Surjective isometries between unitary sets of unital JB

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Linear Algebra and its Applications

www.elsevier.com/locate/laa

Surjective isometries between unitary sets of unital JB

-algebras

María Cueto-Avellaneda^a, Yuta Enami^b, Daisuke Hirota^b, Takeshi Miura^c, AntonioM. Peralta^d,∗

aSchoolofMathematics,StatisticsandActuarialScience,UniversityofKent, Canterbury,KentCT27NX,UK

bGraduateSchoolof ScienceandTechnology,NiigataUniversity,Niigata 950-2181,Japan

cDepartmentofMathematics,FacultyofScience,NiigataUniversity,Niigata 950-2181,Japan

dInstitutodeMatemáticasdelaUniversidaddeGranada(IMAG),Departamento deAnálisisMatemático,FacultaddeCiencias,UniversidaddeGranada,18071 Granada,Spain

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received31May2021 Accepted3February2022 Availableonline7February2022 SubmittedbyP.Semrl

MSC:

primary47B49,46B03,46B20, 46A22,46H70

secondary46B04,46L05,17C65

Keywords:

Isometry

Jordan^∗-isomorphism Unitary

JB^∗-algebra JBW^∗-algebra Extensionofisometries

Thispaperis,inafirststage,devotedtoestablishingatopolo- gical–algebraic characterizationof the principalcomponent, U⁰(M ),ofthesetofunitaryelements,U(M),inaunitalJB^∗- algebraM .Wearrivetotheconclusionthat,asinthecaseof unitalC^∗-algebras,

U⁰(M ) = M₁⁻¹∩ U(M)

=



U_eihn· · · Ue^ih1(1) : n∈ N, hj∈ Msa

∀ 1 ≤ j ≤ n



=

u∈ U(M) : there exists w ∈ U⁰(M ) with

u − w < 2

is analytically arcwise connected. Actually, U⁰(M ) is the smallestquadraticsubsetof U(M) containingthesete^iMsa. Oursecondgoalistoprovide acompletedescriptionof the

* Correspondingauthor.

E-mailaddresses:[email protected],[email protected](M. Cueto-Avellaneda), [email protected](Y. Enami),[email protected](D. Hirota), [email protected](T. Miura),[email protected](A.M. Peralta).

https://doi.org/10.1016/j.laa.2022.02.003

0024-3795/©2022TheAuthor(s). PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

surjective isometries between the principal components of two unitalJB^∗-algebras M andN .Contrary to thecaseof unitalC^∗-algebras,weshalldeducetheexistenceofconnected componentsinU(M) whicharenotisometricasmetricspaces.

We shall also establish necessary and sufficient conditions to guarantee thata surjectiveisometry Δ:U(M)→ U(N) admitsan extensionto a surjectivelinearisometrybetween M and N , a conclusion which is not always true. Among the consequencesit isproved thatM and N areJordan ^∗- isomorphic if, and only if, their principal components are isometric as metric spaces if, and only if, there exists a surjectiveisometryΔ:U(M)→ U(N) mappingtheunitof M toanelementinU⁰(N ).Theseresultsprovideanextension to thesettingofunitalJB^∗-algebrasof theresultsobtained byO.HatoriforunitalC^∗-algebras.

©2022TheAuthor(s).PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Why arewesoattracted bytheunitarygroupofunital C^∗-algebras?Thecelebrated Russo–Dye theorem,assertingthat theconvex hull ofthe unitary elements inaunital C^∗-algebrais norm denseinits closed unitball (cf. [47]),is probably oneofthe eldest toolsleadingourattentiontotheunitarygroup.Itisperhapsunnecessarytorecallthat anelement,u,inaunitalC^∗-algebra,A,liesinthesubgroup,U(A),ofallunitariesinA ifuu^∗= u^∗u= 1.Aswehavealreadyadvanced,U(A) isasubgroupofA containingthe unit element. Applications of the Russo-Dye theorem and subsequent generalizations have been employed along the last fifty five years, recent usages on problems about extensions ofisometriesappear,forexample,in[45] and [6,15].

The unitary groupof aunital C^∗-algebrawas employed as acomplete invariant to classify von Neumann factors and certain unital C^∗-algebras. Highlighting the group structure ofthe set of unitaries, S.Sakai established thatifA and B are AW^∗-factors and ρ : U(A) → U(B) is a uniformly continuous group isomorphism between their respectiveunitary groups,then thereisauniquemap T fromA ontoB which iseither a linear or a conjugate linear ^∗-isomorphism and agrees with ρ on U(A) (see [48]). A similar conclusionwas provedby H.A.Dyefor W^∗-factorsin[19] withanindependent argument whichalsogivesresultswhennocontinuityassumptionismadeonthegroup isomorphism.Morerecently,Al-Rawashdeh,BoothandGiordanoshowthatiftheunitary groupsoftwo simpleunitalAH-algebrasofslowdimensiongrowthandofrealrankzero are isomorphic as abstract groups, then their K₀-ordered groups are isomorphic (cf.

[3]).

In asurprising turn, O. Hatori and L. Molnár considered the set U(A), of unitary elements in a unital C^∗-algebra A, as a metric space equipped with the metric given by the C^∗-norm and forgot about its group structure (see [32]). It should be noted that the metric space (U(A),· ) is almost never compact (cf. [50, Exercise II.2.1]).

Inthe caseof von Neumann algebras,themetric space givenby the set ofunitaries is acomplete invariant since, as shownby Hatori and Molnár, every surjective isometry between the unitary groups of two von Neumann algebras admits an extension to a surjectivereallinearisometrybetweenthesealgebras (see[32,Corollary3]).Actuallyif Δ: U(A)→ U(B) is asurjective isometry betweenthe sets ofunitary elements oftwo unitalC^∗-algebras,thenthereisacentralprojectionp∈ B andaJordan^∗-isomorphism J : A→ B satisfying

Δ(e^ix) = Δ(1)(pJ (e^ix) + (1− p)J(e^ix)^∗),

for allx∈ Asa (see [32, Theorem 1]). Inparticular A and B are Jordan ^∗-isomorphic.

