Berkovich spectra of elements in Banach rings (Q1800475)

\textit{V. G. Berkovich} defined in [Spectral theory and analytic geometry over non-Archimedean fields. Providence, RI: American Mathematical Society (1990; Zbl 0715.14013), Chapter 7] the spectrum of an element in a {non-Archimedean commutative Banach algebra} over a {non-Archimedean field} \(K\), which is a compact subset of \(\mathbb{A}^1_K\). The paper under review defines and studies the spectrum of an element in a {unital Banach algebra} \(R\) over a commutative {unital Banach ring} \(S\). That is, no longer need \begin{itemize} \item the norms of \(R\) and \(S\) be non-Archimedean (that is, ultrametric), \item the algebra \(R\) be commutative, and \item the ring of definition be a non-Archimedean field (but only a commutative unital Banach ring). \end{itemize} This gives applications to a recent generalization of Serre's spectral theory to Banach {modules} (instead of vector spaces) used to study modular forms (see [\textit{R. F. Coleman}, Invent. Math. 127, No. 3, 417--479 (1997; Zbl 0918.11026)], [\textit{J.-P. Serre}, Publ. Math., Inst. Hautes Étud. Sci. 12, 69--85 (1962; Zbl 0104.33601)] and [\textit{E. Urban}, Ann. Math. (2) 174, No. 3, 1685--1784 (2011; Zbl 1285.11081)]). Of particular interest, and what motivated this generalization, is the case of the algebra \(R\) of endomorphisms of a non-Archimedean Banach space \(E\) over a non-Archimedean field \(k\). Motivated by the construction in [Berkovich, loc.\,cit.], a canonical way to define the spectrum of an element \(a\) in the Banach \(S\)-algebra \(R\) is by \[ \begin{multlined} \{ t \in K : K \text{ is a complete valued field including $S$ } \\ \text{ and } a \otimes 1 - 1 \otimes t \text{ is not invertible in } R \otimes_S K \}, \end{multlined} \] where \(\otimes\) will always denote the (Archimedean) completed tensor product. However: \begin{itemize} \item Possibly, \(R \otimes_S K = 0\); thus it must be assumed nonzero. \item If \(L\) is a complete valued field such that \(L \supseteq K\), then \(t\) in \(K\) must be identified with \(t\) in \(L\). \end{itemize} This gives rise to an equivalence relation on the collection of pairs of the form \((K, t)\), where \(K\) is a complete valued field that is a Banach \(S\)-algebra and \(t\) in \(K\): Let \(K_S\) be the set of all complete valued fields (below a certain fixed cardinality such as \(2^{\aleph_0}\)) that are Banach algebras over \(S\). Let \(K^{(t)}\) be the complete valued field in \(K\) generated by \(\{t\}\) and \(S \cdot 1\). For \(K'\) and \(K''\) in \(K_S\), and \(t'\) in \(K'\), \(t''\) in \(K''\), define \((K', t') \sim (K'', t'')\) if there is an isometric \(S\)-algebra isomorphism \(\phi : {K'}^{(t')} \to {K''}^{(t'')}\) such that \(t'' = \phi(t')\). Let \(K_S^\sim\) denote the set of all equivalence classes of such pairs \([K, s]\) for \(K\) in \(K_S\) and \(s\) in \(K\). It is, by Proposition 3.1, in bijection with \(\mathbb{A}^1_S\) by \(((K, |\cdot|), t) \mapsto [p \mapsto |p(t)|]\) (where we recall that the {Berkovich affine analytic line} \(\mathbb{A}^1_S\) over a Banach ring \(S\) is defined as the topological space of multiplicative semi-norms on \(S[t]\) that are contractive on \(S\) with the topology of pointwise convergence). Motivated by the bijection between \(K_S^\sim\) and \(\mathbb{A}^1_S\), Section 3.2 shows that the latter topological space in the case of \(S = \mathbb{Z}^1\) is fibred over the {minimal fields}, defined as follows: {Definition}. A complete valued field \(K\) is {minimal} if the only complete valued subfield included in \(K\) is \(K\) itself. By Ostrowski's theorem, the set \(K^{\mathrm{min}}\) of all minimal fields is \[ K^{\mathrm{min}} = \{\mathbb{Q}\} \cup \{\mathbb{Z}/p\mathbb{Z} : p \in P\} \cup \{\mathbb{R}^\nu : \nu \in (0, 1]\} \cup \{\mathbb{Q}^\omega_p : p \in P; \omega \in (0, \infty)\}, \] where \(\mathbb{Q}\) and \(\mathbb{Z} / p \mathbb{Z}\) have the trivial norm. Let \(M(\mathbb{Z}^1)\) be the topological space of all multiplicative semi-norms \(\| \cdot \|\) on \(\mathbb{Z}\) that satisfy \(\| \cdot \| \leq | \cdot |\) (where the right-hand side is the Euclidean norm with the topology of pointwise convergence). The map \(K^{\mathrm{min}} \to M(\mathbb{Z}^1)\), given by \(| \cdot |_K \mapsto \nu_K\), the restriction of \(| \cdot |_K\) from \(K\) to (the image in \(K\) of) the integers, is bijective. The continuous restriction \(p : \mathbb{A}^1_{\mathbb{Z}^1} \to M(\mathbb{Z}^1)\) given by \(| \cdot | \mapsto | \cdot |_{\mathbb{Z}}\) induces via this bijection the fibration \[ \mathbb{A}^1_{\mathbb{Z}^1} = \bigcup \left\{ p^{-1}(\nu_K) : K \in K^{\mathrm{min}} \right\} \] with fibers \[ p^{-1}(\nu_K) = \bigcup \{ \mu_L(L) : L \text{ a complete valued field containing } K\}, \] where the map \(\mu_K : K \to K_S^\sim\) is defined by \(t \mapsto [(K, t)]\). Section 4 defines the Berkovich spectrum \(\sigma_{R,S}^{\mathrm{Ber}}(a)\) of an element \(a\) in \(R\) and its ultrametric counterpart \(\sigma_{R,S}^{\mathrm{u}}(a)\) (where in the former the abstract tensor product is completed by the Archimedean norm and in the latter by the non-Archimedean norm). The following is motivated by the construction in [Berkovich, loc.