-topology coincide in C(X). 2 Properties of the m A -topology on intermediate algebras. Let A be an intermediate algebra on X. One can define the u A -, k A - and p A - topologies on an intermediate algebra on X in the same way as the u-, k- and p- topologies are defined in C(X) with the only restriction that the functions are now in A. So it is clear that, for example, the neighborhood {g A : f g < ε} of f A in the u A -topology is {g A : f g < ε} = B(f, ε) A. Therefore the u A -topology on A is the topology induced by the u-topology on C(X). The same thing happens for the k- and the p- topologies. Note that if f A, then B(f, ε) A = B(f, ε) and if f / A then B(f, ε) A =. This never happens for the k- or p- topologies since if f C(X)\A and K is a compact subset of X, there exists M R such that f(x) M for all x K. Take g = ( M f) M C (X) A. Then g K = f K, and so B(f, K, ε) = B(g, K, ε), whence B(f, K, ε) A. As any finite subset F is compact, we conclude that B(f, F, ε) A. In a similar way as the m-topology is defined on C(X), one can consider as a fundamental system of neighborhoods of f A the sets B A (f, u) = {g A : f g < u}, where u is a positive unit in A. If no confusion is possible, we shall write B(f, u) for B A (f, u), and we shall write τ ma to denote this topology on A. As any positive unit in A is also a positive unit in C(X), then the m A - 5

topology on A is coarser than the relative m-topology induced by C(X). We shall see later that if A is an intermediate algebra on X other than C(X), then the m A -topology is strictly coarser than the relative m-topology induced by C(X). We note that the set {B(f, u) : u S A }, where S A is the set of bounded units in A, that is the multiplicatively closed subset of C (X) that defines A as a ring of fractions of C (X), is also a fundamental system of neighborhoods of f under the m A -topology. As shown in Theorem 2.1, the m A -topology is a ring topology on any intermediate algebra A. 2.1 Theorem. Let A be an intermediate algebra on X. Then τ ma is a T 2 ring topology, the set of invertible elements is open and the multiplication by invertible elements is a homeomorphism. Proof. We are going to prove that the fundamental system of neighborhoods of zero B 0 = {B(0, u) : u S A }, verifies the conditions in Proposition 1.1. Let V = B(0, v) B 0 and U = B(0, u) where u = v/2. Then U + U V, and so condition (a) holds. To show condition (b), note that V = V. Condition (c) can be seen by taking u = v. If f is any function in A, and v S A, then u = v S 1+f 2 A and fb(0, v) B(0, u), so condition (d) holds. Finally, as B(0, ε) B 0 for every ε > 0 and {B(0, ε) : ε R + } = {0}, then the m A -topology is a Hausdorff ring topology. Even more, it is a Tychonoff space. If u is an invertible element, then one can take v = u /2 and obviously every element in B(u, v) is invertible. Finally, since multiplication is always continuous, then the multiplication for invertible element has a continuous inverse, namely the multiplication by its inverse. 2.2 Remark. As a topological ring, an intermediate algebra is a homogeneous space. Therefore, any local property verified at 0 is also verified at any other point since each translation T a : A A; T a (b) = a + b is a homeomorphism on A. As we said in the above proof, each τ u -neighborhood of 0 on A is also a τ ma - neighborhood of 0, therefore τ u τ ma. As any bounded unit in A is also a unit in C(X) then the m A -topology on A is coarser than the induced τ m -topology by the m-topology on C(X). We are going to prove that 2.3 Proposition. If A contains an unbounded function, i.e. A C (X), then the u-topology is strictly coarser than the m A -topology. If A C(X) the m A -topology is strictly coarser than the relative m-topology induced by C(X). Proof. Let f be an unbounded function in A, and consider u = 1 1+f 2. Then u is a bounded unit in A, and the set B(0, u) fails to be contained in B(0, ε) for any ε > 0. For the second assertion, suppose that v is a bounded unit 6

