Abstract
The set \(\mathcal {E}'(\mathbb {R})\) of all compactly supported distributions, with the operations of addition, convolution, multiplication by complex scalars, and with the strong dual topology is a topological algebra. In this article, it is shown that the topological stable rank of \(\mathcal {E}'(\mathbb {R})\) is 2.
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1 Introduction
The aim of this article is to show that the topological stable rank (a notion from topological K-theory, recalled below) of \(\mathcal {E}'(\mathbb {R})\) is 2, where \(\mathcal {E}'(\mathbb {R})\) is the classical topological algebra of compactly supported distributions, with the strong dual topology \(\beta (\mathcal {E}',\mathcal {E})\), pointwise vector space operations, and convolution taken as multiplication.
We recall some key notation and facts about \(\mathcal {E}'(\mathbb {R})\) in Sect. 2 below, including its strong dual topology \(\beta (\mathcal {E}',\mathcal {E})\), and in Sect. 3, we will recall the notion of topological stable rank of a topological algebra.
We will prove our main result, stated below, in Sects. 4 and 5.
Theorem 1.1
Let \(\mathcal {E}'(\mathbb {R})\) be the algebra of all compactly supported distributions on \(\mathbb {R}\), with
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pointwise addition, and pointwise multiplication by complex scalars,
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convolution taken as the multiplication in the algebra, and
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the strong dual topology \(\beta (\mathcal {E}',\mathcal {E})\).
Then the topological stable rank of \(\mathcal {E}'(\mathbb {R})\) is equal to 2.
2 The topological algebra \(\mathcal {E}'(\mathbb {R})\)
For background on topological vector spaces and distributions, we refer to [2, 6, 7, 11,12,13, 16].
Let \(\mathcal {E}(\mathbb {R})=C^\infty (\mathbb {R})\) be the space of functions \(\varphi :\mathbb {R}\rightarrow \mathbb {C}\) that are infinitely many times differentiable. We equip \(\mathcal {E}(\mathbb {R})\) with the topology of uniform convergence on compact sets for the function and its derivatives. This is defined by the following family of seminorms: for a compact subset K of \(\mathbb {R}\), and \(M\in \{0,1,2,3\cdots \}=\mathbb {Z}_{ \tiny \small \scriptscriptstyle {\ge 0}}\), we define
The space \(\mathcal {E}(\mathbb {R})\) is
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metrizable,
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a Fréchet space, and
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a Montel space;
see e.g. [7, Example 3, p.239].
By a topological algebra, we mean the following:
Definition 2.1
(Topological algebra) A complex algebra \(\mathcal {A}\) is called a topological algebra if it is equipped with a topology \(\mathcal {T}\) making the following maps continuous, with the product topologies on the domains:
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\(\mathcal {A}\times \mathcal {A}\ni \;(a,b)\mapsto a+b\;\in \mathcal {A}\)
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\(\mathbb {C}\times \mathcal {A}\ni \;(\lambda , a)\mapsto \lambda \cdot a \;\;\in \mathcal {A}\)
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\(\mathcal {A}\times \mathcal {A}\ni \; (a,b)\mapsto ab\;\!\;\;\;\;\;\in \mathcal {A}\)
We equip the dual space \(\mathcal {E}'(\mathbb {R})\) of \(\mathcal {E}(\mathbb {R})\) with the strong dual topology \(\beta (\mathcal {E}',\mathcal {E})\), defined by the seminorms
for bounded subsets B of \(\mathcal {E}(\mathbb {R})\). Then \(\mathcal {E}'(\mathbb {R})\), being the strong dual of the Montel space \(\mathcal {E}(\mathbb {R})\), is a Montel space too [12, 5.9, p. 147]. This has the consequence that a sequence in \(\mathcal {E}'(\mathbb {R})\) is convergent in the \(\beta (\mathcal {E}',\mathcal {E})\) topology if and only if it is convergent in the weak dual/weak-\(*\) topology \(\sigma (\mathcal {E}',\mathcal {E})\) of pointwise convergence on \(\mathcal {E}(\mathbb {R})\); see e.g. [16, Corollary 1, p. 358].