Here,and henceforth,theself-adjoint partofaC^∗-algebraA willbedenotedbyAsa. The framework of von Neumann algebras is a very favorable scenario since, among other things, the set of unitaries in a von Neumann algebra W is precisely the set exp(iW_sa),whichisananalyticallyarcwiseconnectedset(cf.[38,Theorem5.2.5]).How- ever,it iswell knownthatinthecaseof ageneralunital C^∗-algebraA theset U(A) is notalwaysconnected(cf. [50,ExerciseI.11.3],[38, Exercises4.6.6and4.6.7], [32,pages 162-164],[28]). Theconnected componentofU(A) containingtheunitelementiscalled theprincipalcomponent,anditisdenotedbyU⁰(A).

HatoriandMolnárpointedoutin[32,Corollary8] theexistenceofsurjectiveisometries betweenthesetsofunitariesintwounitalcommutativeC^∗-algebraswhichdonotadmit an extension to asurjective real linear isometry between these algebras. The problem ofdetermining thestructure ofallsurjective isometriesbetweenthesets ofunitariesin two arbitraryunital C^∗-algebras wascarried out byO.Hatori in[28], where two main resultsareestablished:SupposeA and B aretwounital C^∗-algebras.

(a) SupposeΔ: U⁰(A)→ U⁰(B) isamapping. ThenΔ is asurjective isometryif and onlyifthereexistacentralprojectionp∈ B andaJordan^∗-isomorphismJ : A→ B satisfying

Δ(a) = Δ(1)



pJ (a) + (1− p)J(a)^∗ ,

for all a ∈ U⁰(A). In particular A and B are Jordan ^∗-isomorphic. The principal componentsU⁰(A) andU⁰(B) canbesomehowreplacedbyanytwoconnectedcom- ponents(see[28,Theorem3.1 andCorollary3.5]).

(b) EachsurjectiveisometryΔ:U(A)→ U(B) canbewrittenasadirectsumofafamily ofsurjectivereallinearisometries betweenA and B restrictedto thecorresponding connectedcomponents,andtheextensibilityofΔ toasurjectivereallinearisometry isonlypossibleunderadditionalconditions([28,Theorem 4.1andCorollary5.1]).

From a strict mathematical point of view, C^∗-algebras are contained in thestrictly widerclassofJB^∗-algebras,aclassofJordan-Banachalgebras determinedbyanappro- priate version ofthe Gelfand-Naimark axiom.Theclass of JB^∗-algebras is nowadaysa

consolidated object of study, whose origins go back to the works of P. Jordan, J. von Neumann, E.Wigner,I.Kaplansky,J.D.M. Wright,M.Youngson,H.Hanche-Olsen,E.

Størmer,E.M.AlfsenandF.W.Shultz,amongothers.Detaileddefinitionsandcomplete sourcesofreferencescanbefoundinsubsection1.1andinthemonographs[1,2,26,11,12].

The notions of invertibility, spectrum and unitaries admit aperfect translation to the setting of unital JB^∗-algebras, andthere is aversion of theRusso–Dye theorem estab- lishedby J.D.M.Wrightand M.Youngson (seesubsection1.1for moredetails).In the caseofaunitalJB^∗-algebraM ,thesubset,U(M),ofallunitariesinM isnotaJordan- sub-semigroup ofM –actuallytheJordanproductoftwounitaryelementsneednotbe, ingeneral,aunitaryelement eveninthecaseof unitalC^∗-algebras–.So,intheJordan setting wecannolongerspeakabouttheunitarygroup.

Foranycoupleofelementsa,b inaJordanalgebraM ,weshallwriteUa,b andMafor thelinearoperatorsonM definedby

U_a,b(x) = (a◦ x) ◦ b + (b ◦ x) ◦ a − (a ◦ b) ◦ x, and Ma(x) = a◦ x

forallx∈ M.WeshallsimplywriteUaforUa,a.IfM isobtainedfromanassociativeal- gebra(withproductdenotedbyjuxtaposition)equippedwithitsnaturalJordanproduct a◦ b= ¹₂(ab+ ba),itcanbeeasily checkedthatU_a,b(x)= ¹₂(axb+ bxa) (a,b,x∈ M).

In the same way that a celebrated theorem, due to Sakai, identifies von Neumann algebras with those C^∗-algebras which are dual Banachspaces, in the setting of JB^∗- algebras, those which are dual Banach spaces are called JBW^∗-algebras and enjoy additional geometric properties. For example, each JBW^∗-algebra contains a unit ele- ment, and every unitary element in a JBW^∗-algebra M is of the form exp(ih), where h lies in the self-adjoint part, Msa, of M . In [16], the first and fifth authors of this note extendedthepreviously-commented resultbyHatoriandMolnár,and provedthat the set of unitaries in a JBW^∗-algebra is a complete invariant too. More concretely, suppose Δ : U(M) → U(N) is asurjective isometry, where M and N aretwo JBW^∗- algebras. Thenthere exist aunitary ω inN ,acentralprojection p∈ N,and aJordan

∗-isomorphismΦ: M → N suchthat

Δ(u) = U_ω∗(p◦ Φ(u)) + Uω^∗((1_N− p) ◦ Φ(u)^∗)

= P₂(U_ω∗(p))U_ω∗(Φ(u)) + P₂(U_ω∗(1_N− p))Uω^∗(Φ(u^∗)),

forallu∈ U(M).Consequently,Δ admitsa(unique)extensiontoasurjectivereallinear isometry fromM ontoN (see[16,Theorem 3.9]).

Since unital C^∗-algebras are unital JB^∗-algebras when equipped with the natural Jordan product a◦ b= ¹₂(ab+ ba),andthe sameinvolution andnorm; and whenboth structuresarepossible,thenotionsofunitarycoincide,weknowthatthesetofunitaries inaunitalJB^∗-algebraM neednotbe,ingeneral,connected.Theconnectedcomponent of U(M) containing the unit element will be also called the principal component, and will bedenotedbyU⁰(M ).

