SPECTRAL INTEGRATION AND SPECTRAL THEORY FOR NON-ARCHIMEDEAN BANACH SPACES

http://ijmms.hindawi.com

© Hindawi Publishing Corp.

SPECTRAL INTEGRATION AND SPECTRAL THEORY FOR NON-ARCHIMEDEAN BANACH SPACES

S. LUDKOVSKY and B. DIARRA

Received 22 January 2001 and in revised form 8 August 2001

Banach algebras over arbitrary complete non-Archimedean fields are considered such that operators may be nonanalytic. There are different types of Banach spaces over non- Archimedean fields. We have determined the spectrum of some closed commutative sub- algebras of the Banach algebra ᏸ(E)of the continuous linear operators on a free Ba- nach space E generated by projectors. We investigate the spectral integration of non- Archimedean Banach algebras. We define a spectral measure and prove several proper- ties. We prove the non-Archimedean analog of Stone theorem. It also contains the case of C-algebrasC_∞(X,K). We prove a particular case of a representation of aC-algebra with the help of aL(A, µ,ˆ K)-projection-valued measure. We consider spectral theorems for op- erators and families of commuting linear continuous operators on the non-Archimedean Banach space.

2000 Mathematics Subject Classification: 47A10, 47A25, 47L10.

1. Introduction. This paper is devoted to the non-Archimedean theory of spec- tral integration with the help of the projection-valued measure. Spectral integration plays a very important role in the theory of Banach algebras, theory of operators and has applications to the representation theory of groups and algebras in the clas- sical case of the field of complex numbers C[7, 8, 14, 13, 19, 23]. There are also several works about non-Archimedean Banach algebra theory, which show that there are substantial differences between the non-Archimedean and classical cases [3,5,6, 10,11,12,18,26,27,28,30]. In [3,30], analytic operators overCp were considered and the Shnirelman integration of analytic functions was used, which differs strongly from the non-Archimedean integration theory related to the measure theory [28]. In the non-Archimedean case, the spectral theory differs from the classical results of Gelfand-Mazur, because quotients of commutative Banach algebras over a fieldKby maximal ideals may be fieldsF, which containKas a proper subfield [28]. In general for each non-Archimedean fieldK, there exists its extensionFsuch that a fieldF ≠ K [4,25].

Ideals and maximal ideals of non-Archimedean commutativeE-algebras (seeSection 5.1.1) andC-algebras were investigated in [28,29]. In [5,6], it was shown that the fail- ure of the spectral theory in the non-Archimedean analog of the Hilbert space and it was shown that even symmetry properties of matrices lead to the enlargement of the initial field while a diagonalisation procedure. In [10,11,12], formulas of the spectral radius and different notions of spectrum and analysed some aspects of structures of non-Archimedean Banach algebras. In [28] and the references therein, general theory

of non-Archimedean Banach algebras and their isomorphisms was considered. It was introduced the notion ofC-algebras in the non-Archimedean case apart from the clas- sicalC^∗-algebras. There are principal differences in the orthogonality in the Hilbert space over C and orthogonality in the non-Archimedean Banach space. Therefore, symmetry properties of operators do not play the same role in the non-Archimedean case as in the classical case.

This paper treats another aspect of the non-Archimedean algebra theory and theory of operators. Banach algebras over arbitrary complete non-Archimedean fields are considered such that the operators may be nonanalytic. There are different types of Banach spaces over non-Archimedean fields. In Sections2, 3, and4, are considered specific spaces. InSection 5, are considered general cases.

Let Kbe a field. A non-Archimedean valuation on Kis a function | ∗ | :K→R such that

(1) |x| ≥0 for eachx∈K; (2) |x| =0 if and only ifx=0;

(3) |x+y| ≤max(|x|,|y|)for eachxandy∈K; (4) |xy| = |x||y|for eachxandy∈K.

The fieldKis called topologically complete if it is complete relative to the following metric:ρ(x, y)= |x−y|for eachx andy∈K. A topological vector spaceEoverK with the non-Archimedean valuation may have a norm ∗ such that its restriction on each one-dimensional subspace overKcoincides with the valuation| ∗ |. IfE is complete relative to such norm ∗ , then it is called the Banach space. Such fields and topological vector spaces are called non-Archimedean. An algebraX over Kis called Banach, if it is a Banach space as a topological vector space and the multiplica- tion in it is continuous such thatxy ≤ xyfor eachxandyinX. A finite or infinite sequence(xj:j∈Λ)of elements in a normed spaceEis called orthogonal, if

j∈Λαjxj =max(αjxj:j∈Λ)for eachαj∈Kfor which limjαjxj=0. We con- sider the infinite topologically complete fieldKwith the nontrivial non-Archimedean valuation.

