Operators on a Non-Archimedean Field

Introduction

A non-Archimedean field is a field with an ultrametric absolute value, and its additive operators carry a size — the operator norm — together with a finer invariant than the size: the action induced on the residue field and on the value group. This article treats the operator layer of a non-Archimedean field: the bounded additive operators and their operator norm, the contractive operators that preserve the valuation ring, the residual operator that a contractive operator induces on the residue field, the isometries and the group they form, and the property of spherical completeness, which is the completeness of the nests of balls rather than of the sequences and is exactly the hypothesis under which the extension theorems of the non-Archimedean theory hold.

The article assumes the non-Archimedean field, its absolute value, its valuation ring $\mathcal{O}$, its maximal ideal $\mathrm{M}$, its residue field $k$, its value group $\Gamma$ and the ultrametric geometry of balls from Absolute Values, Valuations and Completions; the valuation as a multiplicative operator and its tropical reading from The Valuation Operator; the residue operator and its multiplicative part from The Residue Operator of a Valued Field; the completion operator and the persistence of the residue field and the value group from The Completion Operator; the bounded linear operators and the operator norm on a normed space from Bounded Operators on a Topological Vector Space, later in this Part; and the continuous additive operators, the homeomorphisms and the continuous automorphisms from Operators on a Topological Ring. The Hahn–Banach theorem over a spherically complete field is named and not proved; the analytic theory of power series over a non-Archimedean field is Part III and is not used. No measure and no form occurs.

Throughout, $(F, \lvert \cdot \rvert)$ is a non-Archimedean field, $\mathcal{O}$, $\mathrm{M}$, $k$ and $\Gamma$ are its valuation ring, maximal ideal, residue field and value group, $\pi$ is a uniformiser when $\Gamma \cong \mathbb{Z}$, and $T$ ranges over the continuous additive operators $F \to F$; the completion is $\widehat{F}$ and the closed ball of radius $r$ about $a$ is $B(a,r) = \{x : \lvert x - a \rvert \leq r\}$.

The Bounded Operators and the Operator Norm

Definition. An additive operator $T : F \to F$ is bounded if the set $\{\lvert Tx \rvert : \lvert x \rvert \leq 1\}$ is bounded above in $\mathbb{R}$, and the operator norm of a bounded $T$ is

$$ \lVert T\rVert = \sup_{\lvert x \rvert \leq 1} \lvert Tx \rvert . $$

Proposition (bounded is continuous). For an additive operator $T$, the following are equivalent: $T$ is bounded; $T$ is uniformly continuous; $T$ is continuous at $0$; there is a constant $C$ with $\lvert Tx \rvert \leq C\lvert x \rvert$ for all $x$. The least such $C$ is the operator norm, and the operator norm is a norm on the space of bounded operators, submultiplicative under composition.

Proof. Boundedness gives the estimate on the unit ball and hence, by scaling, everywhere; an estimate gives continuity at $0$ and therefore, additivity being compatible with the uniform structure, uniform continuity; continuity at $0$ gives a ball $B(0,\rho)$ mapped into $B(0,1)$, whence the estimate with $C = \rho^{-1}$ after scaling. The norm axioms are checked termwise; submultiplicativity is $\lvert(ST)x\rvert \leq \lVert S\rVert\lvert Tx\rvert \leq \lVert S\rVert\lVert T\rVert\lvert x\rvert$ for the least constant.

Proposition (the norm is attained on the unit ball). For a bounded additive $T$ the supremum defining $\lVert T\rVert$ is a maximum whenever the value set $\lvert F^\times\rvert$ is discrete, and in all cases it is the least $C$ with $\lvert Tx\rvert \leq C\lvert x\rvert$; the two-sided ideal of operators of norm zero is $\{T : T(F) \subseteq \overline{\{0\}}\} = 0$ on a Hausdorff field.

Proof. With a discrete value group the values $\lvert Tx\rvert$ form a discrete subset of $\mathbb{R}$, and a bounded subset of a discrete set attains its supremum; in general the supremum need not be attained, as on a dense value group. The kernel of the norm is the set of operators carried into the closure of zero, which is $\{0\}$ for a Hausdorff field, so the norm is a norm in the strict sense.