in C(X) that is not a unit in A (such a v exists if A C(X)) and consider B C(X) (0, v) A = {f A : f v}. If there exists u S A such that B A (0, u) B C(X) (0, v) A, then u/2 B A (0, u) B C(X) (0, v) A, and so u/2 v thus, v 1 2u 1 A. As A is an absolutely convex subalgebra of C(X), it follows that v 1 A, a contradiction. The same argument as that given in the proof of the second part of Proposition 2.3 shows that if A B are different intermediate algebras on X, then the m A -topology on A is strictly coarser than the relative topology that A inherits as a subspace of B (under the m B -topology). From now on A is an intermediate algebra on X other than C (X), and the topology which we refer is the m A -topology on A. 2.4 Proposition. No neighborhood of 0 in A is absorbent. Proof. Let r > 0 a real number, f an unbounded function in A and u a bounded unit in A. If B(0, u) were absorbent then sf u for every scalar s < r. But this is impossible since f is unbounded and u is bounded. 2.5 Corollary. No intermediate algebra A is a topological algebra. Proof. Any topological algebra contains an absorbent neighborhood of zero. 2.6 Remark. As corollary 2.5 above shows, no intermediate algebra is a topological algebra. In order to clarify this, note that if an R-algebra A is a topological algebra, then the map R A A, (λ, f) λf will be continuous. But the usual topology on R is much coarser than the inherited topology from the m-topology if we consider R as a subset of (A, τ ma ). In fact, the topology that R inherits from A is the discrete topology, as follows by the proof of the first assertion in Proposition 2.3. This is the reason why that map is not continuous, and therefore A is not a topological algebra. A complete study on topological algebras is done in [2]. We shall consider now some topological properties such as boundedness and completeness for intermediate algebras (A, τ ma ) on X. 2.7 Proposition. No intermediate algebra A is a bounded topological ring. Proof. For, if u is a unit in A such that 0 < u < 1, and v is any unit in A then w = v/2 B(0, v), g = 2/v A and gw = 1 / B(0, u), so gb(0, v) is not contained in B(0, u). As any compact topological ring is bounded, it follows that no intermediate algebra A is a compact topological ring. Moreover, no intermediate algebra is locally compact as we now show. 7

2.8 Theorem. No intermediate algebra is a locally compact space. Proof. Suppose A is an intermediate algebra that is locally compact, and let U be a compact neighborhood of 0. Then there exists u S A such that B(0, u) has a compact closure. Since the multiplication by the unit u 1 is a homeomorphism on A, we may assume that u = 1. The set {1/n : n N} is a discrete closed subset of the compact B(0, 1), which is a contradiction. There are also some local properties that can be studied on intermediate algebras considered as topological rings under the m-topology. 2.9 Proposition. Any intermediate algebra A is a locally bounded topological ring. Proof. We shall see that B(0, u), u S A, is a bounded subset of A. Let B(0, v), v S A, be any neighborhood of 0, and let w = v A. Then 2u B(0, w)b(0, u) B(0, v). Local convexity can also be defined in any intermediate algebra, in the same way as it is defined for topological vector spaces 2.10 Proposition. Any intermediate algebra A is a locally convex topological ring. Proof. Let B(0, u), u S A, be a neighborhood of zero, f, g B(0, u) and r R, 0 r 1. Then (1 r)f + rg (1 r) f + r g < u, so B(0, u) is convex. Since the uniform topology on A is the topology induced by the uniform topology on C(X), then any uniformly Cauchy net converges in A and so A is uniformly complete. We shall see now that A is also m A -complete. Note that though A is uniformly complete, A need not be m A -complete because the u-topology is coarser than the m A -topology. 