Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 905650, 15 pagesdoi:10.1155/2012/905650

Research Article

A Note on Some Uniform Algebra Generated by Smooth Functions in the Plane

Raymond Mortini¹ and Rudolf Rupp²

¹Département de Mathématiques, LMAM, UMR 7122, Université Paul Verlaine, Ile du Saulcy, 57045 Metz, France²Fakultät Allgemeinwissenschaften, Georg-Simon-Ohm-Hochschule Nürnberg, Keßlerplatz 12, 90489 Nürnberg, Germany

Received 13 March 2010; Accepted 19 October 2010

Academic Editor: N. Kalton

Copyright © 2012 Raymond Mortini and Rudolf Rupp. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We determine, via classroom proofs, the maximal ideal space, the Bass stable rank as well as the topological and dense stable rank of the uniform closure of all complex-valued functions continuously differentiable on neighborhoods of a compact planar set and holomorphic in the interior of . In this spirit, we also give elementary approaches to the calculation of these stable ranks for some classical function algebras on .

1. Introduction

Let be a compact subset of the complex plane . We denote by the uniform closure on of all complex-valued functions continuously differentiable on neighborhoods of and holomorphic in the interior of . As usual, let be the uniform closure on of the set of polynomials the uniform closure on of those rational functions that have no poles in , and the uniform algebra of all continuous functions on that are holomorphic in . Finally, is the algebra of all complex-valued continuous functions on . We obviously have that . Whereas , and are classical objects that are well understood (see, e.g., the books by Browder [1], and Gamelin [9]), we do not know a paper or textbook dealing with this intermediate algebra .

In the present paper we are mainly interested in solving the Bézout equation in these algebras, with particular emphasis on . By giving classroom proofs of our results, we hope that we will make the notions of stable ranks appearing below accessible to a larger group of analysts, especially students willing to work in function theory.

So let be any of the algebras above. The leading questions will be the following.(1)Suppose that is a pair of functions in such that and have no common zeros on . Does there exist a solution of the Bézout equation such that is invertible?(2)Let be arbitrary. Is it possible to uniformly approximate by a pair of functions in such that and have no common zeros on ?

These questions originally stem from algebraic -theory. In the abstract setting of commutative rings or Banach algebras, they run under the heading “Bass stable rank” and “topological stable rank,” and go back to Bass and Rieffel.

The intention of our paper is now twofold. We first present an elementary approach to the calculation of the stable ranks for the classical algebras , and without using sophisticated methods or notions from algebraic topology and without using deep results as the Arens-Taylor-Novodvorsky theory (see the introduction to [5], for details of this theory). These results were known.

Then, we apply our techniques to the new algebra . So, in Section 4 we determine the maximal ideal space of , and the -convex sets. These results will be used in Section 5 to determine the Bass, topological, and dense stable rank of , which represent the main results of our paper.

2. The Central Definitions

Here we recall those definitions necessary to understand this paper. Let be a compact Hausdorff space, and let be the algebra of all complex-valued continuous functions on endowed with the supremum norm. A uniform algebra on is a uniformly closed subalgebra of separating the points of and having as unit the constant function 1. Its maximal ideal space, or spectrum, , is the set of all nonzero, multiplicative linear functionals on . As usual we will identify a function with its Gelfand transform defined on by .

In the sequel, let . A closed subset of the spectrum of is called -convex, if coincides with its -convex hull:

It is well known that can be identified with the spectrum of the algebra (see [9, page 39]). Let us also note that [9, page 27]. In this setting and is called the polynomial convex hull of . Note that is the union of with all the bounded components of , which we will call holes.

An -tuple is said to be invertible (or unimodular) if there exists such that . The set of all invertible -tuples is denoted by . An -tuple is called reducible if there exists such that .

The Bass stable rank of , denoted by , is the smallest integer such that every element in is reducible. If no such exists, then .

The topological stable rank, , of is the least integer for which is dense in , or infinite if no such exists. This notion goes back to Rieffel [19].

The following concept was introduced by Corach and Suárez [7, page 542]. A version from [18] reads as follows.

The dense stable rank, , of is the least integer such that for every -convex set in the character space of the Gelfand transform of any -tuple satisfying on can be uniformly approximated on by the Gelfand transform of -tuples that are invertible in .

It is well known that (see [7, 17]).

Stable ranks of various real or complex function algebras have mainly been determined by Corach and Suárez [4, 6, 7], Rupp [20–22], Rupp and Sasane [24, 25], Mikkola and Sasane [14], and Mortini and Wick [16, 17].