The Contractive Operators and the Residual Operator

Definition. A bounded operator $T$ is contractive if $\lVert T\rVert \leq 1$, equivalently $T(\mathcal{O}) \subseteq \mathcal{O}$; it is infinitesimal if $\lVert T\rVert < 1$, equivalently $T(\mathcal{O}) \subseteq \mathrm{M}$. The contractive operators form a subring $\mathcal{B}_1$ of the ring $\mathcal{B}$ of bounded operators, with the identity as unit, and the infinitesimal operators form a two-sided ideal of $\mathcal{B}_1$.

Proposition (contractive operators and the filtration). For a bounded $T$,

$$ \lVert T\rVert \leq 1 \iff T(\mathcal{O}) \subseteq \mathcal{O}, \qquad \lVert T\rVert < 1 \iff T(\mathcal{O}) \subseteq \mathrm{M} . $$

Consequently an infinitesimal operator satisfies $T(\mathrm{M}) \subseteq \mathrm{M}$ and has zero residual operator; a general contractive operator has a residual operator as soon as $T(\mathrm{M}) \subseteq \mathrm{M}$, which holds when $T$ is multiplicative, when $T$ is a scalar multiplication $x \mapsto cx$ with $c$ a unit, and more generally when $T$ carries the open unit ball into itself.

Proof. The equivalence of $\lVert T\rVert \leq 1$ with $T(\mathcal{O}) \subseteq \mathcal{O}$ is the definition of the operator norm as a supremum; the second equivalence is the same statement with a strict bound, since $\lVert T\rVert < 1$ means $\lvert Tx\rvert < 1$ for all $x$ of $\lvert x\rvert \leq 1$. An infinitesimal operator therefore carries $\mathcal{O}$ into $\mathrm{M}$, hence $\mathrm{M}$ into $\mathrm{M}$, and its residual operator is zero. A multiplicative operator carries $\mathrm{M}$ into $\mathrm{M}$ because the product of an element of $\mathrm{M}$ with any element of $\mathcal{O}$ lies in $\mathrm{M}$; a scalar multiplication by a unit reverses $\mathcal{O}$ and $\mathrm{M}$ onto themselves.

Definition. Let $T$ be a bounded operator with $T(\mathcal{O}) \subseteq \mathcal{O}$. If $T(\mathrm{M}) \subseteq \mathrm{M}$ then $T$ induces a well-defined additive map

$$ \bar T : k \longrightarrow k, \qquad \bar T(x + \mathrm{M}) = Tx + \mathrm{M} , $$

the residual operator of $T$. The assignment $T \mapsto \bar T$ is the reduction of the operator layer.

Proposition (the residual map is a ring homomorphism). On the contractive operators with $T(\mathrm{M}) \subseteq \mathrm{M}$ the assignment $T \mapsto \bar T$ is a ring homomorphism onto the additive endomorphisms of $k$ that it reaches, with $\overline{\mathrm{id}} = \mathrm{id}_k$ and $\overline{ST} = \bar S\bar T$; its kernel is the two-sided ideal of the infinitesimal operators, those with $T(\mathcal{O}) \subseteq \mathrm{M}$, equivalently $\lVert T\rVert < 1$.

Proof. If $x - x' \in \mathrm{M}$ then $Tx - Tx' = T(x - x') \in \mathrm{M}$, so $\bar T$ is well defined; it is additive because $T$ is. For the product, $\overline{ST}(x + \mathrm{M}) = STx + \mathrm{M}$ and $\bar S\bar T(x+\mathrm{M}) = \bar S(Tx + \mathrm{M}) = STx + \mathrm{M}$, so the maps agree, using $S(\mathrm{M})\subseteq \mathrm{M}$; the identity reduces to the identity. The kernel is the set of contractive $T$ with $T(\mathcal{O}) \subseteq \mathrm{M}$, that is the infinitesimal operators $\lVert T\rVert < 1$; this is a two-sided ideal because composition with a contractive operator preserves the condition and the sum of two infinitesimal operators is infinitesimal.

Corollary (the residual operator is the operator on the residue field). The residual operator is the operator-theoretic form of the residue map of The Residue Operator of a Valued Field: reduction of elements $x \mapsto x + \mathrm{M}$ is the case $T = \mathrm{id}$, and the residual operator of a product $ST$ is the product of the residual operators, so the reduction is a homomorphism of the operator layers that carries the contractive operators onto the operators of the residue field.