A non-Archimedean Banach spaceE is said to be free if there exists a family(ej: j∈I)⊂Esuch that any elementx∈Ecan be written in the form of convergent sum x=

j∈Ixjej, that is, lim_j∈Ixjej=0 andx =sup_j∈I|xj|ej (see Section 2). In Section 3, ultrametric Hilbert spaces are considered. InSection 4, we have determined the spectrum of some closed commutative subalgebras of the Banach algebraᏸ^(E)of the continuous linear operators ofEgenerated by projectors.

Section 5is devoted to the spectral integration. We introduce another definition of E-algebras inSection 5.1apart from [29]. In Propositions5.2and5.3we have proved that they are contained in the class ofE-algebras andC-algebras considered in [28,29].

InSection 5.2, a spectral measure is defined. InSection 5.5,Lemma 5.5,Corollary 5.6, Proposition 5.7, andCorollary 5.9its several properties are proved. InTheorem 5.11 the non-Archimedean analog of Stone theorem is proved. It contains also the case ofC-algebrasC_∞(X,K). A particular case of a representation of aC-algebra with the help ofL(A, µ,ˆ K)-projection-valued measure is proved inTheorem 5.14. Spectral the- orems for operators and families of commuting linear continuous operators on a non-Archimedean Banach space are considered in Sections5.8and5.9.

2. Free Banach spaces

2.1. LetE be a free Banach space with an orthogonal base(ej:j∈I). The topo- logical dualE ofE is a Banach space with respect to the norm defined forx ∈E byx =sup_x≠0|x, x|/x. Forx ∈E andy∈E, we define an element(x ⊗y) of the Banach algebra of continuous linear operators ᏸ(E) on the spaceE by set- ting for x ∈ E, (x ⊗y)(x)= x, xy with norm x ⊗y = x y. If E is a free Banach space with base (ej :j∈I), any u∈ᏸ^(E)can be written as a point- wise convergent sumu=

(i,j)∈I×Iαije_j⊗ei. Hence lim_i∈Iαijei=0 for eachj ∈I.

Moreoveru =sup_i,j|αij|e_jei. Notice thate_j =1/ej. Letᏸ0(E)= {u:u=

(i,j)∈I×Iαije_j⊗ei∈ᏸ^{(E); lim}j∈Iαije_j=0 for eachi∈I}.

Theorem2.1. An algebraᏸ0(E)is a closed subalgebra inᏸ^(E)with the unit ele- ment ofᏸ^(E).

Proof. Letu, v∈ᏸ0(E),u=

(i,j)∈I×Iαije_j⊗ei, andv=

(i,j)∈I×Iβije_j⊗ei, then limi∈Iαijei=0=limi∈Iβijeifor eachj∈I, and limi∈Iαijei=0=limi∈Iβijeifor each j∈I. We haveu◦v=

(i,j)∈I×I(

k∈Iαikβkj)e_j⊗ei. Leti∈I, limk∈Iαike_k=0, that is, for eachε >0, there existsJε(i)a finite subset ofI such that for eachk∈Jε(i), αike_k< ε. Hence

k∈I

αikβkj

e_j

=

k∈Jε(i)

αikβkj

e_j+

k∈Jε(i)

αikβkj e_j

≤max

kmax∈J_ε(i)

αike_keiβkjeke_ie_j, sup

k∈Jε(i)

αikβkje_j

≤max

u max

k∈Jε(i)

βkje_jeke_i, εve_i .

(2.1) Since limj∈Iβkjej =0 for eachk∈Jε(i), we have limj∈I(

k∈Iαikβkj)e_j =0 for eachi∈I, thereforeu◦v∈ᏸ0(E). The identity map id being given by id=

i∈Ie_i⊗ei, we have αii=1 and αij =0 if i≠j. Therefore limiαijei=0 for each j ∈I, and limjαije_j =0 for each i∈ I. Hence id∈ᏸ0(E). Letu =

(i,j)∈I×Iαije_j⊗ei be in the closure ofᏸ0(E). For all ε >0, there existsuε=

(i,j)∈I×Iαij(ε)e_j⊗ei∈ᏸ0(E) such thatu−uε =supi,j|αij−αij(ε)|ejei< ε. Hence for alli, j∈I, we have

|αij|e_jei ≤max(ε,|αij(ε)|e_jei). We obtain limiαijei =0 for eachj∈Iand limjαije_j =0 for eachi∈I. Thereforeu∈ᏸ0(E)andᏸ0(E)is closed.