3. The Stable Ranks of : A Classroom Approach

As a major tool, we use the following lemma, due to Corach and Suárez [4, 6].

Lemma 3.1 (see [4, page 636] and [6, page 608]). Let be a commutative unital Banach algebra. Then, for , the set is open-closed inside In particular, for , if is a continuous curve and is reducible, then is reducible.

In the case where a pretty proof goes as follows (see [17]).

Let be so that . Suppose that and is reducible, say . Write

Since , we may choose so big that for all the elements are invertible in . Hence is invertible for some . Thus, is closed within . Now if is invertible, then every small perturbation of is invertible, too. Thus, is open.

3.1. The Algebras   and

Corach and Suárez showed in [4] that . In our opinion, their proof on page 638 contains a gap, since, in general, the boundary of a component of is not accessible via a path staying outside . For example, let be the union of the closed disk and a spiral outside that clusters at .

So, for the reader’s convenience, we present a short proof here (that is very close to the original one, though). We begin with a Lemma, used in [4], without a proof. We think that this needs a proof though, since a priori it is not clear that a domain admits accessible boundary points in the sense described below.

Lemma 3.2. Let    be compact, . If does not vanish at , then there exist and a piecewise -path outside the zero set of joining and within .

Proof. Let be the connected component of in . Since , the maximum principle implies that does not vanish identically on the boundary of , a set that is contained in . Let with . Choose a disk centered at so that on . Let . Since, as an open set, is pathconnected, there is a path in joining with . Since at most finitely many zeros of could belong to such a path, we can easily avoid these zeros by perturbing a little bit the original path . Next, on the segment joining with , there is a first boundary point of , say . The combined path now joins with and stays outside the zero set of .

Theorem 3.3 (Corach-Suárez). Let   be compact. Then, (1); (2)   if and only if ;(3)   if and only if .

Proof. (1) Let be an invertible pair in , and let be a sequence of rational functions with poles outside uniformly approximating on . By Lemma 3.1, in order to show the reducibility of , it suffices to show the reducibility of for large. Now for some polynomials and . We may assume that and have no common zeros. Obviously, is invertible in . Also, since the reducibility of and implies that of , it suffices to show that invertible pairs of the form are reducible. We have to deal with three cases.Case 1. Let . Then, is invertible in and so is reducible.Case 2. Let satisfy . Choose a sequence, , of points outside converging to . Then, the pairs are reducible and hence, by Lemma 3.1, , too.Case 3. Now let and . Choose according to Lemma 3.2 a curve in , joining with a boundary point of such that stays outside the zero set of . The curve , given by , is a continuous curve, and is reducible. Thus, by Lemma 3.1, is reducible. (2), (3) We first note that . In fact, let be a pair of functions in . Then, for any , there exist two rational functions with poles off such that By slightly perturbing, if necessary, the zeros of , we may assume that and have no zeros in common. Thus, is an invertible pair and so .Now if has no interior points, then any can be uniformly approximated on by rational functions without poles and zeros on . Hence, if . If , then, by Rouché’s theorem, the function with cannot be uniformly approximated by holomorphic functions invertible in a neighborhood of . Thus, . Keeping in mind that , we deduce that .

A similar result holds of course for the algebras , too. Since we were unable to find the assertions on the topological stable rank in the literature, we provide them for the reader’s convenience. Recall that for compact sets in the Bass stable rank for was determined by Corach and Suárez in [7].

Theorem 3.4. Let be a compact set. Then, (1); (2) if and only if and has no holes;(3) if and only if or has holes.

Proof. All three assertions follow as above, by noticing that coincides with the polynomial convex hull, , of (i.e., the union of with all its holes) and that .

Our original intention for the present paper was to give such an elementary proof for the algebra . Our method would have been to approximate by functions in that have no zeros on . A “proof” of that claim that appeared in [13, page 268, Theorem  9] does not seem to be correct, though. In fact, even for functions in the disk algebra, one can have that 0 is an interior point of [26], the image of the unit circle under . Hence, contrary to the claim in that paper, there does not exist sufficiently small with (note also the misprint in that paper, where the function in the assertion should have been ). Thus, we were led to consider functions continuously differentiable in a neighborhood of , where such a Peano-curve phenomenon does not occur.