As usual, let \(\mathcal {D}(\mathbb {R})\) denote the space of all compactly supported functions from \(C^\infty (\mathbb {R})\), and \(\mathcal {D}'(\mathbb {R})\) denote the space of all distributions. The vector space \(\mathcal {E}'(\mathbb {R})\) can be identified with the subspace of \(\mathcal {D}'(\mathbb {R})\) consisting of all distributions having compact support. If \(\mathcal {D}'(\mathbb {R})\) is also equipped with its strong dual topology, then one has a continuous injection \( \mathcal {E}'(\mathbb {R})\hookrightarrow \mathcal {D}'(\mathbb {R}). \) For \(T,S\in \mathcal {E}'(\mathbb {R})\), we define their convolution \(T*S\in \mathcal {E}'(\mathbb {R})\) by
The map \(*:\mathcal {E}'(\mathbb {R})\times \mathcal {E}'(\mathbb {R})\rightarrow \mathcal {E}'(\mathbb {R})\) is (jointly) continuous; see for instance [13, Chapter VI, §3, Theorem IV, p. 157].
Thus \(\mathcal {E}'(\mathbb {R})\), endowed with the strong dual topology, forms a topological algebra with pointwise vector space operations, and with convolution taken as multiplication. The multiplicative identity element is \(\delta _{\scriptscriptstyle {0}}\), the Dirac delta distribution supported at 0. In general, we will denote by \(\delta _a\) the Dirac delta distribution supported at \(a\in \mathbb {R}\).
We also recall that the Fourier–Laplace transform of a compactly supported distribution \(T\in \mathcal {E}'(\mathbb {R})\) is an entire function, given by
see e.g. [16, Proposition 29.1, p. 307].
3 Topological stable rank
An analogue of the Bass stable rank (useful in algebraic K-theory) for topological rings, called the topological stable rank, was introduced in the seminal article [10].
Definition 3.1
(Unimodular tuple, Topological stable rank)
Let \(\mathcal {A}\) be a commutative unital topological algebra with multiplicative identity element denoted by 1, endowed with a topology \({\mathcal {T}}\).
We define \(\mathcal {A}^n:=\mathcal {A}\times \cdots \times \mathcal {A}\) (n times), endowed with the product topology.
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(Unimodular n-tuple) Let \(n\in \mathbb {N}:=\{1,2,3,\ldots \}\). We call an n-tuple \((a_1,\ldots ,a_n)\in \mathcal {A}^n\)unimodular if there exists an n-tuple \((b_1,\ldots ,b_n)\in \mathcal {A}^n\) such that the Aryabhatta-Bézout equation \( a_1 b_1+\cdots +a_nb_n=1 \) is satisfied. The set of all unimodular n-tuples is denoted by \(U_n(\mathcal {A})\). Note that \(U_1(\mathcal {A})\) is the group of invertible elements of \(\mathcal {A}\). An element from \(U_2(\mathcal {A})\) is referred to as a coprime pair. It can be seen that if \(U_n(\mathcal {A})\) is dense in \(\mathcal {A}^n\), then \(U_{n+1}(\mathcal {A})\) is dense in \(\mathcal {A}^{n+1}\).
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(Topological stable rank) If there exists a least natural number \(n\in \mathbb {N}\) for which \(U_n(\mathcal {A})\) is dense in \(\mathcal {A}^n\), then that n is called the topological stable rank of \(\mathcal {A}\), denoted by \(\mathrm{tsr}\;\! \mathcal {A}\). If no such n exists, then \({\text {tsr}}\mathcal {A}\) is said to be infinite.
While the notion of topological stable rank was introduced in the context of Banach algebras, the above extends this notion in a natural manner to topological algebras. The topological stable rank of many concrete Banach algebras has been determined previously in several works (e.g. [5, 14, 15]). In this article, we determine the topological stable rank of the classical topological algebra \(\mathcal {E}'(\mathbb {R})\) from Schwartz’s distribution theory.
4 \(\mathrm {tsr}(\mathcal {E}'(\mathbb {R}))\ge 2\)
The idea is that if tsr\((\mathcal {E}'(\mathbb {R}))\) were 1, then we could approximate any T from \(\mathcal {E}'(\mathbb {R})\) by compactly supported distributions whose Fourier transform would be zero-free, and by an application of Hurwitz Theorem, \({\widehat{T}}\) would need to be zero-free too, which gives a contradiction, since we can easily choose T at the outset to not allow this.
Proposition 4.1
\(\text {tsr}(\mathcal {E}'(\mathbb {R}))\ge 2\).