2.11 Theorem. Any intermediate algebra A is a complete topological ring. Proof. Let {f λ } λ Λ a Cauchy net in (A, τ ma ). Then for each u S A, there exists λ 0 Λ such that f λ f µ < u whenever λ, µ λ 0. As the u-topology is coarser than the m A -topology, the net {f λ } λ Λ is a Cauchy net in the u- topology and so {f λ } λ Λ is pointwise convergent to a function f and so it is uniformly convergent in A, thus f A. We show that {f λ } λ Λ converges to f in the m A -topology. Since {f λ } λ Λ is a Cauchy net in the m A -topology, for any u S A there exists λ 0 Λ such that f λ f µ < u/2 whenever λ, µ λ 0, that is to say that for every x X, f λ (x) f µ (x) < u(x)/2 if λ, µ λ 0. Since the net {f µ (x)} λ Λ in R converges to f(x), taking limits over µ it follows that lim µ f λ (x) f µ (x) u(x)/2 < u(x) for each x X. Therefore, λ λ 0 8

implies f λ (x) f(x) < u(x) for all x X, and then {f λ } λ Λ converges to f in the m A -topology, whence (A, τ ma ) is complete. 3 Normed intermediate algebras. The main purpose of this section is to characterize those intermediate algebras (A, τ ma ) that are normed rings. We see what intermediate algebras satisfy the first axiom of countability and further which of them are normed rings. This characterization of the intermediate algebras that are normed rings will provide a tool to distinguish between the two kinds of intermediate algebras having a common property, namely those that contain a countable cofinal subset. Hewitt showed ([11, Theorem 3]) that if C(X) contains an unbounded function, then τ m is not metrizable. Moreover, it fails to satisfy the first axiom of countability at any point. We would like to study this property in arbitrary intermediate algebras. The following definition is needed below. 3.1 Definition. An element c in a topological ring R is a topological nilpotent if for any neighborhood V of zero there exists n 0 N such that c n V whenever n n 0. A subset S of a topological ring R is a topologically nilpotent subset if for any neighborhood V of zero there exists n 0 N such that S n V whenever n n 0. A sufficient condition on normability of topological rings is the next: 3.2 Theorem. [17, 14.4] If A is a Hausdorff ring with identity that possesses a left or right bounded neighborhood V of zero and an invertible topological nilpotent c such that cv = V c, then the topology on A is given by a norm. We single out those intermediate algebras A that satisfy the first axiom of countability. 3.3 Definition. We shall say that an intermediate algebra A is countably generated over C (X) if A may be obtained by adjoining to C (X) a countable family {f 1, f 2,...} of unbounded functions in C(X), so that we get the smallest intermediate algebra on X that contains all the functions f n. The functions f n will be called the generators for A over C (X). We denote by C (X)[f 1, f 2,...] the smallest intermediate algebra containing {f 1, f 2,...}, and we say that {f 1, f 2,...} is a well-chosen set of generators for C (X)[f 1, f 2,...] if 1 < c f 1 f 2..., for some real number c. It is shown in [4] that from any countable set of generators {f 1, f 2,...} we may always obtain a well-chosen one. If the set of generators is finite, then C (X)[f 1, f 2,...] can be generated by only one element (see [5]), and we say that A is a singly generated intermediate 9

algebra; in this case we write A = C (X)[f] if f is a generator for A. The countably generated intermediate algebras have been studied in [4], where they are characterized as those intermediate algebras that contain a countable cofinal subset. We point out that for any countably generated intermediate algebra C (X)[f 1, f 2,...] we can always suppose that the set {f 1, f 2,...} is well-chosen and cofinal. This special property is essential for the proof of Theorem 3.4. 3.4 Theorem. An intermediate algebra A satisfies the first axiom of countability if and only if A is countably generated over C (X). Proof. Suppose that A = C (X)[f 1, f 2,...] where {f 1, f 2,...