Despitenowadayswehaveexcellentandthoroughacademicmonographscoveringthe theoryofJB^∗-algebras (cf., forexample,[1,2,14,26] and therecentbooks[11,12]),some geometricaspectshavenotbeenexploredyet.Thisisthecaseoftheprincipalcomponent of the set of unitaries. Section 2is aimed to complete our knowledge onthe principal component.ThestartingpointisaresultbyO.Hatoriwhichassertsthatforeachunital C^∗-algebraA wehave

U⁰(A) ={e^ih1· . . . · e^ihn1e^ihn· . . . · e^ih1 : n∈ N, h1, . . . , h_n ∈ Asa}

is open, closed, and pathconnected, inthe (relative) normtopology onU(A) (see [28, Lemma3.2]). Theadvantageofthepreviousdescriptionisthatitadmits anexpression intermsofexpressionsgivenbyJordanproducts.InTheorem2.3weprovethatforeach unitalJB^∗-algebraM , theprincipalcomponentoftheset ofunitariesinM admitsthe followingdescription

U⁰(M ) = M₁⁻¹∩ U(M)

={Ue^ihn· · · Ue^ih1(1) : n∈ N, hj ∈ Msa ∀ 1 ≤ j ≤ n}

=

u∈ U(M) : there exists w ∈ U⁰(M ) withu − w < 2,

whereM₁⁻¹ standsfortheprincipalcomponentofthesetM⁻¹ofallinvertibleelements inM . Consequently,U⁰(M ) is analyticallyarcwise connected. A central notionin the theory of Jordan algebras has already appeared inthe previous description where the Uaoperatorassociatedwithanelement a inaJordanalgebraM is definedby

U_a(x) = 2(a◦ x) ◦ a − (a ◦ a) ◦ x, (x ∈ M).

When an associative algebra A is regarded as a Jordan algebra with respect to the naturalJordan productwehaveU_a(x)= axa for alla,x∈ A.

WeshouldadmitthattheinstabilityofU(M) underJordanproductsisahandicapto findadescriptionofthissetintermsofalgebraicproperties.Asimilarinstabilityaffects thesetofinvertible elementsinaunital Jordan-Banachalgebra.Inany case,given two invertible(respectively,unitary)elementsa,b inaunitalJordan-Banachalgebra(respec- tively,inaunitalJB^∗-algebra)M theelementU_a(b) isinvertible(respectively,unitary).

Let M be a unital Jordan-Banach algebra. Following [46], we shall say that a subset M⊆ M⁻¹isaquadraticsubset ifUM(M)⊆ M.LetusobservethatU(M) isaself-adjoint subsetofM wheneverM isaunitalJB^∗-algebra,insuchacasewealsoprovethatU⁰(M ) isthesmallestquadraticsubsetofU(M) containingthesete^iMsa (seeProposition2.6).

Surjectiveisometries between different connected components of the unitary sets of two unital JB^∗-algebras are studiedin section 3. In afirst result we establishthat for each surjective isometry Δ : U⁰(M ) → U⁰(N ) between the principal components of twounital JB^∗-algebras,there exist k₁,. . . ,k_n ∈ Nsa,acentralprojectionp∈ N anda Jordan^∗-isomorphismΦ: M→ N suchthat

Δ(u) = p◦ Ue^ikn. . . U_eik1Φ(u) + (1_N− p) ◦ (Ue^−ikn. . . U_e−ik1Φ(u))^∗,

for all u∈ U⁰(M ). Consequently, M and N areJordan ^∗-isomorphic, and there exists a surjective real linear isometry (i.e., a real linear triple isomorphism) from M onto N whose restriction to U⁰(M ) is Δ (see Theorem 3.4). Surjective isometries between connected components of the unitary sets of two unital JB^∗-algebras are described in Theorem 3.6.

Corollary 3.8in[16] provesthattwo unitalJB^∗-algebrasM andN areisometrically isomorphicas(complex)Banachspacesif,andonlyif,theyareisometricallyisomorphic as real Banach spaces if, and only if, there exists a surjective isometry between their unitary sets. Corollary 3.5 in this note improves this conclusion by showing that the following statementsareequivalent:

(a) M and N areJordan ^∗-isomorphic;

(b) ThereexistsasurjectiveisometryΔ:U(M)→ U(N) satisfyingΔ(1_M)∈ U⁰(N );

(c) ThereexistsasurjectiveisometryΔ:U⁰(M )→ U⁰(N ).

TheextendibilityofasurjectiveisometrybetweentheunitarysetsoftwounitalJB^∗- algebras isdiscussedinsection3.1(seeCorollary3.8 andProposition3.10).

We cannotconcludethissection withoutpointingouttheexceptional particularities naturally linked to the category of JB^∗-algebras. For example, ina unital C^∗-algebra A, anyconnectedcomponentU^c(A) ofU(A) isisometricallyisomorphictotheprincipal componentU⁰(A) –itsufficestoconsidertheleftorrightmultiplicationoperatoronA by anelementinU^c(A).InthesettingofunitalJB^∗-algebrasthisgeometricpropertyisnot always true,combiningour resultswithaconstructiondue to R.Braun,W.Kaupand H. Upmeier in[8], we shall exhibit anexample of aunital JB^∗-algebraM admitting a unitaryw suchthattheconnectedcomponentofU(M) containingw isnotisometrically isomorphictotheprincipalcomponent(seeRemark3.7).Thiscounterexamplereinforces thefactthataunitalJB^∗-algebraadmitsmanydifferentJordanproductsandinvolutions, atleastasmanyasunitaries,andsomeofthemproducestructureswhicharenotJordan

∗-isomorphic.However,byafascinatingresultduetoW.Kaup,eachJB^∗-algebraadmits an essentially unique structure of JB^∗-triple (see [39, Proposition 5.5] and subsection 1.1).ThisisessentiallythereasonforwhichweemployJB^∗-tripletheoryinsomeofour arguments.