Proof
Suppose on the contrary that \({\text {tsr}}(\mathcal {E}'(\mathbb {R}))=1\). Let
By our assumption, \(U_1(\mathcal {E}'(\mathbb {R}))\) is dense in \((\mathcal {E}'(\mathbb {R}),\beta (\mathcal {E}',\mathcal {E}))\). But then the set \(U_1(\mathcal {E}'(\mathbb {R}))\) is also sequentially dense: This is a consequence of the fact that a subset F of \(\mathcal {E}'(\mathbb {R})\) is closed in \(\beta (\mathcal {E}',\mathcal {E})\) if and only if it is sequentially closed. (See [9, Satz 3.5, p. 231], which says that \(E'\), with the \(\beta (E',E)\)-topology, is sequential whenever E is Fréchet–Montel. A locally convex space F is sequential if any subset of F is closed if and only if it is sequentially closed. If F has this property, then the closure of any subset equals its sequential closure, and therefore being dense is the same as being sequentially dense. In our case, \(E=\mathcal {E}(\mathbb {R})\) is Fréchet–Montel, and so \(\mathcal {E}'(\mathbb {R})\) is sequential. In fact, in the remark following [9, Satz 3.5], the case of \(\mathcal {E}'(\mathbb {R})\) is mentioned as a corollary.)
Thus there exists a sequence \((T_n)_{n\in \mathbb {N}}\) in \(U_1(\mathcal {E}'(\mathbb {R}))\) such that \(T_n{\mathop {\longrightarrow }\limits ^{n\rightarrow \infty }} T\) in \(\mathcal {E}'(\mathbb {R})\). But since each \(T_n\) is invertible in \(\mathcal {E}'(\mathbb {R})\), there exists a sequence \((S_n)_{n\in \mathbb {N}}\) in \( \mathcal {E}'(\mathbb {R})\) such that
Taking the Fourier–Laplace transform, we obtain
In particular, the entire functions \(\widehat{T_n}\) are all zero-free.
But as \(T_n{\mathop {\longrightarrow }\limits ^{n\rightarrow \infty }} T\) in \(\mathcal {E}'(\mathbb {R})\), we now show that \((\widehat{T_n})_{n\in \mathbb {N}}\) converges to \({\widehat{T}}\) uniformly on compact subsets of \(\mathbb {C}\) as \(n\rightarrow \infty \). The pointwise convergence of \((\widehat{T_n})_{n\in \mathbb {N}}\) to \({\widehat{T}}\) is clear by taking the test function \(x\mapsto e^{-2\pi i x z}\):
Now for any \(\varphi \in \mathcal {E}(\mathbb {R})\), we know that the sequence \((\langle T_n,\varphi \rangle )_{n\in \mathbb {N}}\) converges to \(\langle T,\varphi \rangle \), and in particular, the set
is bounded, for every \(\varphi \in \mathcal {E}(\mathbb {R})\). By the Banach–Steinhaus Theorem for Fréchet spaces (see for example [11, Theorem 2.6, p. 45]), applied in our case to the Fréchet space \(\mathcal {E}(\mathbb {R})\), we conclude that
is equicontinuous. Thus for every \(\epsilon >0\), there exists a neighbourhood V of 0 in \(\mathcal {E}(\mathbb {R})\) such that \(T_n(V)\subset B(0,\epsilon ):=\{z\in \mathbb {C}:|z|<\epsilon \}\) for all \(n\in \mathbb {N}\). From here it follows that there exist \(M\in \mathbb {Z}_{\scriptscriptstyle {\ge 0}}\), \(R>0\) and \(C>0\) such that
By taking \(\varphi =(x\mapsto e^{-2\pi i x z})\) in the above, we obtain
Also, by the Payley–Wiener–Schwartz Theorem [2, Theorem 4.12, p. 139] for \(T\in \mathcal {E}'(\mathbb {R})\), we have
It now follows that for some constants \(C_*,M_*,R_*\) that
But this means that the pointwise convergent sequence \((\widehat{T_n})_{n\in \mathbb {N}}\) of entire functions is uniformly bounded on compact subsets of \(\mathbb {C}\) (that is, the sequence constitutes a normal family). Then it follows from Montel’s Theorem (see e.g. [17, Exercise 9.4, p. 157]) that \((\widehat{T_n})_{n\in \mathbb {N}}\) converges to \({\widehat{T}}\) uniformly on compact subsets of \(\mathbb {C}\) as \(n\rightarrow \infty \).
But now by Hurwitz Theorem (see e.g. [17, Exercise 5.6, p.85]), and considering, say, the compact set \(K=\{z\in \mathbb {C}:|z|\le 1\}\), we conclude that \({\widehat{T}}\) must be either be identically zero on K or that it must be zero-free in K. But \({\widehat{T}}\) is neither:
a contradiction. Hence \({\text {tsr}}(\mathcal {E}'(\mathbb {R}))\ge 2\). \(\square \)
An alternative Proof of Proposition 4.1, suggested by Peter Wagner, is as follows. The theorem of supports ([6, Theorem 4.3.3]) implies that \(U_1(\mathcal {E}'(\mathbb {R}))\) equals the set of nonzero multiples of \(\delta _a\) for arbitrary \(a\in \mathbb {R}\), and this set is not dense in \(\mathcal {E}'(\mathbb {R})\). We give the details below. First, one can show the following structure result for \(U_1(\mathcal {E}'(\mathbb {R}))\).