2.2. Suppose that the orthogonal basis is orthonormal, that is,ej =1 for each j∈I. Thenu=

(i,j)∈I×Iαije_j⊗ei∈ᏸ0(E), if and only if limiαij=0 for eachj∈I and limjαij = 0 for eachi∈I. Setting for u=

(i,j)∈I×Iαije_j⊗ei∈ ᏸ0(E), u^∗ =

(i,j)∈I×Iαjie_j⊗ei, we see thatu^∗∈ᏸ0(E), called the adjoint ofu. We verify easily the following proposition.

Proposition2.2. An elementu∈ᏸ^(E)has an adjointu^∗if and only ifu∈ᏸ0(E).

Letu, v∈ᏸ0(E), λ∈K. Then (u+λv)^∗=u^∗+λv^∗; (u◦v)^∗=v^∗◦u^∗; u^∗∗=u.

Moreover,u^∗ = u.

As usual, we say thatu∈ᏸ0(E)is normal (resp., unitary) ifu◦u^∗=u^∗◦u(resp., u◦u^∗=id=u^∗◦u). Anduis selfadjoint ifu=u^∗, this is equivalent here to say that the matrix ofuis symmetric.

Note2.3. (i) We haveu = u^∗. However, in generalu◦u^∗≠u². For ex- ample, ifI is the set of positive integers, andEwith orthogonal base(en:n≥1), let a, b∈K. The operatorudefined byu(e1)=ae1+be2,u(e2)=be1−ae2,u(e3)=ce3, and u(en)=0 forn≥4. We see thatu is selfadjoint. Ifi=√

−1∈K; then taking b=iaand|c|<|a|, we see thatu² = |c|²<|a|²= u².

(ii) It should be interesting to characterize the elements ofᏸ0(E)that are normal, unitary. Considering, whenever the base ofEis orthonormal, the bilinear formf on Edefined byf (x, y)=

i∈Ixiyi, we obtain that the above definition of an adjointu^∗ of an elementu∈ᏸ0(E)is equivalent to say thatf (u(x), y)=f (x, u^∗(y))for each xandy∈E. In fact, here the adjoint of an operator is its transposition. This example is related to ultrametric Hilbert spaces.

3. Ultrametric Hilbert spaces. For the so-called ultrametric Hilbert spaces we can also define the adjoint of an operator with respect to an appropriate bilinear symmet- ric form.

3.1. Remark and definition. Ochsenius and Schikhof write in [24] “as a slogan:

There are nop-adic Hilbert spaces.” Nevertheless we will give a definition ofp-adic Hilbert spaces (cf. [20,21] for some fields with infinite rank valuation). Letω=(ωi)_i≥0 be a sequence ofnonzero elements ofK. We consider the free Banach space Eω= c0(N,K, (|ωi|^1/2)i≥0)= {x:x=(xi)i≥0⊂K; limi→+∞|xi||ωi|^1/2=0}. Thenx=(xi)i≥0

∈Eω ↔lim_i→+∞x_i²ωi= 0. Settingei =(δi,j)_j≥0 (Kronecker symbol), we have that (ei: i≥0) is an orthogonal base of Eω: for allx∈ Eω, x =

i≥0xiei and x = sup_i≥0|xi|ei =sup_i≥0|xi||ωi|^1/2, in particular, ei = |ωi|^1/2 for eachi≥0. Let fω:Eω×Eω→Kbe defined by fω(x, y)=

i≥0ωixiyi. It is readily seen thatfω

is a bilinear symmetric form onEω, with|fω(x, y)| ≤ xy, that is, the bilinear formfωis continuous. Moreover,fωis nondegenerate, that is,fω(x, y)=0 for each y∈Eω⇒x=0. Furthermore,fω(x, x)=

i≥0ωix²_i andfω(ei, ej)=ωiδi,j foriand j≥0. The spaceEωis called ap-adic Hilbert space.

Note3.1. (i) It may happen that |fω(x, x)|<x² for some x∈ Eω and even worse,Eωcontains isotropic elementsx≠0, that is,fω(x, x)=0.

(ii) Let V be a subspace ofEω and V^⊥= {x∈Eω:fω(x, y)=0,for ally ∈V}. The fundamental property on subspaces of the classical Hilbert spaceH:V=V^⊥⊥⇒ V⊕V^⊥=Hfails to be true in thep-adic case. This explains the claim of Ochsenius and Schikhof.

Remark 3.2. A free Banach space E with an orthogonal base(ei:i≥0) can be given a structure of ap-adic Hilbert space if and only if there exists(ωi:i≥0)⊂K such thatei = |ωi|^1/2for each i≥0. Furthermore, ifKcontains a square of any of its element, then anyp-adic Hilbert is isomorphic, in a natural way, to the space c0(N,K).