1.1. Definitions andbackground

A complex (respectively, real) Jordan algebra M is a (non-necessarily associative) algebraoverthecomplex(respectively,real)fieldwhoseproduct“◦”iscommutativeand satisfies theso-calledJordan identity:

(a◦ b) ◦ a²= a◦ (b ◦ a²) (a, b∈ M). (1)

AnormedJordanalgebra isaJordanalgebraM equippedwithanorm,.,satisfying

a◦b≤ ab (a,b∈ M).AJordan-Banachalgebra isanormedJordanalgebrawhose norm is complete. Every real or complex associative Banach algebrais areal Jordan- Banachalgebrawithrespect to itsnaturalJordan product.AllJordan algebras inthis paperareassumedto beunital.

TheJordanidentityimpliesthatJordan algebrasarepowerassociative,thatis,each subalgebragenerated byasingleelement a inaJordan algebraM isassociative.More concretely,letusseta⁰= 1,a¹= a,andaⁿ⁺¹= a◦ aⁿ (n≥ 1).Inthiscasetheidentity a^m+n= a^m◦ aⁿ,holds forallnaturalnumbersn,m (cf. [26,Lemma2.4.5]).Inthecase thatM isaJordan-Banachalgebra,thecompletenessofthenormassuresthattheseries exp(a)= e^a=

∞ n=0

aⁿ

n! is uniformly convergentonbounded subsetsof M . Furthermore, themappinga→ e^adefinesananalyticmappingonM .TheJordan-Banachsubalgebra C ofM generatedbyanelementa isacommutativeassociativesubalgebrawithrespect to theinherited Jordan product [36, 1.1].SupposeA is anassociative Banachalgebra.

For each element a ∈ A, the powers of A with respect to the associative and to the Jordan product coincide, and thus e^a is just the usual exponential in the usual sense for associative Banach algebras. It follows that exp(a) has its usual meaning in the Jordan-BanachsubalgebraC,and,inparticularwehave

e^sa◦ e^ta= e^(s+t)a, for all s, t∈ C. (2) Twoelementsa,b inaJordanalgebraM aresaidtooperatorcommute iftheidentity (a◦ c)◦ b= a◦ (c◦ b) holdsforallc∈ M (cf.[26,4.2.4]).Thecentre ofM isformed by allelementsinM which operatorcommutewithanyother elementinM .

InaJordanalgebratheleft or rightmultiplication operatorbyafixedelement does not play the role played by the left or right multiplication in an associative Banach algebra.TheprotagonistintheJordansettingisassociatedwiththeU operator.LetM beaJordan-Banachalgebra.OneofthefundamentalidentitiesinJordantheoryassures that

UUa(b)= UaUbUa, for all a, b in a Jordan algebra M (3) (see[26,2.4.18]).Itcanbe easilyproved,viaaninductionargument,thatthefollowing generalisationof(3) holdsineveryJordanalgebraM :

U_{Uan···Ua1(b)}= U_an· · · Ua1U_bU_a1· · · Uan, (4) foralln> 0,andalla_n,. . . ,a₁,b∈ M.

An element a in a unital Jordan-Banach algebra M is called invertible whenever there exists b ∈ M satisfying a◦ b = 1 and a²◦ b = a. The element b is unique and it will be denoted by a⁻¹ (cf. [26, 3.2.9] and [11, Definition 4.1.2]). We know from [11, Theorem 4.1.3] thatan element a∈ M isinvertible if andonly ifUa is abijective

mapping,andinsuchacaseU_a⁻¹= U_a−1 (cf.(3)).IfanassociativeBanachalgebraA is regardedwithitsnaturalJordanproduct,thenotionofinvertibilityintheJordansetting ispreciselytheusualnotionintheassociativecontext.Asintheassociativesetting,the setM⁻¹ofallinvertibleelementsinM isopen(cf.[11,Theorem4.1.7]),anditsprincipal component(i.e. theconnectedcomponentcontainingtheunitelement) willbe denoted byM₁⁻¹.

AdditionalgeometricaxiomsarerequiredtodefineJB- andJB^∗-algebras.Thedefini- tionofJB^∗-algebrasisessentiallyduetoI.Kaplansky,whointroducedthemasaJordan generalization of C^∗-algebras. A JB^∗-algebra is a complex Jordan-Banach algebra M equipped withanalgebrainvolution ^∗ satisfyingthefollowing geometricaxiom:

Ua(a^∗) = a³, (a∈ M). (5) Every C^∗-algebra A is a JB^∗-algebra when equipped with its natural Jordan product and theoriginalnormandinvolution.Weobserve thatinthis casethegeometricaxiom (5) writes in the form aa^∗a = a³ (a ∈ A), which is known to be an equivalent reformulationof theGelfand-Naimark axiom.M.A.Youngson provedin[54, Lemma4]

thattheinvolutionofeveryJB^∗-algebraisinfactanisometry.

A JB-algebra is areal Jordan-BanachalgebraJ in whichthe norm satisfies thefol- lowing twoaxiomsforalla,b∈ J:

(i) a²=a²; (ii) a²≤ a²+ b².

As pointed outbyKaplansky,the hermitianpart, Msa,of aJB^∗-algebra,M , isalways a JB-algebra. The converse implication was open for sometime, and it was shown to be trueinacelebratedresultduetoJ.D.M. Wright,assertingthatthecomplexification of every JB-algebrais a JB^∗-algebra(see [51]). The readerwho mayfeel the necessity of additionaldetailsand explanations forany ofthe basicresultsonJB-algebras,JB^∗- algebras and JB^∗-triples (whosedefinition appearsbelow) canconsultthemonographs [26,14,1,2,11] and [12].

Let M and N be JB^∗-algebras. A linear mapping Φ : M → N is called a Jordan homomorphism ifΦ(a◦ b)= Φ(a)◦ Φ(b) (a,b∈ M),andaJordan^∗-homomorphism ifit isaJordanhomomorphismandΦ(a)^∗= Φ(a^∗) foralla∈ M.AJordan^∗-isomorphismis aJordan^∗-homomorphismwhichisalsoabijection.AreallinearmappingfromM toN preservingJordanproductswillbecalledareallinearJordanhomomorphism.Reallinear Jordan^∗-homomorphismsaresimilarlydefined.AmappingbetweenunitalJB^∗-algebras is calledunital ifitsendstheunitinthedomaintotheunitinthecodomain.