Proposition 4.2
\(U_1(\mathcal {E}'(\mathbb {R}))=\{c\delta _a: a\in \mathbb {R}, \;0\ne c\in \mathbb {C}\}\).
Proof
It is clear that \(\{c\delta _a:a\in \mathbb {R},\;0\ne c\in \mathbb {C}\}\subset U_1(\mathcal {E}'(\mathbb {R}))\) since
Now suppose that \(T\in U_1(\mathcal {E}'(\mathbb {R}))\). Then there exists an \(S\in \mathcal {E}'(\mathbb {R})\) such that \(T*S=\delta _0\). By the Theorem on Supports [6, Theorem 4.3.3, p. 107], we have
where, for a distribution \(E\in \mathcal {E}'(\mathbb {R})\), the notation \(\text {c.h.supp}(E)\) is used for the closed convex hull of \(\text {supp}(E)\), that is, the intersection of all closed convex sets containing \(\text {supp}(E)\). So we obtain
from which it follows that \(\text {c.h.supp}(T)=\{a\}\) and \(\text {c.h.supp}(S)=\{-a\}\) for some \(a\in \mathbb {R}\). But then also \(\text {supp}(T)=\{a\}\) and \(\text {supp}(S)=\{-a\}\). As distributions with support in a point p are linear combinations of the Dirac delta distribution \(\delta _p\) and its derivatives \(\delta _p^{(n)}\) [16, Theorem 24.6, p. 266], we conclude that S and T have the form
for some integers \(N,M\ge 0\) and some complex numbers \(t_n,s_m\) (\(0\le n\le N\), \(0\le m\le M\)). Now \(T*S=\delta _0\) implies that \(N=M=0\) and \(t_0s_0=1\), thanks to the linear independence of the set
in the complex vector space \(\mathcal {E}'(\mathbb {R})\). In particular \(t_0\ne 0\). Thus
Consequently, \(U_1(\mathcal {E}'(\mathbb {R}))=\{c\delta _a: a\in \mathbb {R}, \;0\ne c\in \mathbb {C}\}\). \(\square \)
Based on the above, we can now give the following alternative proof of Proposition 4.1.
Proof
We show \(U_1(\mathcal {E}'(\mathbb {R}))\) is not dense in \((\mathcal {E}'(\mathbb {R}),\beta (\mathcal {E}',\mathcal {E}))\). If it were, then it would be sequentially dense too, and so for each element T of \(\mathcal {E}'(\mathbb {R})\), there would exist a sequence in \(U_1(\mathcal {E}'(\mathbb {R}))\) that converges to T in the \(\beta (\mathcal {E}',\mathcal {E})\) topology, and hence also in the \(\sigma (\mathcal {E}',\mathcal {E})\) topology. But we now show that \(\delta _0'\in \mathcal {E}'(\mathbb {R})\) cannot be approximated in the \(\sigma (\mathcal {E}',\mathcal {E})\) topology by elements from \(U_1(\mathcal {E}'(\mathbb {R}))=\{c\delta _a: a\in \mathbb {R}, \;0\ne c\in \mathbb {C}\}\). Suppose, on the contrary, that \((c_n\delta _{a_n})_{n\in \mathbb {N}}\) converges to \(\delta _0'\) in \((\mathcal {E}'(\mathbb {R}),\sigma (\mathcal {E}',\mathcal {E}))\).
We first note that \((a_n)_{n\in \mathbb {N}}\) is bounded. For if not, then there exists a subsequence \((a_{n_k})_{k\in \mathbb {N}}\) of \((a_n)_{n\in \mathbb {N}}\) such that \(|a_{n_k}|>2\) for all \(k\in \mathbb {N}\). Now choose a \(\varphi \in \mathcal {D}(\mathbb {R})\) such that \(\varphi '(0)=1\) and \(\varphi \equiv 0\) on \(\mathbb {R}\setminus (-1,1)\). Then we arrive at the contradiction that
So \((a_n)_{n\in \mathbb {N}}\) is bounded.