We will need the following classical result based on Sard’s Lemma [12, page 81] that tells us that the image of the set of critical points of a -map in has Lebesgue measure zero. As it is less time consuming to present a proof here than to browse through monographs not readily available, we add these few lines.

Theorem 3.5. Let   be open and a -map on . Suppose that   is compact and nowhere dense. Then, is nowhere dense, too.

Proof. Let be in the target space. We have to show that is not an interior point of . If , then we are done. So let . Consider the image of the set of critical points of within a fixed compact neighborhood of . Then, is a compact set having Lebesgue measure zero by Sard’s Lemma. Thus, there exist closed disks with centers and radii such that , and . It suffices to show that each of these disks contains points that do not belong to . Let . Note that each point in is a regular point for . Thus, we may cover by a finite number of closed disks such that is a diffeomorphism of onto . By our hypothesis, is nowhere dense in . Thus, is nowhere dense in . Therefore, is a finite union of nowhere dense, compact sets. Hence is nowhere dense, too. Note that and so  . Thus, is nowhere dense. Since the disks are eventually arbitrarily close to , we conclude that is not an interior point of .

3.2. The Algebra 

For arbitrary compact Hausdorff spaces , Vasershtein [27] and Rieffel [19] gave the following formula for the Bass and topological stable ranks of and (see also [15], for a self-contained, easy proof).

Theorem 3.6. Let be a compact Hausdorff space. Then,

Using the additional fact from dimension theory that the covering dimension (or Čech-Lebesgue dimension) of is less than or equal to one if and only if has no interior points, and two otherwise, it follows that the Bass stable rank of is one whenever and two if . The proofs of these facts on the covering dimension are rather technical. We would like to present two elementary approaches, independent of dimension theory. The first one determines directly the Bass stable rank of . The second approach determines the topological stable rank and then deduces the Bass stable rank. But first we present an elementary proof of Rieffel’s result [19] that the Bass and topological stable ranks of coincide.

Theorem 3.7. Let be a compact Hausdorff space. Then, .

Proof. Let , and let be an -tuple in . If is invertible, we are done. So we may assume that the have at least one zero in common. Consider the sets Choose by Urysohn’s Lemma a function with such that vanishes identically on and is constant one on . Then, the -tuple is invertible in . Since , this tuple is reducible. Hence, there exist so that is invertible in . Now on where whenever (this can easily be seen by considering the cases , and ). Thus, .The reverse inequality holds for any Banach algebra , (see [19]). In fact, let , and suppose that . Choose so that . Approximating by an invertible -tuple yields that on . Moreover, there is with . Hence, This shows that   and so .The cases where or immediately follow.

Theorem 3.8 (see [27]). Let   be compact and suppose that . Then, .

Proof. Let be an invertible pair in . Hence, on . By [2, Theorem 4.29], there exists a rational function without poles and zeros in such that on we have for some continuous function . Since has no interior points, we may shift the poles and zeros of so that the new rational function, , has no zeros and poles in and satisfies Thus, and, hence, by a standard reasoning with series in Banach algebras, on . So We may assume that and are continuously extended to . Thus, is a continuous zero-free extension of to .Next we use a simple version of a technique in [23]. Let the closed neighborhood of be chosen so that Hence, again, and so, on for some function .Now let Then, and on . Therefore, the pair is reducible.

We will now determine with our classroom proofs the stable ranks of for arbitrary compacta in .

Theorem 3.9. Let   be a compact set. Then, (1)   if and only if ;(2)   if and only if .

Proof. In view of Theorem 3.7, it suffices to show the assertions for the topological stable rank. (1) Suppose that , and let . By Weierstrass’ approximation theorem, there exists for each a polynomial in the real variables and such that . Since , we may use Theorem 3.5 to conclude that is nowhere dense in . In particular, there exists with arbitrarily small modulus, say , such that has no zeros on . Thus, Hence, has been approximated by invertible functions on and so .Now if , then, after a suitable affine transformation, we may assume that . But the identity function cannot be uniformly approximated on by invertible functions in . Otherwise for all ; hence, by Brouwer’s fixed-point theorem, [2, page 108], has a fixed point in and so has a zero, a contradiction. Thus, implies that .(2) We show that . Having done this, (2) is the logical negation of (1).So let . As above, by Weierstrass’ approximation theorem, there exist for each two polynomials and in the real variables and such that . Now we consider the set as a subset in and look upon the map as a map from to . Since has no interior points, we may conclude with Theorem 3.5 as in (1) that there exist with such that is an invertible pair in . Hence, has been approximated by invertible pairs in .

