A NOTE ON THE UNIQUENESS OF INVOLUTION IN LOCALLY C*-ALGEBRAS 2 ALEXANDERA.KATZ 1 0 2 Abstract. In the present note we show that the involution in locally C*- n algebrasisuniquelydetermined. a J 3 1. Introduction ] A One ofthe importantbasicfacts of the theory ofC∗-algebras is that the unary O operation of involution in a C∗-algebra is uniquely determined. This property was . first observed in 1955 by Bohnenblust and Karlin in [2] (see as well [7] for a nice h t exposition). a The Hausdorffprojectivelimits ofprojectivefamilies ofBanachalgebrasas nat- m ural locally-convex generalizations of Banach algebras have been studied sporadi- [ callybymanyauthorssince1952,whentheywerefirstintroducedbyArens[1]and 1 Michael [6]. The Hausdorff projective limits of projective families of C∗-algebras v were first mentioned by Arens [1]. They have since been studied under various 4 namesby many authors. Developmentofthe subjectis reflectedin the monograph 7 of Fragoulopoulou [3]. We will follow Inoue [4] in the usage of the name locally 5 0 C∗-algebras for these algebras. . Thepurposeofthepresentnotesistoshowthattheunaryoperationofinvolution 1 in locally C∗-algebras is uniquely determined. 0 2 1 2. Preliminaries : v First, we recall some basic notions on topological ∗-algebras. A ∗-algebra (or i X involutive algebra) is an algebra A over C with an involution r ∗ a :A→A, such that ∗ ∗ ∗ (a+λb) =a +λb , and ∗ ∗ ∗ (ab) =b a , for every a,b∈A and λ∈ C. A seminormk.k ona ∗-algebraA is a C∗-seminormif it is submultiplicative, i.e. kabk≤kakkbk, Date:January2,2012. Keywords and phrases. C*-algebras,locallyC*-algebras,projectivelimitofprojectivefamily ofC*-algebras. 2010AMSSubjectClassification: Primary46K05. 1 2 ALEXANDERA.KATZ and satisfies the C∗-condition, i.e. ∗ 2 ka ak=kak , for every a,b ∈ A. Note that the C∗-condition alone implies that k.k is submulti- plicative, and in particular ∗ ka k=kak, for every a∈A (cf. for example [3]). Whenaseminormk.kona∗-algebraAisaC∗-norm,andAiscompleteininthe topologygeneratedbythisnorm,AiscalledaC∗-algebra. Thefollowingtheorem is valid. Theorem 1 (Bohnenblust andKarlin[2]). The unary operation of involution in a C∗-algebra is uniquely determined. Proof. See for example [7] for details. (cid:3) Atopological∗-algebraisa∗-algebraAequippedwithatopologymakingtheop- erations (addition, multiplication, additive inverse, involution) jointly continuous. Foratopological∗-algebraA,oneputsN(A)forthesetofcontinuousC∗-seminorms on A. One can see that N(A) is a directed set with respect to pointwise ordering, because max{k.k ,k.k }∈N(A) α β for every k.k ,k.k ∈N(A), where α,β ∈Λ, with Λ being a certain directed set. α β For a topological ∗-algebra A, and k.k ∈N(A), α∈Λ, α kerk.k ={a∈A:kak =0} α α is a ∗-ideal in A, and k.k induces a C∗-norm (we as well denote it by k.k ) on α α the quotient A = A/kerk.k , and A is automatically complete in the topology α α α generated by the norm k.k , thus is a C∗-algebra (see [3] for details). Each pair α k.k ,k.k ∈N(A), such that α β β (cid:23)α, α,β ∈Λ, induces a natural (continuous) surjective ∗-homomorphism gβ :A →A . α β α Let, again, Λ be a set of indices, directed by a relation (reflexive, transitive, antisymmetric) ”(cid:22)”. Let {A ,α∈Λ} α be a family of C∗-algebras, and gβ be, for α α(cid:22)β, the continuous linear ∗-mappings gβ :A −→A , α β α so that gα(x )=x , α α α for all α∈Λ, and gβ ◦gγ =gγ, α β α whenever α(cid:22)β (cid:22)γ. THE UNIQUENESS OF INVOLUTION IN LOCALLY C*-ALGEBRAS 3 Let Γ be the collections {gβ} of all such transformations.Let