Next we show that \((c_n)_{n\in \mathbb {N}}\) converges to 0. Let \(R>0\) be such that \(|a_n|<R\) for all \(n\in \mathbb {N}\). Let \(\psi \in \mathcal {D}(\mathbb {R})\) be such that \(\psi \equiv 1\) on \([-R,R]\). Then we have
Finally, we show that \((c_n\delta _{a_n})_{n\in \mathbb {N}}\) converges to 0 in \((\mathcal {E}'(\mathbb {R}),\sigma (\mathcal {E}',\mathcal {E}))\). For any \(\chi \in \mathcal {D}(\mathbb {R})\), we have
So \((c_n\delta _{a_n})_{n\in \mathbb {N}}\) converges to 0 in \((\mathcal {E}'(\mathbb {R}),\sigma (\mathcal {E}',\mathcal {E}))\). But this is a contradiction, since \(0\ne \delta '_0\) in \(\mathcal {E}'(\mathbb {R})\). Consequently, \(U_1(\mathcal {E}'(\mathbb {R}))\) is not dense in \((\mathcal {E}'(\mathbb {R}),\beta (\mathcal {E}',\mathcal {E}))\), and so \({\text {tsr}}(\mathcal {E}'(\mathbb {R}))\ge 2\). \(\square \)
5 \(\mathrm {tsr}(\mathcal {E}'(\mathbb {R}))\le 2\)
The idea is to reduce the determination of tsr\((\mathcal {E}'(\mathbb {R}))\) to tsr(\(\mathbb {C}[z]\)) of the polynomial ring \(\mathbb {C}[z]\) as follows. Given a pair from \(\mathcal {E}'(\mathbb {R})\), we use mollification to make a pair in \(\mathcal {D}(\mathbb {R})\), and then approximate the resulting smooth functions by a linear combination of Dirac distributions with uniform spacing. The uniform spacing affords the identification of the linear combination of Dirac deltas with the ring of polynomials.
For \(n\in \mathbb {N}\), we define the collection \({{\mathbf {D}}}_{n}\) of all ‘finitely supported Dirac delta combs’ with spacing 1 / n by
where ‘span’ means the set of all (finite) linear combinations.
Lemma 5.1
(Approximating a pair of Dirac combs by a unimodular pair) Let \(n\in \mathbb {N}\)and \(T,S\in {{\mathbf {D}}}_{n}\). Then there exist sequences \((T_k)_{k\in \mathbb {N}}\)and \((S_k)_{k\in \mathbb {N}}\)in \({{\mathbf {D}}}_n\), which converge to T, S, respectively, in \((\mathcal {E}'(\mathbb {R}),\sigma (\mathcal {E}',\mathcal {E}))\), and henceFootnote 1also in \((\mathcal {E}'(\mathbb {R}),\beta (\mathcal {E}',\mathcal {E}))\), and are such that for each k, \((T_k,S_k)\in U_2(\mathcal {E}'(\mathbb {R}))\).
Proof
Write \(T=\sum _{\ell =-L}^{L}t_{\scriptscriptstyle {\ell }} \delta _{\scriptscriptstyle {\ell /n}}\), and \( S=\sum _{\ell =-L}^{L}s_{\scriptscriptstyle {\ell }} \delta _{\scriptscriptstyle {\ell /n}}, \) for some \(L\in \mathbb {N}\), \(t_{\scriptscriptstyle {\ell }},s_{\scriptscriptstyle {\ell }} \in \mathbb {C}\).