Note3.3. Letu, v∈ᏸ(Eω); we haveu=

i,jαije_j⊗eiandv=

i,jβije_j⊗eiwith lim_i→+∞|αij||ωi|^1/2=0=lim_i→+∞|βij||ωi|^1/2for eachj≥0. Furthermore, the norm ofu∈ᏸ(Eω)is given by

u =sup

i,j

ωi^1/2αij

ωj^1/2 . (3.1)

The operatorvis said to be an adjoint ofuwith respect tofωfω(u(x), y)= fω(x, v(y)), for allx, y∈Eω. Sincefωis symmetric,uis an adjoint ofv.

Sincefωis nondegenerate, if an operatoruhas an adjoint, this adjoint is unique and is denoted byu^∗. Since(ei:i≥0)is an orthogonal base ofEω, we have thatv is an adjoint ofuif and only iffω((u(ei), ej)=fω(ei, v(ej)))for eachiandj≥0.

That is, fω(

k≥0αkiek, ej)=αjiωj=fω(ei,

k≥0βkjek)=βijωi, for alli, j ≥0 βij=ω⁻_i¹ωjαji, for alli, j≥0. Furthermore, we must have limi→+∞|βij||ωi|^1/2=0 for eachj≥0, that is,

i→+∞lim ωi^1/2ω⁻¹_j ωjαji=ωjlim

i→+∞ωi^−1/2αji=0, ∀j≥0. (3.2) Hence lim_i→+∞|ωi|^−1/2|αji| =0 for eachj≥0. We have proved the following theorem.

Theorem 3.4. Let (ωi)_i≥0 ⊂ K^∗ and Eω = c0(N,K, (|ωi|^1/2)_i≥0) be the p-adic Hilbert space associated withω. Letu=

i,jαijej⊗ei∈ᏸ(Eω). Thenuhas an ad- jointv=u^∗∈ᏸ^(Eω)if and only iflimj→+∞|ωj|^−1/2|αij| =0for eachi≥0. In this condition,u^∗=

i,jω⁻_i¹ωjαjiej⊗ei.

It follows from this theorem that not any continuous linear operator ofEωhas an adjoint: it is another difference with classical Hilbert spaces. Letᏸ0(Eω)= {u:u=

i≥0

j≥0αije_j⊗ei∈ᏸ^(Eω); limj→+∞|ωj|^−1/2|αij| =0,∀i≥0}. We remember that u=

i,jαije_j⊗ei∈ᏸ^(Eω)is equivalent to limi→+∞|ωi|^1/2|αij| =0 for eachj≥0. It is readily seen, as inTheorem 2.1, thatᏸ0(Eω)is a closed unitary subalgebra ofᏸ(Eω).

Corollary3.5. An elementu∈ᏸ(Eω)has an adjointu^∗if and only ifu∈ᏸ0(Eω).

Letu, v∈ᏸ0(Eω),λ∈K. Then(u+λv)^∗=u^∗+λv^∗;(u◦v)^∗=v^∗◦u^∗;u^∗∗=u.

Moreover,u^∗ = u.

Proof. We only prove thatu^∗ = u. Since foru=

i,jαije_j⊗ei∈ᏸ0(Eω), we haveu =sup_i,j(|ωi|^1/2|αij|/|ωj|^1/2)andu^∗=

i,jωjω⁻¹_i αjie_j⊗ei, we obtain u^∗=sup

i,j

ωi^1/2

ωj^1/2ωjω⁻¹_i αij=sup

i,j

ωj^1/2

ωi^1/2αji= u. (3.3) Remark3.6. (i)u=

i,jαije_j⊗ei∈ᏸ0(Eω)is selfadjoint, that is,u=u^∗ if and only ifαji=ωiω⁻¹_j αij, for eachi≥0 and eachj≥0.

(ii) Examples of selfadjoint operators on ultrametric Hilbert spaces and study of their spectrum are given in [1,2,6,22].

4. Closed subalgebras generated by projectors

4.1. LetJ be a subset ofI and E be a free Banach space with orthogonal basis (ej:j∈I). The linear operatorpJ=

i∈Je_i⊗eiofEbelongs toᏸ0(E). LetᏰ= {u:u=

i∈Iλie_i⊗ei∈ᏸ0(E); sup_i∈I|λi|<+∞}. It is clear thatᏰis isometrically isomorphic to the algebra of bounded families^∞(I,K). Let Homalg(Ᏸ,K)denotes the family of all algebra homomorphisms ofᏰintoK. Consider the spectrumᐄ(Ᏸ)=Homalg(Ᏸ,K) in a topology inherited from the Tihonov topology of the productK^Ᏸof copies ofK.

Proposition4.1. (i)An elementu=

i∈Iλie_i⊗ei∈Ᏸis an idempotent if and only if there existsJ⊂Isuch thatu=pJ.