Proof. The case $T = \mathrm{id}$ gives $\bar T = \mathrm{id}_k$, which is the identity residue map; multiplicativity of the reduction is the proposition. The image is the set of induced maps, which is a subring of $\operatorname{End}(k)$.

Isometries

Definition. An additive operator $T$ is an isometry if $\lvert Tx \rvert = \lvert x \rvert$ for all $x$, equivalently if $\lVert T\rVert = 1$ and $T$ is injective; the isometries of $F$ onto itself form a group $\operatorname{Isom}(F)$, the isometry group. A scaling is a map $x \mapsto c x$ for $c \in F^\times$, and a rotation is a map $x \mapsto ux$ with $u \in \mathcal{O}^\times$.

Proposition (the isometry group). Every isometry preserves the valuation ring and the maximal ideal, $T(\mathcal{O}) = \mathcal{O}$ and $T(\mathrm{M}) = \mathrm{M}$, so it has a residual operator $\bar T$, and $T \mapsto \bar T$ is a homomorphism from $\operatorname{Isom}(F)$ onto a subgroup of the isometry group of $k$. The rotations form a subgroup $\mathcal{O}^\times$ of $\operatorname{Isom}(F)$ with residual image $k^\times$ acting by multiplication, and the scalings form a subgroup $F^\times$ of $\operatorname{Isom}(F)$ with residual image trivial on the classes of the value group.

Proof. An isometry maps $\{x : \lvert x\rvert \leq 1\}$ onto itself, giving $T(\mathcal{O}) = \mathcal{O}$, and maps $\mathrm{M}$ onto $\mathrm{M}$; hence it has a residual operator. The map $T\mapsto\bar T$ is a homomorphism by the previous section, and it takes values in the additive automorphisms of $k$ preserving the multiplication by the residue of the rotation, whence the statements about the two subgroups.

Proposition (the isometries are a closed subgroup). In the topology of uniform convergence on the balls, the isometry group $\operatorname{Isom}(F)$ is a closed subgroup of the group of homeomorphisms of $F$, and it is contained in the group of uniformly continuous bijections.

Proof. An isometry is uniformly continuous and bijective onto $F$ by the inverse function theorem for isometries of a complete field (the inverse is the isometry that reverses it); the isometry condition is closed, being the intersection of the closed conditions $\lvert Tx\rvert = \lvert x\rvert$ for all $x$, so the group is closed. The uniform continuity is immediate from the definition.

Spherical Completeness

Definition. A non-Archimedean field $F$ is spherically complete if every decreasing sequence of closed balls $B(a_0, r_0) \supseteq B(a_1, r_1) \supseteq \cdots$ has nonempty intersection. Equivalently, every nest of closed balls, any two of which are nested, has nonempty intersection.

Proposition (spherical completeness is stronger than completeness). A spherically complete field is complete; the converse fails, and $\mathbb{C}_p$ is complete but not spherically complete.

Proof. If $F$ were not complete there would be a Cauchy sequence with no limit; its tail balls form a nest with empty intersection, by the ultrametric estimate $\lvert x_m - x_n\rvert \leq \max_{n\leq k

Theorem (characterisations). For a non-Archimedean field $F$ the following are equivalent: $F$ is spherically complete; $F$ has no nontrivial immediate extension, where an immediate extension is one with the same value group and the same residue field; every additive operator from a subspace of a normed $F$-space to $F$ that is bounded extends to the whole space with the same norm (the non-Archimedean Hahn–Banach theorem, or Ingleton's theorem).

Proof sketch. The equivalence of the first two is the standard theorem of Krull: a maximal immediate extension of $F$ is obtained by adjoining the limits of the nests, and it is nontrivial exactly when a nest has empty intersection, so an immediate extension can be built from a nest with no common point and conversely a missing point of a nest yields an immediate extension. The third is Ingleton's theorem: spherical completeness is exactly the hypothesis that makes the ultrametric Hahn–Banach argument close, because the argument produces a nest of balls whose common point is the value to be assigned.

Corollary (spherical completeness is stable under the standard operations). A discretely valued complete field is spherically complete, so $\mathbb{Q}_p$ and $k((t))$ are; the completion of a spherically complete field is spherically complete; and a finite extension of a spherically complete field is spherically complete.