Define
For a given \(k\in \mathbb {N}\), let \(\epsilon =1/(2^k \cdot 2L)>0\). Then we can perturb the coefficients of the polynomials \(p_{\scriptscriptstyle {T}}, p_{\scriptscriptstyle {S}}\) within a distance of \(\epsilon \) to make them have no common zeros, that is after perturbation of coefficients they are coprime in the ring \(\mathbb {C}[z]\). Indeed any polynomial \(p_{\scriptscriptstyle {T}}, p_{\scriptscriptstyle {S}}\) can be factorized as
and if there is some common zero \(\alpha _{\scriptscriptstyle {\ell }}=\beta _{\scriptscriptstyle {\ell '}}\), we simply replace \(\beta _{\scriptscriptstyle {\ell '}}\) by \(\beta _{\scriptscriptstyle {\ell '}}+\epsilon '\) with an \(\epsilon '\) small enough so that the final coefficients (of this new perturbed polynomial obtained from \(p_{\scriptscriptstyle {S}}\)), which are polynomial functions of the zeros, lie within the desired \(\epsilon \) distance of the coefficients of \(p_{\scriptscriptstyle {S}}\). So we can choose \({\widetilde{t}}_{\scriptscriptstyle {-L,k}},\ldots ,{\widetilde{t}}_{\scriptscriptstyle {L,k}}\) and \({\widetilde{s}}_{\scriptscriptstyle {-L,k}},\ldots ,{\widetilde{s}}_{\scriptscriptstyle {L,k}}\) such that for all \(\ell =-L,\ldots , L\), we have
and so that
have no common zeros. Thus \({\widetilde{p}}_{\scriptscriptstyle {T,k}},{\widetilde{p}}_{\scriptscriptstyle {S,k}}\) are coprime in \(\mathbb {C}[z]\), and hence there exist polynomials \(q_{\scriptscriptstyle {T,k}},q_{\scriptscriptstyle {S,k}}\in \mathbb {C}[z]\) ([1, Corollary 8.5, p. 374]) such that
Set \(Q_{\scriptscriptstyle {T,k}}:=z^L q_{\scriptscriptstyle {T,k}}\) and \(Q_{\scriptscriptstyle {S,k}}:=z^L q_{\scriptscriptstyle {S,k}}\), and
Then in the ring \(\mathbb {C}[z,z^{-1}]\) of linear combinations of monomials \(z^n\), where \(n\in \mathbb {Z}\) (i.e. the Laurent polynomial ring \(\mathbb {C}[z,z^{-1}] =\mathbb {C}[z,w]/\langle zw-1 \rangle \); see for example [1, p. 367]), we have
Suppose that \(Q_{\scriptscriptstyle {T,k}}\) and \(Q_{\scriptscriptstyle {S,k}}\) have the expansions
Finally, set
and
Then it follows from (1) that
To see this, we note that \(\Phi :\mathbb {C}[z,z^{-1}]\rightarrow {\mathbf {D}}_n\) given by
defines a ring homomorphism, and then (2) above follows by applying \(\Phi \) on both sides of (1). Hence \((T_k,U_k)\in U_2(\mathcal {E}'(\mathbb {R}))\). Also, for any \(\varphi \in \mathcal {E}(\mathbb {R})\), we have
Hence \(T_k{\mathop {\longrightarrow }\limits ^{k\rightarrow \infty }}T\) in \((\mathcal {E}'(\mathbb {R}),\sigma (\mathcal {E}',\mathcal {E}))\) as \(k\rightarrow \infty \). But then this convergence is also valid in \((\mathcal {E}'(\mathbb {R}),\beta (\mathcal {E}',\mathcal {E}))\), by [16, Corollary 1, p. 358], since \(\mathcal {E}'(\mathbb {R})\) is a Montel space. Similarly, \(S_k{\mathop {\longrightarrow }\limits ^{k\rightarrow \infty }}S\) in \((\mathcal {E}'(\mathbb {R}),\sigma (\mathcal {E}',\mathcal {E}))\) as \(k\rightarrow \infty \), and again, the convergence holds in \((\mathcal {E}'(\mathbb {R}),\beta (\mathcal {E}',\mathcal {E}))\). This completes the proof. \(\square \)
Lemma 5.2
(Approximation in \(\mathcal {E}'(\mathbb {R})\) by Dirac combs)
Let \(T\in \mathcal {E}'(\mathbb {R})\). Then there exists a sequence \((T_n)_{n\in \mathbb {N}}\)such that
-
for all \(n\in \mathbb {N}\), \(T_n\in {{\mathbf {D}}}_n\), and
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\(T_n{\mathop {\longrightarrow }\limits ^{n\rightarrow \infty }} T\)in \((\mathcal {E}'(\mathbb {R}),\sigma (\mathcal {E}',\mathcal {E}))\), and hence also in \((\mathcal {E}'(\mathbb {R}),\beta (\mathcal {E}',\mathcal {E}))\).