\,cit.]: {Definition}. The {Archimedean spectrum} \(\sigma^{\mathrm{Ber}}_{R,S} (a)\) of an element \(a\) is the set of all \(\lambda \in K_S^\sim\) for which there is \(K \in K_S\) and \(s \in K\) such that \begin{itemize} \item \(\lambda = [K, s]\) and \item \(1 \otimes s - a \otimes 1\) is not a unit in \(R \otimes_S K\), \end{itemize} where \(R \otimes_S K\) denotes the completion of the abstract tensor product by the Archimedean norm. The {non-Archimedean spectrum} \(\sigma^{\mathrm{u}}_{R,S} (a)\) is defined likewise with the only difference that \(R \otimes_S K\) denotes the completion of the abstract tensor product by the non-Archimedean norm. By a proof similar to that of the compactness of \(\Sigma_f\) in [Berkovich, loc.\,cit.], Theorem 4.5 proves that both \(\sigma_{R,S}^{\mathrm{Ber}}(a)\) and \(\sigma_{R,S}^{\mathrm{u}}(a)\) are compact subsets of \(\mathbb{A}^1_S\). Moreover: \begin{itemize} \item \(\sigma_{R,S}^{\mathrm{Ber}}(a)\) is nonempty if there is a nonzero contractive \(S\)-module map from \(R\) to an element in \(K_S\), \item while \(\sigma_{R,S}^{\mathrm{u}}(a)\) is nonempty if and only if there exists a nonzero contractive \(S\)-module map from \(R\) to a non-Archimedean element in \(K_S\). \end{itemize} Section 4.2 studies the case \(S = \mathbb{Z}^1\), that is, \(R\) is an ultrametric unital Banach ring. By Theorem 4.8: \begin{itemize} \item If \(|mx| = |m \cdot 1| |x|\) or \item if \(R\) is equipped with the trivial norm, \end{itemize} then \(\sigma_{R,S}^{\mathrm{u}}(a)\) is always a nonempty subset of \(\mathbb{A}_{\mathbb{Z}}^1\). Section 4.3 studies the case of a commutative algebra: If \(R\) is commutative and generated by \(a\) as a unital Banach S-algebra, then by Proposition 4.10(b), \(\sigma_{R,S}^{\mathrm{Ber}}(a)\) coincides with the Berkovich spectrum \(M(R)\) as a topological space. Section 4.4 studies the case of a unital Banach algebra over a complete valued field, that is, \(S\) is a complete valued field \(K\). By Theorem 4.5.(c), we have \(\sigma^{\mathrm{Ber}}_{A, K}(a) \neq \emptyset\) when \(K\) is Archimedean. If \(K\) is non-Archimedean but \(A\) is Archimedean, then \(\sigma^{\mathrm{u}}_{A, K}(a) \neq 0\) if and only if the ``ultrametrization'' \(A \otimes_K^{\mathrm{u}} K \neq 0\). To handle a non-unital algebra \(B\) over a valued field \(K\), it is made unital by \(D = B \oplus K\). Then \(\sigma^{\mathrm{Ber}}_{D,K}(a) \neq 0\). The sets \(\sigma^{\mathrm{Ber}}_{B, K}(a)\) and \(\sigma^{\mathrm{Ber}}_{D, K}(a)\) only possibly differ in the single point \(\mu_K(0)\), and likewise for \(\sigma^{\mathrm{u}}\) instead of \(\sigma^{\mathrm{Ber}}\). Section 5 considers the case that \begin{itemize} \item \(S = k\) is a non-Archimedean complete valued field, \item \(R = L(E)\) is the Banach \(k\)-algebra of endomorphisms over an infinite-dimensional ultrametric Banach \(k\)-space \(E\) with an orthogonal basis, and \item \(u\) in \(R\) is completely continuous. \end{itemize} The aim is a description of the zeros of \(\det(1 - T u)\) in all complete valued field extensions of \(k\) by the spectrum of \(u\). First, by Proposition 5.3, \(\sigma_{R,S}^{\mathrm{Ber}}(a) = \sigma_{R,S}^{\mathrm{u}}(a)\). By [Serre, loc.\,cit., Proposition 11], respectively, [Berkovich, loc.\,cit., Theorem 1.3.1], Theorem 5.4 shows \begin{itemize} \item all the ``nonzero'' elements of \(\sigma_{R,S}^{\mathrm{Ber}}(a)\) to be the equivalence classes of the inverses of all the zeros of the Fredholm determinant \(\det(1 - ta)\) of \(a\), and \item the spectral radius of \(a\) to be the maximum among the absolute values of the zeros of the Fredholm determinant \(\det(1 - Ta)\) of \(a\). \end{itemize} In particular, if the spectral radius is \(0\), that is, the operator is contractive, then the only zero of the Fredholm determinant would be \(0\), which is impossible. Therefore, as concludes Example 5.6, there are no zeros of the Fredholm determinant.