(ii)The spectrumᐄ(Ᏸ)is homeomorphic to the subset of ultrafilters onI:Φc= {ᐁ: ᐁis an ultrafilter onI,such that for allu=

i∈Iλie_i⊗ei∈Ᏸ,the limit lim_ᐁλi exists inK}.

Proof. (i) Letu=

i∈Iλie_i⊗ei; then u◦u=u if and only if

i∈Iλi2

e_i⊗ei=

i∈Iλie_i⊗ei, if and only ifλ²_i=λifor eachi∈I, if and only ifλi=0 orλi=1. Setting J= {i:i∈I;λi=1}, we haveu=pJ.

(ii) Letχbe a character ofᏰ, that is, an algebra homomorphism (necessarily contin- uous) ofᏰintoK. For allJ, L⊂Iwe havepJ◦pL=pJ∩L, hencepJ◦pJ^c=p_∅=0, where J^c=I\J. Furthermore,χ(pJ)=χ(pJ)χ(pJ)implies thatχ(pJ)=0 or 1. Letᐁχ= {J: J⊂I;χ(pJ)=1}. This family of subsets is an ultrafilter. Indeed,∅ ∈ᐁχ. IfJ⊂Lwith J∈ᐁχ, then 1=χ(pJ)=χ(pJ∩L)=χ(pJ)χ(pL)=χ(pL), henceL∈ᐁχ. On the other hand, forJ⊂I, we have 1E=pJ+pJ^c, and 1=χ(1E)=χ(pJ)+χ(pJ^c)withχ(pJ)=1 or 0 andχ(pJ^c)=1 or 0. Ifχ(pJ)=1, thenχ(pJ^c)=0, and ifχ(pJ^c)=1, we haveχ(pJ)= 0. HenceJ∈ᐁχorJ^c∈ᐁχ. Letu=

i∈Iλie_i⊗ei∈Ᏸ. Putχ(u)=λ∈K; then for all J∈ᐁχ,χ(up_J)=χ(u)=λχ(pJ). Thereforeχ(up_J−λpJ)=0, that is, up_J−λpJ∈kerχ.

Setφ_ᐁχ(u)=lim_ᐁχ|λi|. It is well known and readily seen thatφ_ᐁχ is a multiplicative semi-norm onᏰand that kerφ_ᐁχ= {u:u∈Ᏸ;φ_ᐁχ(u)=0}is a maximal ideal ofᏰ, sinceᏰis isomorphic to^∞(I,K). On the other hand|χ(up_J)| ≤ up_J =sup_i∈J|λi|for eachJ∈ᐁχ. It follows that|χ(u)| = |χ(up_J)| ≤inf_J∈ᐁχsup_i∈J|λi| =φ_ᐁχ(u). Hence, kerφ_ᐁχ⊂kerχand kerφ_ᐁχ=kerχ. LetJ∈ᐁχ, we deduce from(up_J−λpJ)∈kerχ= kerφ_ᐁχ, that 0=φ_ᐁχ(up_J−λpJ)=lim_ᐁχ|λi−λ|. It follows that lim_ᐁχλi=λexists in K. Moreover,χ(u)=λ=lim_ᐁχ|λi|, and we see thatχ=χ_ᐁχ. Reciprocally, ifᐁis an ultrafilter onIsuch that for allu=

i∈Iλie_i⊗ei∈Ᏸ, lim_ᐁλiexists inK; then setting χ_ᐁ(u)=lim_ᐁλi, it is readily seen thatχ_ᐁis a character ofᏰ. Moreover, for allJ∈ᐁ^, χ_ᐁ(pJ)=lim_ᐁ1=1, that is,J∈ᐁχ_ᐁ andᐁ=ᐁχ_ᐁ. The proposition is proved if we consider onᐄ(Ᏸ)the weak^∗-topology and onΦcthe topology induced by the natural topology on the space of ultrafilters, which is the weakest topology onΦcrelative to which the mapping lim :Φc→Kis continuous.

Remark4.2. (i) IfKis locally compact, then for any bounded family(λi)_i∈I⊂K, the limit lim_ᐁλiexists inK. Therefore,Φc is equal to the entire set of all ultrafilters onIandᐄ⁽Ᏸ⁾is compact, homeomorphic to the Stone-ˇCech compactificationβ(I)of the discrete topological spaceI.

(ii) IfKis not spherically complete andIis a small set, that is, the cardinal ofIis nonmeasurable, it is well known that the continuous dual of^∞(I,K)is equal to the

spacec0(I,K)of the families converging to zero (cf. [28, Theorem 4.21]). Then, we can prove thatᐄ(Ᏸ)is homeomorphic withI.