Proof
Let \(k\in \mathbb {N}\) be such that the support of T is contained in \((-k,k)\). We first produce a mollified approximating sequence for T. Let \(\varphi :\mathbb {R}\rightarrow [0,\infty )\) be any test function in \(\mathcal {D}(\mathbb {R})\) with support in \([-a,a]\) for some \(a>0\), and such that
Then we know that if we define \( \varphi _m(x):=m\cdot \varphi (mx)\) (\(m\in \mathbb {N}\)), then for each m,
is a smooth function having a compact support, and moreover,
in \((\mathcal {E}'(\mathbb {R}),\beta (\mathcal {E}',\mathcal {E}))\); see for example [2, Theorem 3.3, p.97]. So the convergence is also valid in \((\mathcal {E}'(\mathbb {R}), \sigma (\mathcal {E}',\mathcal {E}))\). Moreover, as the support of \( f_m= T*\varphi _m\) is contained in the sum of the supports of \(\varphi _m\) and of T, for all m large enough, say \(m\ge M\), we have
From now on, we will assume that \(m\ge M\), so that \(\text {supp}(f_m)\subset [-k,k]\). Now we will approximate \(f_m\) by Dirac comb elements. To this end, we define
We will show that \(T_{m,n}{\mathop {\longrightarrow }\limits ^{n\rightarrow \infty }} f_m\) in \((\mathcal {E}'(\mathbb {R}), \sigma (\mathcal {E}',\mathcal {E}))\). Let \(\psi \in \mathcal {E}(\mathbb {R})\). Then
Thus \( \langle T_{m,n},\psi \rangle \) gives a Riemann sum for the integral of the continuous function \(f_m\psi \) with compact support contained in \([-k,k]\), giving
Hence \(T_{m,n}{\mathop {\longrightarrow }\limits ^{n\rightarrow \infty }} f_m\) in \((\mathcal {E}'(\mathbb {R}),\sigma (\mathcal {E}',\mathcal {E}))\). As \(\mathcal {E}'(\mathbb {R})\) is a Montel space, this convergence is also valid in \((\mathcal {E}'(\mathbb {R}),\beta (\mathcal {E}',\mathcal {E}))\), and the proof is completed. \(\square \)
Proposition 5.3
\(\text {tsr}(\mathcal {E}'(\mathbb {R}))\le 2\).
Proof
Let \(T,S\in \mathcal {E}'(\mathbb {R})\). Throughout this proof, \(\mathcal {E}'(\mathbb {R})\) is endowed with the strong dual topology \(\beta (\mathcal {E}',\mathcal {E})\), and then \((\mathcal {E}'(\mathbb {R}))^2=\mathcal {E}'(\mathbb {R})\times \mathcal {E}'(\mathbb {R})\) is equipped the product topology. Let V be a neighbourhood of (T, S) in \((\mathcal {E}'(\mathbb {R}))^2\). By Lemma 5.2, it follows that \( \bigcup \limits _{n\in \mathbb {N}} ({{\mathbf {D}}}_n\times {{\mathbf {D}}}_n) \) is sequentially dense, and hence dense, in \((\mathcal {E}'(\mathbb {R}))^2\).
Thus there exists a pair \((T_*,S_*)\in V\cap ({{\mathbf {D}}}_n\times {{\mathbf {D}}}_n)\) for some \(n\in \mathbb {N}\). By Lemma 5.1, there exists a sequence \((T_k,S_k)_{k\in \mathbb {N}}\) in \(({{\mathbf {D}}}_n\times {{\mathbf {D}}}_n)\cap U_2(\mathcal {E}'(\mathbb {R}))\) that converges to \((T_*,S_*)\) in \((\mathcal {E}'(\mathbb {R}))^2\). Since V is also a neighbourhood of \((T_*,S_*)\) in \((\mathcal {E}'(\mathbb {R}))^2\), there exists an index K large enough so that for all \(k>K\), \((T_k,S_k)\in V\).
Consequently, \(U_2(\mathcal {E}'(\mathbb {R}))\) is dense in \((\mathcal {E}'(\mathbb {R}))^2\). \(\square \)
Proof of Theorem 1.1
It follows from Propositions 4.1 and 5.3 that the topological stable rank of \((\mathcal {E}'(\mathbb {R}), +,\cdot ,*, \beta (\mathcal {E}',\mathcal {E}))\) is equal to 2. \(\square \)
Remarks 5.4
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1.