4. The Algebra : Its Spectrum, Its -Convex Sets

Suppose that is given. Does the algebra resemble or ? For which the algebra is strictly contained in ? We refer the reader to the books by Gaier [8] and Gamelin [9] for a thorough study of the algebras and . In particular, they give descriptions of those compacta for which both coincide (Vitushkin’s theorem).

Concerning our algebra , since every continuous function can be uniformly approximated on compact planar sets by polynomials in the real variables and , it would be natural to guess that coincides with . For instance, if , then, by the Stone-Weierstrass theorem just mentioned, (note that ). It will follow from our representation of (Theorem 4.2) that this is not true in general. Indeed, will be “closer” to than to , and this is due to the following theorem that will be the main tool for our study of .

Theorem 4.1 (see [1, page 160] or [9, page 26]). Let for some neighborhood of , and suppose that on . Then, .

We start now with a representation of in terms of the algebras and for some specified compact sets and . Let be the set of all holomorphic functions in an open set .

Theorem 4.2. If   is compact, then .

Proof. (1) Let for a neighborhood of , and suppose that is holomorphic in . Then, on , and so, by Theorem 4.1, . Since is uniformly closed, we obtain that .(2) To show the reverse inclusion, let . We may think of as being continuous on the whole plane (Tietze). Let . Choose a rational function without poles in such that . Note that possible poles of belong to or to . By shifting the boundary poles, if necessary, we may assume that has no poles on . Let be a neighborhood of , so that , too. By the Stone-Weierstrass theorem, let be a polynomial of the two real variables and such that . Now choose such that on on , and . Finally let Then, for some open set with . Also, on ; hence, is holomorphic in , and so . Moreover, on and on we have Thus, we have uniformly approximated each by functions in . Thus, .

The proof shows that a set of generators of is given by

Corollary 4.3. The following assertions hold. (a) If , then ,   and if  , then .(b) There exist compact sets    with   for which .(c) There exist compact sets   with  .(d) There exist compact sets   with  .

Proof. (a) This immediately follows from Theorem 4.2.(b) A Swiss cheese with for which appears in [8, page 103]. Hence, .(c) Let be the Swiss cheese above (with ), and let be a Swiss cheese satisfying , (see [8, page 104]). We may assume that . Then, satisfies .(d) Let be as above, and choose any with such that . If , then satisfies .

Theorem 4.4. The maximal ideal space of can be identified with via point evaluations. Also, the Shilov boundary of coincides with , the topological boundary of .

Proof. We use Theorem 4.2 and the fact that all functions defined on , respectively, , and appearing here admit continuous extensions to the whole plane. Let satisfy on . Since for every compact set (see [9, page 27]), there exists such that Since , there is such that Let be a neighborhood of so that on . Choose a function , , with on and on . Let . Then, . Moreover, on . In fact, on we have and on we have Since zero-free functions in are obviously invertible, we obtain that is invertible, too. Hence, the ideal generated by coincides with and so .That the Shilov boundary of is works in the same manner as for (see [9, page 27]).

Next we determine the -convex subsets of for a large class of function algebras. It applies in particular to , and . We do not claim originality of this result, but we could not find a reference.

Recall that a hole of a compact set is a bounded component of .

Theorem 4.5. Let   be compact, and let be a uniform algebra with and such that (via point evaluation). Suppose that is a closed subset of . Then is -convex if and only if each hole of contains a hole of .

Proof. Let   be -convex. Suppose that there exists a hole of with . Note that . By virtue of the maximum principle for holomorphic functions, we see that for any and Thus, is contained in the -convex hull of , which coincides, by our hypothesis, with ; a contradiction. Hence, if is -convex, then each hole of contains a hole of .To prove the converse, let satisfy the condition above. We first note that our hypothesis on the spectrum actually implies that . In fact, for , the function has no zeros on the spectrum of the algebra, so is invertible. Thus, each rational function with poles outside belongs to . So actually .Choose any point , and let in a neighborhood of and in a neighborhood of . By Runge’s theorem, there exists a rational function with poles off such that on . Since each hole of contains a hole of and the unbounded component of contains the unbounded component of , we may assume, by Runge again, that the poles of are outside . So and Thus, does not belong to the -convex hull of . Since was arbitrary, is -convex.