Note4.3. ForKspherically complete, not locally compact, it is interesting to find explicit conditions on an ultrafilterᐁin such a way that lim_ᐁλiexists for any bounded family(λi:i∈I)⊂K. We can try to use Banach limits, that is, continuous linear forms on^∞(I,K)that extend the usual continuous linear form of the limit operation defined on the subspacecv(I,K)of convergent families.

Let(Jν:ν∈Λ)be a family of subsets ofI, such thatJν∩Jµ= ∅forν≠µ. Putting pν=

i∈Jνe_i⊗ei, we obtainpν◦pµ=δν,µpν, forν≠µ. Hence the subalgebra with the unityᏮ^of ᏸ0(E), generated by(pν:ν∈Λ) is equal toK·id⊕(⊕ν∈ΛK·pν). In- deed ifu=α0id+u1 and v=β0id+v1with u1=

ν∈Λανpν andv1=

ν∈Λβνpν

(finite sums), we haveu◦v=α0β0id+α0v1+β0u1+u1◦v1=α0β0id+

ν∈Λ(α0βν+ ανβ0+ανβν)pν∈Ꮾ. On the other hand, sinceu=α0id+

ν∈Λpν withΓ = {ν:ν∈ Λ;αν≠0}finite andI=(

ν∈ΓJν) (

ν∈ΓJνc)(a partition), we haveu=α0

i∈Ie_i⊗ ei+

ν∈Γαν

i∈Jνe_i⊗ei= α0

i∈∩ν∈ΓJνce_i⊗ei+

ν∈Γ

i∈Jν(α0+αν)e_i⊗ei. Hence u =max(|α0|,maxν|α0+αν|).

Lemma 4.4. Let u = α0id+

ν∈Λανpν ∈ Ꮾ ^and Λ0 = Λ∪ {0}. Then u = max_ν∈Λ0|αν|. That is,{id}∪{pν:ν∈Λ}is an orthonormal family inᏸ0(E).

Proof. Since

u =maxα0,max

ν∈Λα0+αν, maxν α0+αν≤maxα0,max

ν∈Λαν.

(4.1)

We haveu ≤maxν∈Λ0|αν|. Moreover,|α0| ≤ u. Hence forν∈Λ, we have|αν| =

|αν+α0−α0| ≤max(|αν+α0|,|α0|)≤ u. It follows that max_ν∈Λ0|αν| ≤ u, and Lemma 4.4is proved.

Lemma4.5. Assume that(ei:i∈I)is an orthonormal basis ofEorEis an ultra- metric Hilbert space. Then anyu∈Ꮾis selfadjoint, that is,u^∗=u, andu² = u².

Proof. That any element of Ꮾ is selfadjoint is easy to verify. Let u= α0id+ ν∈Λανpν ∈Ꮾ^{, we have u2} =α02id+

ν∈Λ(2α0αν+αν2)pν ∈Ꮾ^{. Hence} u² = max(|α0|²,max_ν∈Λ|α02+2α0αν+αν2|) = (max(|α0|,max_ν∈Λ|α0+αν|))² = u².

Note4.6. In fact,Lemma 4.5is true foru∈Ᏸ^{. LetE}be a free Banach space with orthogonal basis(ei:i∈I). Fixπ∈Ksuch that 0<|π|<1. There exists for any i∈Ian integerni∈Zsuch that|π|ⁿⁱ⁺¹<ei ≤ |π|ⁿⁱ. Forx=

i∈Ixiei, we have limi∈Ixiπⁿⁱ =0. Hence we define on E a norm by setting xπ =sup_i∈I|xi||π|ⁿⁱ; this norm is equivalent to with|π|xπ≤ x ≤ xπ. Furthermore, settingx=

i∈Ixieiandy=

i∈Iyiei∈E,fπ(x, y)=

i∈Iπ²ⁿⁱxiyi, we have a continuous, non degenerated, bilinear form onE such that|fπ(x, y)| ≤ xπyπ≤ |π|⁻²xy. Therefore, we obtain onE, a structure of ultrametric Hilbert spaceEπ=(E, π, fπ).

Since the norms and π are equivalent,ᏸ(E)=ᏸ(Eπ)and ᏸ0(E)=ᏸ0(Eπ).

The norms onᏸ(E)induced by and π are equivalent with|π|uπ ≤ u ≤

|π|⁻¹uπ. As in Note 3.3, we define the adjoint u^∗ ofu ∈ᏸ(E) with respect to fπ. We obtain the results stated in Theorem 3.4, that is,u admits an adjoint with respect tofπ if and only ifu∈ᏸ0(E). Furthermore, ifu=

i,jαijej⊗ei∈ᏸ0(E), thenu^∗ =

i,jπ^nj−niαjiej⊗ei, anduis selfadjoint, that is,u^∗ =uif and only if πⁿⁱαij=π^njαji, for alli, j∈I.