From the proofs, it is clear that we have shown that \(U_1(\mathcal {E}'(\mathbb {R}))\) is not dense in \((\mathcal {E}'(\mathbb {R}),\sigma (\mathcal {E}',\mathcal {E}))\), while \(U_2(\mathcal {E}'(\mathbb {R}))\) is sequentially dense, and hence dense, in \((\mathcal {E}'(\mathbb {R}))^2\) endowed with the product topology with \(\mathcal {E}'(\mathbb {R})\) bearing the \(\sigma (\mathcal {E}',\mathcal {E})\) topology. However, we note that \(*:\mathcal {E}'(\mathbb {R})\times \mathcal {E}'(\mathbb {R})\rightarrow \mathcal {E}'(\mathbb {R})\) is not continuous if we use the \(\sigma (\mathcal {E}',\mathcal {E})\) topology on \(\mathcal {E}'(\mathbb {R})\): For example, in \((\mathcal {E}'(\mathbb {R}),\sigma (\mathcal {E}',\mathcal {E}))\), we have that \( \delta _{\pm n}{\mathop {\longrightarrow }\limits ^{n\rightarrow \infty }} 0, \) so that in the product topology on \((\mathcal {E}'(\mathbb {R}))^2\), we have \( (\delta _n,\delta _{-n}){\mathop {\longrightarrow }\limits ^{n\rightarrow \infty }} (0,0) \). But on the other hand, we have \(\delta _n*\delta _{-n}=\delta _{n-n}= \delta _0{\mathop {\longrightarrow }\limits ^{n\rightarrow \infty }} \delta _0\ne 0=0*0\). So \((\mathcal {E}'(\mathbb {R}), +,\cdot ,*,\sigma (\mathcal {E}',\mathcal {E}))\) is not a topological algebra in the sense of our Definition 2.1.
-
2.
We remark that in higher dimensions, with a similar analysis, it can be shown that \({\text {tsr}}(\mathcal {E}'(\mathbb {R}^d))\le d+1\).
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3.
The Bass stable rank (a notion from algebraic K-theory, recalled below) of \(\mathcal {E}'(\mathbb {R})\) is not known. If \(\mathcal {A}\) is a commutative unital ring, then \((a_1,\dots ,a_n,b)\in U_{n+1}(\mathcal {A})\) is called reducible if there exists an n-tuple \((\alpha _1,\ldots ,\alpha _n)\in \mathcal {A}^n\) such that \( (a_1+\alpha _1 b,\ldots , a_n+\alpha _n b)\in U_n(\mathcal {A}). \) It can be seen that if every element of \(U_{n+1}(\mathcal {A})\) is reducible, then every element of \(U_{n+2}(\mathcal {A})\) is reducible too. The Bass stable rank of \(\mathcal {A}\), denoted by \(\text {bsr} \;\!\mathcal {A}\), is the smallest \(n\in \mathbb {N}\) such that every element in \(U_{n+1}(\mathcal {A})\) is reducible, and if no such n exists, then \(\text {bsr} \;\!\mathcal {A}:=\infty \). It is known that for commutative unital Banach algebras \(\mathcal {A}\), \(\text {bsr}\;\! \mathcal {A}\le {\text {tsr}}\;\!\mathcal {A}\) [4, Theorem 3]. But the validity of such an inequality in the context of topological algebras does not seem to be known. We conjecture that \(\text {bsr}(\mathcal {E}'(\mathbb {R}))=2\).
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4.
There are also several other natural convolution algebras of distributions on \(\mathbb {R}\), for example
and we leave the determination of the stable ranks of these algebras as open questions.
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5.
[8, Corollary 3.1] gives a ‘corona-type’ pointwise condition for coprimeness in \(\mathcal {E}'(\mathbb {R})\), reminiscent of the famous Carleson corona conditionFootnote 2 of coprimeness in the Banach algebra \(H^\infty (\mathbb {D})\): \(T_1, T_2\in U_2(\mathcal {E}'(\mathbb {R}))\) if and only if there exist positive C, N, M such that for all numbers \(z\in \mathbb {C},\;\; |\widehat{T_1}(z)|+|\widehat{T_2}(z)|\ge C(1+|z|^2)^{-N} e^{-M|\text {Im}(z)|}. \)
Notes
Because \(\mathcal {E}'(\mathbb {R})\) is a Montel space; see [16, Corollary 1, p. 358].
The Hardy algebra \(H^\infty (\mathbb {D})\) is the Banach algebra of all bounded and holomorphic functions on the unit disk \(\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}\). The Carleson Corona Theorem [3] says that \((f_1,f_2)\in U_2(H^\infty (\mathbb {D}))\) if and only if there exists a \(\delta >0\) such that for all \(z\in \mathbb {D}\), \(|f_1(z)|+|f_2(z)|>\delta \).
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Acknowledgements
I thank Professor Michael Kunzinger (University of Vienna) for answering my query on whether density in \(\mathcal {E}'\) implies sequential density, for the reference [9], and for useful comments. I also thank Professor Peter Wagner (University of Innsbruck) for showing me the alternative approach to establishing Proposition 4.1, which we have included in this article (Proposition 4.2).
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Sasane, A. Topological stable rank of \(\mathcal {E}'(\mathbb {R})\). Math. Z. 296, 313–323 (2020). https://doi.org/10.1007/s00209-019-02418-2
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DOI: https://doi.org/10.1007/s00209-019-02418-2