We construct explicit strictly ascending chains of dense subalgebras of length 𝔠 in every separable infinite-dimensional complex Banach algebra. For large classes of commutative C*-algebras we also construct strictly descending chains of the same length. The constructions rely on algebraic independence, Stone–Weierstrass arguments, and transfinite recursion.
Although separable Banach algebras are topologically small, they often possess remarkably rich algebraic substructures. One manifestation of this phenomenon is the existence of large families of dense subalgebras with strict inclusion relations.
The purpose of this paper is to study the maximal possible length of chains of dense subalgebras in separable Banach algebras. Our main results show that the continuum \(\mathfrak c\) is the optimal length in this setting and that such chains can be constructed explicitly under mild hypotheses.
Our contribution is threefold:
(i) explicit constructions of chains of dense subalgebras;
(ii) realization of the maximal possible chain length \(\mathfrak c\) in separable Banach algebras;
(iii) a transparent sufficient condition guaranteeing the existence of descending chains.
Throughout, all Banach algebras are complex unless stated otherwise.
We begin with a basic analytic independence result that underlies the algebraic constructions in the commutative case.
Lemma 1. Let \(\mu_1,\dots,\mu_m\in\mathbb C\) be distinct and let \(r_j\ge0\) be integers. If \[\sum_{j=1}^m\sum_{\ell=0}^{r_j} c_{j,\ell} x^\ell e^{\mu_j x}\equiv0 \quad\text{on an interval,}\] then all coefficients \(c_{j,\ell}\) vanish.
Proof. This is a standard consequence of the linear independence of exponential polynomials. One may differentiate sufficiently many times to obtain a linear system whose Wronskian determinant is nonzero due to the distinctness of the \(\mu_j\). We refer to [1] for a detailed treatment. ◻
We now treat commutative \(C^*\)-algebras of the form \(C(K)\).
Theorem 1. Let \(K\) be an infinite compact Hausdorff space admitting a countable point-separating family \(\{u_n\}\subset C(K,\mathbb R)\). Then \(C(K,\mathbb C)\) admits strictly ascending and strictly descending chains of dense subalgebras of length \(\mathfrak c\).
Proof. Let \(\Lambda\subset(0,\infty)\) be a \(\mathbb Q\)-linearly independent set of cardinality \(\mathfrak c\). For \(\lambda\in\Lambda\) and \(n\in\mathbb N\), define \[v_{\lambda,n}(x)=e^{\lambda u_n(x)}.\]
Let \(G\) consist of \(\{1\}\), all functions \(u_n\), and all \(v_{\lambda,n}\). Any nontrivial polynomial relation among finitely many elements of \(G\) yields, upon restriction to suitable subsets of \(K\), an identity of the form treated in Lemma 1, forcing all coefficients to vanish. Hence \(G\) is algebraically independent.
Let \(A\) be the algebra generated by \(G\). It is self-adjoint, contains the constants, and separates points of \(K\). By the Stone–Weierstrass theorem, \(A\) is dense in \(C(K,\mathbb C)\).
To obtain a strictly ascending chain, enumerate \(\Lambda\) as \(\{\lambda_\alpha:\alpha<\mathfrak c\}\) and let \(A_\alpha\) be the algebra generated by \(\{u_n\}\) together with \(\{v_{\lambda_\beta,n}:\beta<\alpha\}\). Algebraic independence ensures \(A_\alpha\subsetneq A_\beta\) for \(\alpha<\beta\), and each \(A_\alpha\) remains dense.
For the descending chain, define \(B_\alpha\) by removing the generators indexed by \(\{\lambda_\beta:\beta<\alpha\}\). Density follows from the presence of the \(u_n\), while strictness again follows from algebraic independence. ◻
Corollary 1. Every infinite compact metrizable space \(K\) satisfies the hypotheses of Theorem 1.
Proof. Separability of \(C(K)\) implies the existence of a countable point-separating family; see [2]. ◻
We now turn to the noncommutative setting.
Theorem 2. Every separable infinite-dimensional Banach algebra admits a strictly ascending chain of dense subalgebras of length \(\mathfrak c\).
Proof. Let \(E_0\) be a fixed countable dense subalgebra. Since the ambient algebra has cardinality \(\mathfrak c\), we may construct by transfinite recursion a family \(\{x_\alpha:\alpha<\mathfrak c\}\) such that \(x_\alpha\) does not belong to the algebra generated by \(E_0\cup\{x_\beta:\beta<\alpha\}\).
Define \(E_\alpha\) to be the algebra generated by \(E_0\cup\{x_\beta:\beta<\alpha\}\). Each \(E_\alpha\) is dense since it contains \(E_0\), and strict inclusion follows from the choice of the \(x_\alpha\). ◻
Theorem 3. If a separable Banach algebra contains an algebraically independent family of cardinality \(\mathfrak c\), then it admits a strictly descending chain of dense subalgebras of length \(\mathfrak c\).
Proof. Let \(\{y_\alpha:\alpha<\mathfrak c\}\) be algebraically independent and fix a countable dense subalgebra \(E_0\). For each \(\alpha<\mathfrak c\), define \(D_\alpha\) as the algebra generated by \(E_0\cup\{y_\beta:\beta\ge\alpha\}\).
Density follows from the inclusion of \(E_0\). If \(\alpha<\beta\), then \(y_\alpha\in D_\alpha\setminus D_\beta\), since otherwise algebraic independence would be violated. Hence the chain is strictly descending. ◻
Remark 1. The independence hypothesis in the preceding theorem is sufficient but not known to be necessary. It would be of interest to characterize precisely those separable Banach algebras admitting descending chains of maximal length.
The author has no relevant financial or non-financial interests to disclose.
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Markushevich, A. I. (2013). Theory of Functions of a Complex Variable. American Mathematical Soc..
Murphy, G. J. (2014). \(C^*\)-Algebras and Operator Theory. Academic press.