} is well-chosen and cofinal. Then a function f C(X) is in A if and only if f f n for some n N. Therefore the set of neighborhoods of zero given by the sets B(0, 1 f n ) is a countable fundamental system of neighborhoods of zero. Conversely, if A satisfies the first axiom of countability, then there is a countable fundamental system {V n : n N} of neighborhoods of zero. For each n, there exists a positive bounded unit u n such that B(0, u n ) V n, and so {B(0, u n )} is a countable fundamental system of neighborhoods of zero. We only need to see that {1/u n : n N} is a cofinal subset of A. To do this, as A is the ring of fractions of C (X) with respect to the multiplicatively closed subset S A consisting of the bounded units in A, it suffices to prove that the set {u n : n N} is a coinitial subset of S A. So, let u be any positive unit in A. Since the set {B(0, u n )} is a fundamental system of neighborhoods of zero, there is some n N such that B(0, u n ) B(0, u); hence u n u. As it is known, a first countable topological group is normed. But this property is not enough to assure that the ring A is a normed topological ring. We are going to characterize those intermediate algebras that are normed topological rings. 3.5 Proposition. An intermediate algebra A contains an invertible topological nilpotent element if and only if A is a singly generated intermediate algebra. Proof. If A = C (X)[f] with 1 < c f for some c R then the set {1/f n : n N} gives rise to a countable fundamental system of neighborhoods of zero, and so 1/f is an invertible topological nilpotent element in A. Conversely, if v is a topological nilpotent unit in A, for each positive bounded unit u in A, there exists n N such that v n u, and then {v n : n N} is a coinitial subset of S A, so {1/v n : n N} is cofinal in A. Therefore, the intermediate algebra A is a countably generated intermediate algebra whose set of generators is {1/v n : n N}, whence A = C (X)[1/v]. 3.6 Theorem. An intermediate algebra A is a normed topological ring if and only if A is a singly generated intermediate algebra. 10

Proof. Suppose that A is a normed topological ring, and let N denote its norm. From Proposition 3.5 we only need to show that A contains an invertible topological nilpotent. As A is a normed ring, A satisfies the first axiom of countability, whence from Theorem 3.4 A is a countably generated intermediate algebra. We can then assume that A = C (X)[f 1, f 2...] where {f 1, f 2,...} is well-chosen and cofinal. For n N, let V n = {f A : N(f) < 1/n}. Then {V n : n N} and {B(0, 1/f n )} are both fundamental systems of neighborhoods of zero, so there exists n N such that B(0, 1/f n ) V 2, that is N(1/f n ) < 1/2. Hence (1/f n ) k V 2 k for all k N. Since {V n : n N} is a fundamental system of neighborhoods of zero, 1/f n is an invertible topologically nilpotent element of A, whence A is singly generated. The converse follows from Proposition 3.5 and Theorem 3.2 Theorem 3.6 gives the answer to a question suggested from [4] which was to decide whether an intermediate algebra containing a countable cofinal subset is a singly generated intermediate algebra or not. 4 Algebra norms vs. ring norms For a non-pseudocompact space X, Yood [18] showed that there is no Banach algebra norm on C(X), the algebra of all real-valued continuous functions on X. With the help of a result due to Kaplansky [12], Pruss [15] was able to show that, in fact, C(X) does not admit any algebra norm, even permitting C(X) to be incomplete. We have seen in Theorem 3.6 that under the m-topology some intermediate algebras are normed rings. We have also seen (Corollary 2.5) that no intermediate algebra that contain an unbounded function is a topological algebra under this topology, therefore, the m-topology is not a normed algebra topology. The main goal in this section is to prove that an important difference can be established between the ring normability and the algebra normability in the realm of intermediate algebras. We will show in Theorem 4.3 that no intermediate algebra containing an unbounded function admit a normed algebra norm. Let us now explain some results from Kaplansky and Pruss which will be needed later. By C 0 (X) we denote the algebra of continuous functions on X vanishing at infinity, equipped with the supremum norm. 4.1 Proposition. [12] Let (A,. ) be a commutative Banach algebra which is isometrically isomorphic to C 0 (X) for some locally compact X. Then any normed algebra norm. on A satisfies f f for any f A. 11