Theorem 4.6. Let    be compact and an -convex subset of . Then,

Proof. Let . To show the inclusion , we may assume, by Theorem 4.2, that is a rational function without poles in . Thus, . So To show the reverse inclusion, let . Note that . We may extend continuously to . Let be a rational function without poles in so that on . By moving possible poles on a little bit, we may assume that has no poles in . Note that . Since is -convex, by Theorem 4.5, each hole of contains a hole of . Also, the unbounded component of contains the unbounded component of . Thus, we may assume, using Runge’s theorem, that has no poles in . Let be a neighborhood of so that on .Also, let be a polynomial of the real variables and with on . In particular, on . Choose , , so that on and on . Then, is in for some neighborhood of . Moreover, is holomorphic in . Thus, . Now, on , we have that (see the estimate in Theorem 4.2). Thus, .

Remark 4.7. Let us mention the following interesting and related result by Izzo [11, Theorem 2.2 and Corollary 2.5]. Let be a closed subset of . Then, is dense in if and only if has no interior points and each component of contains a component of . Note that the second condition is, in view of Theorem 4.5, equivalent to being -convex.

5. The Stable Ranks of

Let us call an algebra of holomorphic functions on a planar open set  stable if whenever and . It is clear that the algebras , and are all stable, where for we take .

Lemma 5.1. For all compact subsets ,  one has .

Proof. Let be a pair of functions in . By definition, for , there is a neighborhood of and functions and in so that . Since has no interior points, we conclude from Theorem 3.5 that the images and have no interior points. In particular, the value zero is not an interior point of and . Thus, we may choose and in with so that and . Therefore, and have no zeros on . Since both functions are holomorphic in , they have only finitely many zeros at all. Hence, there exist (holomorphic) polynomials and and functions and zero-free on such that and . Since is a stable algebra, and actually belong to . By moving, if necessary, the zeros of a little bit, we may assume that and have no common zeros. By construction we have and so Since the functions and have no common zeros on , they form, by Theorem 4.4, an invertible pair in .

We are now able to prove the main results of this paper.

Theorem 5.2. Let   be a compact set. Then, (1); (2)  if and only if ;(3)  if and only if .

Proof. (1) follows from [21, Corollary 1.3] (that was based on the Arens-Taylor-Novodvorsky theory) and Theorem 4.4. It can also be seen in the following way, using the simpler results that for every compact set and whenever is a compact set in with empty interior (see Theorem 3.8). So let be an invertible pair in . By Theorems 4.2 and 3.3, there exists so that We may assume that is a rational function without poles in . By moving the poles a little bit, if necessary, we may also assume that has no poles on .Let be taken so that on and that where . Then, is an invertible pair in . Since , there is so that Now by Theorem 4.2, and (note that vanishes identically on ). Thus, . Moreover, by (5.3) and (5.5), we obtain that on . Since, by Theorem 4.4, , we conclude that .(2), (3) By Lemma 5.1 we have that . Thus, it suffices to show (2). So assume that . Then, by Corollary 4.3, . Hence, by Theorem 3.9, . If , then, by Rouché’s theorem, the function cannot be uniformly approximated by functions holomorphic and invertible in a neighborhood of . Hence, .

Our final theorem determines the dense stable rank of a large class of function algebras whose spectrum is, via point evaluation, a compact set in . In particular it applies to , and . For compact sets in the dense stable rank for has been determined by Corach and Suárez in [7].

Theorem 5.3. Let   be a compact set. Suppose that is a uniform algebra with and that . Then, .

Proof. As in the proof of Theorem 4.5, we observe that the hypothesis implies that . Let be an -convex set, and suppose that does not vanish on . Since is -convex, by Theorem 4.5, each hole of contains a hole of . Moreover, the unbounded component of contains the unbounded component of . Hence, by [2, Corollary 4.30], there exists a rational function with zeros and poles off and some continuous function so that . Now is invertible in and so the function belongs to and admits a continuous logarithm on . Note that , because is -convex. Thus, by [9, Corollary 6.2, page 88], actually admits a logarithm in ; that is, for some . Uniformly approximating on by functions in now shows that, on , is the uniform limit of functions of the form that are invertible in . Hence, .

We remark that Theorem 5.3 yields another proof that for these algebras, since the Bass stable rank is always dominated by the dense stable rank (see [7, 17]). In particular, we therefore have a rather short proof for (see [5, 25]). We do not know, though, whether for every compact set .