Proof. In a discretely valued complete field a nest of balls has radii taking finitely many values or tending to a limit in the value group, so it is eventually constant or its centres form a Cauchy sequence, whose limit lies in the intersection; hence the field is spherically complete. The completion statement is the same argument applied to the denser field; the finite-extension statement follows from the equivalence with the absence of immediate extensions, because an immediate extension of a finite extension restricts to an immediate extension of the base.

Remark (the boundary). Spherical completeness is used in the non-Archimedean theory of normed spaces — the Hahn–Banach extension theorem, the structure of the dual and the theory of orthogonal bases — which are the modules and vector spaces of this Part, treated by Topological Modules and Vector Spaces and Bounded Operators on a Topological Vector Space. This article only fixes the property and its operator-theoretic content; the normed-space theory that uses it is deferred, and the analytic theory over a non-Archimedean field is Part III.

Examples

Example ($\mathbb{Q}_p$). The value group is $\mathbb{Z}$, discrete, and $\mathbb{Q}_p$ is complete, so it is spherically complete; the residue field is $\mathbb{F}_p$, the contractive operators reduce to the additive operators of $\mathbb{F}_p$, and the rotations are $\mathbb{Z}_p^\times$.

Example ($k((t))$). Spherically complete, with the same argument; the residual operator of a contractive operator is an additive endomorphism of $k$, and the rotation by $u = 1 + t$ is a residue $1$ isometry.

Example ($\mathbb{C}_p$). Complete but not spherically complete; the value group is $\mathbb{Q}$ and the residue field is $\overline{\mathbb{F}_p}$, both infinite, and the nest of balls of radii tending to a limit outside the value group witnesses the failure. Consequently the non-Archimedean Hahn–Banach theorem does not apply over $\mathbb{C}_p$ in the same form.

Summary

On a non-Archimedean field the bounded additive operators are exactly the continuous ones, they carry the operator norm $\lVert T\rVert = \sup_{\lvert x\rvert\leq1}\lvert Tx\rvert$, submultiplicative under composition and attained on the unit ball for a discrete value group, and the contractive operators $\lVert T\rVert \leq 1$, equivalently $T(\mathcal{O}) \subseteq \mathcal{O}$, form a unital subring. A contractive operator with $T(\mathrm{M}) \subseteq \mathrm{M}$ induces a residual operator $\bar T$ on the residue field $k$, and $T \mapsto \bar T$ is a ring homomorphism whose kernel is the ideal of the infinitesimal operators, so the residue operator of a valued field extends from the elements to the operators. The isometries preserve $\mathcal{O}$ and $\mathrm{M}$, they form a closed subgroup $\operatorname{Isom}(F)$ containing the rotations $\mathcal{O}^\times$ and the scalings $F^\times$, and reduction maps them onto a subgroup of the operators of the residue field.

A non-Archimedean field is spherically complete when every nest of closed balls has nonempty intersection; this implies completeness, fails for $\mathbb{C}_p$, holds for the discretely valued complete fields $\mathbb{Q}_p$ and $k((t))$, and is characterised by the absence of a nontrivial immediate extension and by the validity of the non-Archimedean Hahn–Banach theorem. It is the property of the completeness of the nests, finer than the completeness of the sequences, and the normed-space theory that uses it is later in this Part.

Summary of Notation

Symbol	Meaning
$(F, \lvert \cdot \rvert)$	The non-Archimedean field
$\mathcal{O}$, $\mathrm{M}$, $k$, $\Gamma$	Valuation ring, maximal ideal, residue field, value group
$\pi$	A uniformiser when $\Gamma \cong \mathbb{Z}$
$\lVert T\rVert = \sup_{\lvert x\rvert\leq1}\lvert Tx\rvert$	The operator norm of a bounded additive $T$
$\mathcal{B}$, $\mathcal{B}_1$	Bounded operators, and the contractive subring $\lVert T\rVert\leq1$
$\bar T$	The residual operator on $k$, defined when $T(\mathrm{M})\subseteq\mathrm{M}$
$T\mapsto\bar T$	The reduction, a ring homomorphism with kernel the infinitesimal operators
$\operatorname{Isom}(F)$	The isometry group, closed in the homeomorphism group
$B(a,r)$	Closed ball of radius $r$ about $a$
Spherical completeness	Every nest of closed balls has nonempty intersection
Immediate extension	An extension with the same value group and residue field