4.2 Corollary. [15] Let X be any topological space and let. be a normed algebra norm on C (X). Then for every f C (X) we have f sup f(x). x X With these two facts in mind, we are going to see our main result on normability of intermediate algebras. 4.3 Theorem. Let A be an intermediate algebra on X containing an unbounded function. Then there is no normed algebra norm on A. Proof. Suppose. is a normed algebra norm on A. Then it is also a normed algebra norm on C (X), and according to Corollary 4.2, it follows f { f(x), f C (X)}. Let g be an unbounded unit in A such that g 1. sup x X If r > 0, there exists (g 1 + r) 1, the multiplicative inverse of the function (g 1 + r), and it is a bounded function whence (g 1 + r) 1 sup (g 1 (x) + r) 1. x X As in any normed algebra inversion is continuous wherever defined, then ( g 1 + 1/n ) 1 g 0. n Since g 1 is a bounded function, we have that sup (g 1 + 1/n) 1 = M < n and so sup (g 1 (x) + 1/n) 1 M for each x X and n N. But it is clear x X( that lim (g(x) 1 + 1/n) 1) = g(x) whence we follow that g(x) M for n every x X which contradicts the fact that g is unbounded. References [1] V. I. Arnautov, S. T. Glavatsky and A. V. Mikhalev, Introduction to the theory of topological rings and modules, Pure and Applied Mathematics 197, Marcel- Dekker, New-York, 1996. [2] E. Beckenstein, L. Narici and C. Suffel, Topological algebras, North-Holland, Amsterdam, 1977. [3] G. Di Maio, L. Holá, D. Holý and R. A. McCoy, Topologies on spaces of continuous functions, Topology Appl. 86 (1998), 105-122. [4] J. M. Domínguez and J. Gómez-Pérez, Countably generated intermediate algebras between C (X) and C(X), Topology Proc., 24 Summer 1999 (2001), 129-138. 12

[5] J. M. Domínguez, J. Gómez-Pérez and M. A. Mulero, Intermediate algebras between C (X) and C(X) as rings of fractions of C (X), Topology Appl. 77 (1997), 115-130. [6] E. van Douwen, Nonnormality or hereditary paracompactness of some spaces of real functions, Topology Appl. 39 (1991), 3-32. [7] R. Engelking. General topology. Sigma Series in Pure Mathematics, 6. Heldermann Verlag, Berlin, 1989. [8] L. Gillman and M. Jerison, Rings of continuous functions, Springer-Verlag, New York, 1976. [9] J. Gómez Pérez and W. Wm. McGovern, The m-topology on C m (X) revisited, preprint. [10] M. Henriksen and D. G. Johnson, On the structure of a class of lattice-ordered algebras, Fund. Math. 50 (1961), 73-94. [11] E. Hewitt, Rings of real-valued continuous functions, I, Trans. Amer. Math. Soc. 64 (1948), 45-99. [12] I. Kaplansky, Normed Algebras, Duke Math. J. 16 (1949), 399 418. [13] R. A. McCoy, Fine topology on function spaces, Internat. J. Math. & Math. Sci. 3 (1986), 417-424. [14] R. A. McCoy and I. Ntantu, Topological properties of spaces of continuous functions, Lecture Notes in Mathematics 1315, Springer-Verlag, Berlin Heidelberg, 1988. [15] A. R. Pruss, A remark on the non-existence of an algebra norm for the algebra of continuous functions on a topological space admitting an unbounded function, Studia Math. 116 (1995), 295 297. [16] S. Warner, Topological fields, North-Holland Mathematics Studies 157, North- Holland, Amsterdam, 1989. [17] S. Warner, Topological rings, North-Holland Mathematics Studies 178, North- Holland, Amsterdam, 1993. [18] B. Yood, On the non-existence of norms for some algebras of functions, Studia Math. 111 (1994), 97 101. 13