Acknowledgments

The authors thank Joel Feinstein for some comments concerning Corollary 4.3 and, in particular, for reminding them of Theorem 4.1. They also thank the referee for his remarks that improved the exposition of our paper.

References

A. Browder, Introduction to Function Algebras, W. A. Benjamin, New York, NY, USA, 1969.

R. B. Burckel, An Introduction to Classical Complex Analysis, Birkhäuser, Basel, Switzerland, 1979.

G. Corach and A. R. Larotonda, “Stable range in Banach algebras,” Journal of Pure and Applied Algebra, vol. 32, no. 3, pp. 289–300, 1984.

G. Corach and F. D. Suárez, “Stable rank in holomorphic function algebras,” Illinois Journal of Mathematics, vol. 29, no. 4, pp. 627–639, 1985.

G. Corach and F. D. Suárez, “Extension problems and stable rank in commutative Banach algebras,” Topology and Its Applications, vol. 21, no. 1, pp. 1–8, 1985.

G. Corach and F. D. Suárez, “On the stable range of uniform algebras and ,” Proceedings of the American Mathematical Society, vol. 98, no. 4, pp. 607–610, 1986.

G. Corach and F. D. Suárez, “Dense morphisms in commutative Banach algebras,” Transactions of the American Mathematical Society, vol. 304, no. 2, pp. 537–547, 1987.

D. Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser, Basel, Switzerland, 1980.

T. W. Gamelin, Uniform Algebras, Chelsea Pub. Company, New York, NY, USA, 1984.

J. B. Garnett, Bounded Analytic Functions, vol. 96, Academic Press, New York, NY, USA, 1981.

A. J. Izzo, “Uniform approximation by holomorphic and harmonic functions,” Journal of the London Mathematical Society, vol. 47, no. 1, pp. 129–141, 1993.

W. Kaballo, Einführung in die Analysis III, Spektrum Akademischer, Heidelberg, Berlin, Germany, 1999.

N. G. Lloyd, “Remarks on generalising Rouché's theorem,” The Journal of the London Mathematical Society, vol. 20, no. 2, pp. 259–272, 1979.

K. Mikkola and A. Sasane, “Bass and topological stable ranks of complex and real algebras of measures, functions and sequences,” Complex Analysis and Operator Theory, vol. 4, no. 2, pp. 401–448, 2010.

R. Mortini and R. Rupp, “Real symmetric extensions of invertible tuples of multivariable continuous functions,” Complex Analysis and Operator Theory. In press

R. Mortini and B. D. Wick, “The Bass and topological stable ranks of and ,” Journal für die Reine und Angewandte Mathematik, vol. 636, pp. 175–191, 2009.

R. Mortini and B. D. Wick, “Spectral characteristics and stable ranks for the Sarason algebra ,” Michigan Mathematical Journal, vol. 59, no. 2, pp. 395–409, 2010.

A. Nicolau and D. Suárez, “Approximation by invertible functions of ,” Mathematica Scandinavica, vol. 99, no. 2, pp. 287–319, 2006.

M. A. Rieffel, “Dimension and stable rank in the K-theory of C*-algebras,” Proceedings of the London Mathematical Society, vol. 46, no. 2, pp. 301–333, 1983.

R. Rupp, “Stable rank of subalgebras of the disc algebra,” Proceedings of the American Mathematical Society, vol. 108, no. 1, pp. 137–142, 1990.

R. Rupp, “Stable rank of finitely generated algebras,” Archiv der Mathematik, vol. 55, no. 5, pp. 438–444, 1990.

R. Rupp, “Stable rank and boundary principle,” Topology and its Applications, vol. 40, no. 3, pp. 307–316, 1991.

R. Rupp, “Zerofree extension of continuous functions on a compact Hausdorff space,” Topology and its Applications, vol. 93, no. 1, pp. 65–71, 1999.

R. Rupp and A. Sasane, “On the stable rank and reducibility in algebras of real symmetric functions,” Mathematische Nachrichten, vol. 283, no. 8, pp. 1194–1206, 2010.

R. Rupp and A. Sasane, “Reducibility in , and ,” Canadian Journal of Mathematics, vol. 62, no. 3, pp. 646–667, 2010.

R. Salem and A. Zygmund, “Lacunary power series and Peano curves,” Duke Mathematical Journal, vol. 12, pp. 569–578, 1945.

L. N. Vasershtein, “Stable rank of rings and dimensionality of topological spaces,” Functional Analysis and Its Applications, vol. 5, no. 2, pp. 102–110, 1971.