A JBW^∗-algebra is a JB^∗-algebra which is also a dual Banach space. Norm-closed Jordan ^∗-subalgebras of C^∗-algebras are called JC^∗-algebras. JC^∗-algebras which are also dual Banach spaces are called JW^∗-algebras. Any JW^∗-algebra is a weak^∗-closed Jordan^∗-subalgebraofavonNeumannalgebra.Thereadershouldbewarnedaboutthe

existenceofexceptionalJB^∗-algebraswhichcannotbeembeddedasJordan^∗-subalgebras ofsomeB(H) (see[26,Corollary 2.8.5],[11,Example 3.1.56]).

Werecallthatanelement u inaunital JB^∗-algebraM is aunitary if itisinvertible anditsinversecoincideswithu^∗.Asintheassociativesetting,weshalldenotebyU(M) thesetofallunitaryelementsinM .Itisknownthattheidentities

Uu(u^∗) = u, and Uu((u^∗)²) = 1,

holdforeveryunitaryelementu inM (cf.[11,Theorem4.1.3]).Furthermore,M becomes aunital JB^∗-algebra,withunitu,fortheJordanproduct andinvolutiondefinedby

x◦uy := Ux,y(u^∗) and x^∗u := Uu(x^∗), respectively. (6) ThisnewunitalJB^∗-algebra(M,◦u,∗u) iscalledtheu-isotope ofM , andwillbedenoted byM (u) (see[11,Lemma4.2.41(i)]).Foreacha∈ M,weshallwrite a^(n,u) forthen-th powerofa intheJB^∗-algebraM (u).LetusobservethatifaunitalC^∗-algebraisregarded asaJB^∗-algebrabothnotionsofunitariescoincide.Anelements inaunitalJB-algebra J iscalled a symmetry ifs² = 1_J. IfM is aunital JB^∗-algebra,the symmetriesin M aredefinedasthesymmetriesinitsself-adjointpartM_sa.

In a JBW^∗-algebra M the set U(M) is path connected and coincides with the set {e^ih : h∈ Msa} (cf.[16,Remark3.2(9)]).However,asinthecaseofunitalC^∗-algebras,in ageneralunitalJB^∗-algebrathesetofunitariesmightcontainmanydifferentconnected components.TheprincipalcomponentofU(M) istheconnectedcomponentcontaining theunit,and itwillbedenotedbyU⁰(M ).

Beside the deepconnections with holomorphic theory onarbitrary complexBanach spacesandtheclassificationofboundedsymmetricdomainsinthiscomplexsettingwhich ledW.KauptointroducethosecomplexBanachspacescalledJB^∗-triplesin[39],forthe purposesofthisnote,weshallemploytheJB^∗-triplestructureassociatedwitheachJB^∗- algebrato simplifyallpossible Jordanproducts and involutionsunderatriple product which encompasses all of them. A JB^∗-triple is a complex Banach space E equipped with a continuous triple product {., ., .} : E × E × E → E, (a,b,c) → {a,b,c}, which isbilinear and symmetricin(a,c) andconjugate linearinb,andsatisfies the following axiomsforalla,b,x,y∈ E:

(a) L(a,b)L(x,y)= L(x,y)L(a,b)+L(L(a,b)x,y)−L(x,L(b,a)y),whereL(a,b): E → E is theoperatordefinedbyL(a,b)x={a, b, x};

(b) L(a,a) isahermitian operatorwith non-negativespectrum;

(c) {a,a,a}=a³.

ExamplesofJB^∗-triplesincludeallC^∗-algebrasandJB^∗-algebraswiththetripleprod- uctsoftheform

{x, y, z} = 1

2(xy^∗z + zy^∗x), (7)

and

{x, y, z} = (x ◦ y^∗)◦ z + (z ◦ y^∗)◦ x − (x ◦ z) ◦ y^∗, (8) respectively ([11, Fact4.1.41 and Theorem 4.1.45]).Actually, when aC^∗-algebrais re- gardedasaJC^∗-algebrawithitsnaturalJordanproduct,theexpressionin(8) coincides with thetripleproductin(7).Theproductin(7) isactuallyvalid todefine astructure of JB^∗-triple on spacesB(H,K) of bounded linear operators betweencomplex Hilbert spaces (infinitedimensionalexamplesof rectangularmatrices),complexHilbertspaces, and J^∗-algebras inthe sense introduced by L.Harris in [27]–i.e. closed complex-linear subspacesofB(H,K) whichareclosed forthetripleproduct in(7)–.

Thetripleproduct ofeveryJB^∗-tripleisanon-expansivemapping,thatis,

{a, b, c} ≤ abc for all a, b, c (9) (see [25,Corollary3]).

A (real linear)triple homomorphism betweenJB^∗-triplesE and F is a(real) linear mappingΦ: E → F preservingtripleproducts,i.e.,

T{a, b, c} = {T (a), T (b), T (c)}, for all a, b, c ∈ E.

It iswell knownthateveryunital triplehomomorphism betweenunital JB^∗-algebrasis aJordan^∗-homomorphism.

AmongtheamazinggeometricpropertiesofthosecomplexBanachspacesintheclass ofJB^∗-tripleswefindaBanach-Stonetypetheorem,provedbyW.Kaup,assertingthat a linear bijection T betweentwo JB^∗-triples is anisometry ifand only if it is atriple isomorphism(cf. [39,Proposition5.5],seealso [11,Theorem 2.2.28] or[18] and[23] for alternative proofs). Thanks to this result we can understand now that, despite of the existenceofmanydifferentJordanproductsandinvolutionsonaunitalJB^∗-algebra,they allproducethesametriplestructure.Namely,ifu1isanyunitaryinaunitalJB^∗-algebra M , weknowthatthetripleproduct onM givenin(8) coincideswiththeonegivenby

{x, y, z}1= (x◦u1y^∗u1)◦u1z + (z◦u1y^∗u1)◦u1x− (x ◦u1z)◦u1y^∗u1, becausetheidentitymappingisasurjectivelinearisometry.