Note4.7. Letπ be another element ofKsuch that 0<|π|<1. Also let(mi:i∈ I)⊂Z be defined by|π |^mi+1<ei ≤ |π |^mi. Then the adjoint u^†=

i,jπ^mj−mi αjiej⊗ei of u with respect to fπ coincides with u^∗ if and only if π^nj−niαji = π ^mj−miαji, for eachiandj∈I. If this is true for allu∈ᏸ0(E), we haveπ^nj−ni= π ^mj−mi, fori, j∈I. Hence, log|π|/|log|π | =(mj−mi)/(nj−ni)=m/n >0 and the sets(mj−mi)i≠jand(nj−ni)i≠jmust be finite.

IfJis a subset ofI, the projectorpJ=

i∈Je_i⊗eiis selfadjoint with respect to any bilinear symmetric formfπ andpJ =1= pJπ.

Lemma4.8. LetEbe a free Banach space with orthogonal basis(ei:i∈I). Defining an adjoint of a continuous operator with respect tofπ, then anyu∈Ꮾ(resp.,Ᏸ) is selfadjoint andu² = u².

Proof. It is the same as inLemma 4.5. Since for any u=α0id+

ν∈Λανpν∈Ꮾ we have u =max_ν∈Λ0|αν|, that is, {id, pν :ν ∈ Λ}is an orthonormal family in ᏸ0(E), we see that the closureᏭ=ᏮofᏮis the subspace ofᏸ0(E)of all elementsu which can be written in the unique form of summable familiesu=α0id+

ν∈Λανpν

withα0, αν∈Kand limναν=0. It is readily seen that Ꮽis a closed unitary subal- gebra ofᏸ0(E), contained inᏰ, such that any elementuofᏭis selfadjoint. More- over for the pointwise convergence,u=α0

i∈∩J^c_νe_i⊗ei+

ν∈Λ

i∈J_ν(α0+αν)e_i⊗ ei. Hence, if

ν∈ΛJ_ν^c = ∅, thenu=

ν∈Λ

i∈Jν(α0+αν)e_i⊗ei and id=

ν∈Λpν. Example4.9. IfΛ=IandJi= {i}for eachi∈I, we haveᏭ= {α0id+

i∈Iαie_i⊗ei: αi∈K,lim_i∈Iαi=0}. As an element ofᏰanyu∈Ꮽis in the formu=

i∈Iaie_i⊗ei

with lim_i∈Iai=α0exists inK.

Proposition4.10. (i)Any elementuof the Banach algebraᏭwith the unit element idis selfadjoint with respect to any bilinear symmetric formfπ andu² = u².

(ii)The spectrumᐄ(Ꮽ)=Homalg(Ꮽ,K)ofᏭ, equipped with the weak^∗-topology, is homeomorphic to the Alexandroff compactification of the discrete spaceΛ.

Proof. The first part is an easy consequence ofLemma 4.8. Letχ∈ᐄ⁽Ꮽ^{), thenχis} a continuous linear form with normχ =1. Furthermore,χ(id)=1 andχ(pνpµ)= χ(pν)χ(pµ)=δν,µχ(pν), forν, µ∈Λ. It follows that for anyν ∈Λ, χ(pν)=1 or χ(pν)=0. Hence (a) there existsν∈Λsuch thatχ(pν)=1 andχ(pµ)=0 forµ≠ν, or (b)χ(pν)=0 for allν∈Λ. In case (a), we putχ=χν and in case (b),χ=χ0. We verify that foru=α0id+

ν∈Λανpν∈Ꮽ, we haveχ0(u)=α0andχν(u)=α0+αν, ν∈Λ. It follows thatᐄ(Ꮽ)= {χ0, χν:ν∈Λ}andᐄ(Ꮽ)is in a bijective correspondence with the setΛ0=Λ∪ {0}. LetW (χ;ε, u1, . . . , un)= {η:η∈ᐄ⁽Ᏸ^);|χ(uj)−η(uj)|< ε,

uj ∈ Ꮽ,1≤ j ≤ n} be a fundamental neighborhood of χ∈ ᐄ(Ꮽ) for the weak^∗- topology. Since foruj=α0jid+

µ∈Λαµjpµ∈Ꮽ, lim_µ∈Λαµj=0, there exists a finite subsetΓεofΛ, such that for anyµ∈Γε,|αµj|< εfor each 1≤j≤n. Ifχ=χν,ν∈Λ, we have for 1≤j≤n,µ∈Λ,χν(uj)−χµ(uj)=ανj−αµj. Choosing(uj: 1≤j≤n) such thatεν=min_1≤j≤n|ανj|>0, there existsΓν⊂Λ, Γν finite such that|αµj|< εν

for 1≤j≤nand for allµ∈Γν. Hence|αµj|<|ανj|and|ανj−αµj| = |ανj| ≥εν, for 1≤j≤nandµ∈Γν. Therefore, ifε < εν, thenW (χν;ε, u1, . . . , un)= {χν}, that is,{χν} is open inᐄ(Ꮽ). Hence{χν:ν∈Λ}is a discrete subset ofᐄ(Ꮽ). On the other hand, ifχ=χ0, thenχ0(uj)−χµ(uj)= −αµj. Hence forε >0, there exists a finite subsetΓε

ofΛsuch that forµ∈Γε,|χ0(uj)−χµ(uj)| = |αµj|< εfor each 1≤j≤n. In other words,W (χ0;ε, u1, . . . , un)= {χµ:µ∈Γε}. Furthermore, χ0=lim_µ∈Λχµ in ᐄ(Ꮽ)for the weak^∗-topology. It follows thatᐄ(Ꮽ)is weak^∗-compact. Consider onΛ0=Λ∪{0}

the topology such thatΛis a discrete subset ofΛ0and the neighborhoods of 0 are W_Γ(0)=Λ0\Γ, whereΓ⊂Λis finite. It becomes clear thatΛ0is homeomorphic to the Alexandroff compactification of the discrete spaceΛ. Identifyingᐄ(Ꮽ)with Λ0, we conclude the proof of the proposition.

4.2. Let Ꮿ(ᐄ(Ꮽ),K) be the K-Banach algebra of the continuous functions f on the compact spaceᐄ(Ꮽ)with values inK. It is readily seen that f∈Ꮿ(ᐄ(Ꮽ),K)is defined by the family(f (χν):ν∈Λ0)⊂Ksuch that limν∈Λf (χν)=f (χ0). Hence Ꮿ(ᐄ(Ꮽ),K)is isometrically isomorphic to the algebracv(Λ0,K)= {a:a=(aν:ν∈ Λ0)⊂K; limν∈Λaν=a0}: oncv(Λ0,K), we consider the usual multiplication defined pointwise and the norm(aν:ν∈Λ0) =sup_ν∈Λ0|aν|.

Corollary4.11. The Banach algebraᏭwith the unit elementidis isometrically isomorphic to the algebracv(Λ0,K)= {a:a=(aν)_ν∈Λ0⊂K; lim_ν∈Λaν=a0}.

Proof. LetᏳ:Ꮽ→Ꮿ(ᐄ(Ꮽ),K)be the Gelfand transformᏳ(u)(χ)=χ(u). As usual, Ᏻis continuous. Since foru=α0id+

ν∈Λανpν∈Ꮽ^{, we haveχ}0(u)=α0andχν(u)= α0+αν, ν∈Λ, and obtainu =max(|χ0(u)|,sup_ν∈Λ|χν(u)|)=sup_{χ∈ᐄ(Ꮽ)}|χ(u)|. Hence, Ᏻ(u) = u. Furthermore, Ᏻ(id)(u)=1, that is, Ᏻ(id)=f0 the constant function equal to 1. On the other hand, for ν ∈Λ, Ᏻ^(pν)(χ)=1 if χ=χν and 0 otherwise. Hence, setting forν∈Λ,fν:ᐄ(Ꮽ)→Ksuch thatfν(χµ)=δν,µ,µ∈Λ, we haveᏳ(pν)=fν. Letu=α0id+

ν∈Λανpν∈Ꮽ, we haveᏳ(u)=α0f0+

ν∈Λανfν. Since anyf∈Ꮿ⁽ᐄ⁽Ꮽ^),K)can be written in the unique convergent sumf=f (χ0)f0+

ν∈Λ(f (χν)−f (χ0))fν with lim_ν∈Λ(f (χν)−f (χ0))=0, we havef=Ᏻ(u)withu= f (χ0)id+

ν∈Λ(f (χν)−f (χ0))pν. Hence,Ᏻis surjective. Together withᏳ(u) = u, the corollary is proved.

5. Spectral integration

5.1. Suppose thatXand Y are Banach spaces over a topologically complete non- Archimedean fieldKwith a nontrivial valuation andᏸ(X, Y )denotes the Banach space of bounded linear operators E : X → Y supplied with the operator norm: E := sup_0≠x∈XExY/xX. ForX=Y we denoteᏸ^{(X, Y )simply by}ᏸ^{(X). LetX andY} be isomorphic with the Banach spacesc0(α,K)andc0(β,K)and let them be supplied