It shouldberemarkedthatKaup’sBanach–Stonetheoremisnot,ingeneral,truefor surjective real linearisometries between JB^∗-triples (cf. [17, Remark 2.7],[33, §4] and [22]). However, surjective real linear isometries between C^∗-algebras and JB^∗-algebras areallreallineartripleisomorphisms(see[17,Corollaries3.2and3.3] and[22,Corollary 3.4]). Wefurtherknow that everyunital surjective real linearisometry between unital JB^∗-algebrasisareallinearJordan^∗-isomorphism[17,Corollary3.2].

Foreachcoupleofelementsa,b inaJB^∗-tripleE,wedenotebyQ(a,b) theconjugate linear operatoronE defined byQ(a,b)(x)={a,x,b}.Weshall write Q(a) forQ(a,a).

InthecaseofaJB^∗-algebraM ,theQ operatorisintrinsicallyrelatedtotheU operator bytheidentity

Q(a, b)(x) ={a, x, b} = Ua,b(x^∗), for all x∈ M.

We have already mentioned that the involution of every unital JB^∗-algebra M is anisometry. We canactually apply theprevious Kaup’sBanach-Stone theorem to the involution as asurjective linear isometry from M ontothe Banach space M equipped withthesamenormbutreplacingtheproductbyscalarsbytheonegivenbyλ x= λx, oritcanbedirectlycheckedthattheinvolutionisa(conjugatelinear)tripleisomorphism.

Consequently,foranya andb inM ,we havethat

(U_a(b))^∗= ({a, b^∗, a})^∗={a^∗, b, a^∗} = Ua^∗(b^∗). (10) Byaninductionargumentwehave

(Uan· · · Ua1(a0))^∗= Ua^∗_n· · · Ua^∗₁(a^∗₀), (11) forallnaturaln andan,. . . ,a1,a0∈ M.

Each element e in a JB^∗-triple E satisfying {e, e, e} = e is called a tripotent. If a C^∗-algebra A is regarded as a JB^∗-triple with the triple product in (7), the partial isometries in A are precisely the tripotents of A. Each tripotent e ∈ E determines a Peirce decomposition ofE in theform

E = E2(e)⊕ E1(e)⊕ E0(e),

whereEj(e)={x∈ E : {e, e, x} = ^j₂x} (j = 0,1,2)iscalled thePeircej-subspace.

TripleproductsamongelementsinPeircesubspacessatisfythefollowingPeircearith- metic:

{Ei(e), Ej(e), Ek(e)} ⊆ Ei−j+k(e) if i− j + k ∈ {0, 1, 2}, {Ei(e), Ej(e), Ek(e)} = {0} if i − j + k /∈ {0, 1, 2},

and {E2(e), E₀(e), E} = {E0(e), E₂(e), E} = {0}. Consequently, each Peirce subspace E_j(e) isaJB^∗-subtripleofE.

Thenaturalprojection,Pk(e),of E ontoEk(e) iscalled thePeircek-projection.It is knownthatP2(e)= Q(e)²,P1(e)= 2(L(e,e)−Q(e)²),andP0(e)= IdE−2L(e,e)+Q(e)². Furthermore,Pk(e)≤ 1 forallk = 0,1,2 (cf.[24,Corollary1.2]).Itisworthremarking thatif e is atripotent in aJB^∗-triple E, the Peirce 2-subspace E2(e) is aunital JB^∗- algebrawithunite,productx◦ey :={x, e, y} andinvolutionx^∗e :={e, x, e},respectively (cf.[11,Theorem 4.1.55]).

The adjective “unitary” is also employed to denote some special tripotents. More precisely,atripotent e inaJB^∗-tripleE is calledunitary if E2(e)= E.However,there

is no risk of controversy, since, as shown by R. Braun, W. Kaup and H. Upmeier in [8, Proposition 4.3], the unitaries ina unital JB^∗-algebraM are precisely the unitary tripotents inM whenthelatteris regardedas aJB^∗-triple.Wecantherefore conclude thateverysurjective reallinearisometry T betweentwo unital JB^∗-algebras M andN mapsU(M) ontoU(N).Inparticular,foreachunitaryu∈ M,themappingUu: M → M is atripleisomorphismsatisfyingUu(U(M)) = U(M).

2. Rangetripotentsandthe principalcomponent oftheunitarysetina JB^∗-algebra Theprincipalcomponentsofthesets ofinvertible elements,A⁻¹, andoftheunitary group, U(A), of aunital C^∗-algebra A are the connected components of these subsets containing the unit 1 ∈ A. These principal components will be denoted by A⁻¹₁ and U⁰(A),respectively.Thesetwo setsarewell known,studiedanddescribed inthelitera- ture.Moreconcretely,forany(associative)unitalrealorcomplexBanachalgebraA,A⁻¹ isanopensubsetofA whichisstableunderassociativeproducts.Theprincipalcompo- nentof A⁻¹ ispreciselytheleast subgroupofA⁻¹ containingexp(A)={e^a: a∈ A},it is pathconnected,anditcanbe algebraicallydescribedinthefollowingterms:

A⁻¹₁ ={e^a1· . . . · e^an : n∈ N, a1, . . . , a_n∈ A}

(see [7,Propositions8.6and8.7],[50,page12] and [42,Theorem2.3.1]).

TheprincipalcomponentoftheunitarygroupofaunitalC^∗-algebraA hasbeenalso studiedanddescribed bydifferentauthors.Itis knownthat

U⁰(A) ={e^ih1· . . . · e^ihn: n∈ N, h1, . . . , hn∈ Asa}

={e^ih1· . . . · e^ihn1e^ihn· . . . · e^ih1 : n∈ N, h1, . . . , hn ∈ Asa} (12) is open,closed, and pathconnected,inthe(relative)norm topologyonU(A) –thefirst equality in(12) canbe foundinKadison–Ringrosebook[38, Exercises4.6.6and4.6.7], while thesecond equality,which is closer to the Jordan structure,is due to O.Hatori [28,Lemma3.2].Anotherinterestingresultassertsthat,bythecontinuityofthemodule mappingx→ |x|:= (x^∗x)¹²,we have

A⁻¹₁ ∩ U(A) = U⁰(A) (see [50, Exercise 2, page 56]).

Thereadershouldbewarnedthat“polardecompositions”arenotavailableforelements in a JB^∗-algebra, so these arguments are not valid inthe Jordan setting. To the best of our knowledge, inthe setting of real C^∗-algebras a full description of the principal componentoftheunitarygroupremainsunknown.

In thecaseof aunitalC^∗-algebraA, thesets A⁻¹, A⁻¹₁ , U(A),U⁰(A) are closedfor theassociativeproductbuttheyarenot,ingeneral,stableunderJordanproducts.

In the case of Jordan-Banach algebras, B. Aupetit posed the question whether the principalcomponentofthesetofinvertibleelementsinaJordan-Banachalgebraisana- lyticallyarcwiseconnectedin[4,commentsafter theproofofTheorem3.1].Acomplete positive solution to this question was found byO. Loos in[43] (see also [11, Theorem 4.1.111]).However,aslongasweknowthedescriptionoftheprincipalcomponentofthe setofunitariesinaunitalJB^∗-algebrashasnotbeenexplicitlytreatedintheliterature.

Byemployingsometoolsintherecentnote[16],weshallcompletethestate-of-the-artin theJordansetting.OneofthemaingeometricpropertiesderivedfromthelocalGelfand theory in C^∗-algebras assures that for each unitary u in a unital C^∗-algebra A with

1 − u < 2 we can always find a hermitian element h ∈ Asa satisfying u= e^ih (see [38,Exercise4.6.6]).BuildinguponthecelebratedShirshov-Cohntheorem,whichproves thatthe JB^∗-subalgebra of a JB^∗-algebragenerated by two self-adjoint elements (and the unit element) is a JC^∗-algebra, that is, a JB^∗-subalgebra of some B(H) (cf. [26, Theorem 7.2.5] and [51, Corollary 2.2]), thejust commentedresult has been extended tothesettingofunital JB^∗-algebrasin[16,Lemma2.2].

LetM be aunital JB^∗-algebra. Letus consider aunitary element u in M , and the u-isotope M (u) := (M,◦u,∗u) as defined insubsection 1.1. It is known that U(M)= U(M(u)) (cf. [16, Lemma 2.1], [49, Lemma 4.2(ii) and Theorem 4.6] or [11, Lemma 4.2.41(ii)]). Let us observe thatthe unital JB^∗-algebras M and M (u) share the same underlying Banach spaces, the quoted unitary sets are equipped with the same norm anddistance.Let ussuppose thatu liesinU⁰(M ).Since theconnected componentsof atopological spacearedisjoint,wecanaffirmthat

U⁰(M (u)) =U⁰(M ), for every u∈ U⁰(M ). (13) Section3belowisdevotedtostudyinganddescribingthesurjectiveisometriesbetween theprincipalcomponentsoftheunitarysetsoftwounitalJB^∗-algebras.Forthispurpose, weshallfirstestablishanalgebraiccharacterizationofthesetofunitaryelementsinterms ofU operators ofexponentials ofskew symmetricelements,like theonecommentedin thecaseof unitalC^∗-algebras(cf.(12)).

Let us recall the local Gelfand theory to the readers. In a C^∗-algebra, the C^∗- subalgebraA generatedbyanon-normalelementcannotbeidentified,viaGelfandtheory, withacommutativeC^∗-algebraoftheformC0(L),foralocallycompactHausdorffspace L.However,ifweregardA asaJB^∗-triple, theJB^∗-subtriplegenerated byasingleele- mentisalwaysrepresentable as aC₀(L) space.Let0= a beanelementinaJB^∗-triple E. The JB^∗-subtriple of E generated by a (denoted by E_a) coincides with the norm closure of the linear span of the odd powers of a defined as a^[1] = a, a^[3] ={a, a, a}, anda^[2n+1]:=

a, a, a^[2n−1]

,(n∈ N). ThelocalGelfandtheory forJB^∗-triplesassures that Ea is isometrically JB^∗-triple isomorphic to some C0(Ωa) for a unique compact HausdorffspaceΩ_acontainedintheset[0,a],suchthat0 cannotbeanisolatedpoint in Ω_a, where, along this note, the symbol C₀(Ω_a) will stand for the Banach space of allcomplex-valuedcontinuous functionsonΩa vanishing at0 in case that0∈ Ωa, and

all continuous functionson Ωaotherwise. It is furtherknownthatthere exists atriple isomorphism Ψ : Ea → C0(Ωa) satisfying Ψ(a)(t) = t (t ∈ Ωa) (cf. [40, Lemma 3.2, Corollary 3.4 and Proposition3.5],[39, Corollary 1.15],see also [11, Theorem4.2.9] or [14,Theorem 3.1.12]).The set Ω_a is called thetriple spectrum of a (inE),and it does not changewhencomputed withrespect to any JB^∗-subtripleF ⊆ E containing a [40, Proposition 3.5].In some reference, like [14, Theorem 3.1.12] and [11, Theorem 4.2.9], thelocalGelfandtheoryisbuiltuponthelocallycompactsetΩ\{0}.Herewefollowthe notationin[40],where,inamorenaturalterminology,thetriplespectrumisacompact set.

As in the associative setting, the local Gelfand theory opens the door to apply a continuous triple functional calculus.Given an elementa in aJB^∗-tripleE, letΩaand Ψ: Ea→ C0(Ωa) denotethetriplespectrumofa andthetripleisomorphismgiveninthe previous paragraph.Foreachcontinuous functionf ∈ C0(Ω_a),weset f_t(a):= Ψ⁻¹(f ), andwecallitthecontinuoustriple functionalcalculus off ata.Ifp(λ)= α1λ+ α3λ³+ . . . + α2n−1λ²ⁿ⁻¹ isanoddpolynomialwithcomplexcoefficients,itiseasytocheckthat pt(a)= α1a+ α3a^[3]+ . . . + α2n−1a^[2n−1].However,formoregeneralfunctionsinC0(Ωa) the continuous triple functional calculus is not so obvious. For example if a is a non- self-adjoint elementinaC^∗-algebraA,theelement a^[2]= h_t(a) withh(t)= t² doesnot coincidewith a² inA. Anotherexample:thefunctiong(t)=√³

t producesgt(a)= a^[13]. Letusnotethatgt(a)= a^[13] istheuniquecubic root ofa inEa,i.e.theuniqueelement a^[13]∈ Easatisfying

a^[13], a^[13], a^[13]

= a. (14)

The sequence (a^[3n1])n can be recursivelydefined by a^{[3n+11 ]} =

 a^{[3n1 ]}

[¹₃]

, n ∈ N.It is easy to see thatthe sequence (a^{[3n1 ]})_n need notbe, in general, norm convergent in E (consider,forexample,theelementa(t)= t inE = C([0,1])).

The functional analysis provides another topology to assure the convergence of the above sequence. Let us recall that a JBW^∗-triple is a JB^∗-triple which is also a dual Banachspace(withauniqueisometricpredual[5]).ThetripleproductofeveryJBW^∗- tripleisseparatelyweak^∗-continuous(cf.[5]),andtheseconddual,E^∗∗,ofaJB^∗-tripleE isalwaysaJBW^∗-tripleunderatripleproductwhichextendstheoriginaltripleproduct inE (cf.[12, Proposition5.7.10]).

IfW isaJBW^∗-triple(inparticularwhenweregardE asaJB^∗-subtripleofE^∗∗),the sequence(a^[3n1])_n convergesintheweak^∗-topologyofW toa(unique)tripotentdenoted byr(a).Thetripotentr(a) iscalledtherange tripotent ofa in W .Thisrangetripotent r(a) canbecharacterizedas thesmallesttripotent e∈ W satisfyingthata ispositivein theJBW^∗-algebraW2(e) (compare[20,Lemma3.3]).

Givenanelementa inaJB^∗-tripleE,weshalldenotebyr(a) therangetripotentofa inE^∗∗.There existexamplesinwhichwecanguaranteethattheelementr(a) liesinE.

Accordingtotheusualnotationfollowedin[10,21,40,37],anelementa inaJB^∗-tripleE

iscalledvonNeumann regular ifa∈ Q(a)(E);ifa∈ Q(a)²(E) wesaythata isstrongly vonNeumannregular.ForeachvonNeumannregularelement a inaJB^∗-tripleE there mightexistmanydifferentelementsc inE satisfyingQ(a)(c)= a.However,itisknown (cf.[21,Theorem1] and[40,Lemma4.1,Lemma3.2andcommentsafteritsproof],[10, Theorem2.3and Corollary2.4])thatthefollowing statementsareequivalent:

(a) a isvonNeumannregular;

(b) There exists b ∈ E such that Q(a)(b) = a, Q(b)(a) = b and [Q(a),Q(b)] :=

Q(a)Q(b)− Q(b)Q(a)= 0;

(c) 0 doesnotbelongtothetriplespectrum,Ω_a,ofa;

(d) Q(a) hasnorm-closedrange;

(e) Therangetripotent r(a) of a lies inE anda is positive andinvertible inthe JB^∗- algebraE₂(r(a)).

The element b appearing in statement (b) above is unique, and it will be called the generalized inverse of a inE (denoted by a^†).Henceforth we shallwrite Reg_vN(E) for theset of all von Neumannregular elements inE. It is furtherknownthatL(a,a^†)= L(a^†,a)= L(r(a),r(a)) and Q(a)Q(a^†)= Q(a^†)Q(a)= P2(r(a)) (see, forexample,[37, page589],[10,commentsinpage192] or[41,Lemma3.2andsubsequentcomments]).

Ifa is aninvertible element inaunital JB^∗-algebraM andthelatter isregardedas aJB^∗-triple, thena is von Neumann regularand itsrange tripotent,r(a),is aunitary elementinM ,thatis,

a∈ M⁻¹⇒ r(a) ∈ U(M) (cf. [37, Lemma 2.2 and Remark 2.3]). (15) Namely,inthiscase,bydenotingz⁻¹ theinverseofz inM we have

Q(z)((z⁻¹)^∗) ={z, (z⁻¹)^∗, z} = Uz(z⁻¹) = z;

Q((z⁻¹)^∗)(z) ={(z⁻¹)^∗, z, (z⁻¹)^∗} = U(z⁻¹)^∗(z^∗) = (z⁻¹)^∗; Q(z)Q((z⁻¹)^∗)(x) = U_z

U_(z−1)^∗(x^∗)_∗

= U_zU_z−1(x) = U_zU_z⁻¹(x) = x;

Q((z⁻¹)^∗)Q(z)(x) = U_(z−1)^∗(Uz(x^∗))^∗= U_(z∗)⁻¹Uz^∗(x) = U_z⁻¹∗Uz^∗(x) = x, forallx∈ M,whichproves thatz isvonNeumannregular withz^†= (z⁻¹)^∗= (z^∗)⁻¹, andsinceP₂(r(z))= Q(z)Q(z^†)= Id,thetripotent r(z) isaunitaryinM .

Weexplorenextcertainpropertiesof thecontinuous triplefunctionalcalculuswhich might resultsurprising. Letus fix an element a in aJB^∗-triple E. Let us consider the isometric triple isomorphism Ψ : E_a → C0(Ω_a) satisfying Ψ(a)(t) = t (t ∈ Ωa). The functiong(t)= t²liesinC₀(Ω_a),andhenceg_t(a)= a^[2]isanelementinE_a.Thereader shouldbe warnedthata^[2] isonlyasquareforalocal binaryproductdependingonthe element a.Suppose, additionally,thata isvon Neumann regular,thatis, Ωa⊂ R⁺. In this casethefunctionh(t)= ¹_t liesinC0(Ωa) anda^[−1]= ht(a)∈ Ea. It iseasy tosee that