A be a ∗-subalgebra α of the direct product algebra Y Aα, α∈Λ so that for its elements x =gβ(x ), α α β for all α(cid:22)β, where x ∈A , α α and x ∈A . β β Definition1. The∗-algebraAconstructedaboveiscalledaHausdorffprojective limit of the projective family {A ,α∈Λ}, α relatively to the collection Γ={gβ :α,β ∈Λ:α(cid:22)β}, α and is denoted by limA , ←− α and is called the Arens-Michael decomposition of A. Itiswellknown(see,forexample[8])thatforeachx∈A,andeachpairα,β ∈Λ, such that α(cid:22)β, there is a natural projection π :A−→A , β β defined by π (x)=gβ(π (x)), α α β and each projection π for all α∈Λ is continuous. α Definition 2. A topological ∗-algebra A over C is called a locally C∗-algebra if there exists a projective family of C∗-algebras {A ;gβ;α,β ∈Λ}, α α so that A∼=l←im−Aα, i.e. A is topologically ∗-isomorphic to a projective limit of a projective family of C∗-algebras, i.e. thereexitsitsArens-Michaeldecomposition ofAcomposedentirely of C∗-algebras. Atopological∗-algebraAoverCisalocallyC∗-algebraiffAisacompleteHaus- dorfftopological∗-algebrainwhichtopologyisgeneratedbyasaturatedseparating family of C∗-seminorms (see [3] for details). Example 1. Every C∗-algebra is a locally C∗-algebra. Example 2. A closed ∗-subalgebra of a locally C∗-algebra is a locally C∗-algebra. Example 3. The product Y Aα of C∗-algebras Aα, with the product topology, is α∈Λ a locally C∗-algebra. 4 ALEXANDERA.KATZ Example 4. Let X be a compactly generated Hausdorff space (this means that a subset Y ⊂ X is closed iff Y ∩K is closed for every compact subset K ⊂ X). Then the algebra C(X) of all continuous, not necessarily bounded complex-valued functions on X, with the topology of uniform convergence on compact subsets, is a locallyC∗-algebra. Itiswellknownthatallmetrizablespacesandalllocallycompact Hausdorff spaces are compactly generated (see [5] for details). Let A be a locally C∗-algebra. Then an element a∈A is called bounded, if kak ={supkak ,α∈Λ:k.k ∈N(A)}<∞. ∞ α α The set of all bounded elements of A is denoted by b(A). It is well-known that for each locally C∗-algebra A, its set b(A) of bounded elements of A is a locally C∗-subalgebra, which is a C∗-algebra in the norm k.k , ∞ such that it is dense in A in its topology (see for example [3]). 3. The uniqueness of involuton in locally C*-algebras Here we present the main theorem of the current notes. Theorem2. Theunaryoperation ofinvolutioninanylocally C∗-algebraisunique, i.e., if(A,∗,k.k ,α∈Λ)and(A,#,k.k ,α∈Λ)aretwolocallyC∗-algebras, means α α that each seminorm k.k ,α∈Λ, satisfies the C∗-property for both operations, ”∗” α and ”#”, then ∗ =# on A. Proof. Let now A be a locally C∗-algebra,and let A=limA , ←− α α ∈ Λ, be its Arens-Michael decomposition, built using the family of seminorms k.k ,α∈Λ, so that for each α∈Λ, α (A ,∗ ,k.k ) α α α and (A ,# ,k.k ) α α α areC∗-algebras,where the unary operations ”∗ ” and”# ” on A are defined as α α α follows: π (x∗)=(π (x))∗α, α α and π (x#)=(π (x))#α, α α for each x∈A and α∈Λ. Let us now assume, to the contrary to the statement of the theorem, that there exists some x∈A, such that x∗ =y 6=z =x#. Then there must exist α ∈Λ, such that 0 π (y)6=π (z). α0 α0 In fact, if it is not the case, and π (y)=π (z) α α THE UNIQUENESS OF INVOLUTION IN LOCALLY C*-ALGEBRAS 5 for each α∈Λ, implies that y =z, which contradicts the assumption. So, α must be such that 0 π (x∗)6=π (x#), α0 α0 which means that for π (x)=x ∈A , α0 α0 α0 x∗α 6=x#α, α0 α0 which contradicts Theorem 1. Found contradiction proves the theorem. (cid:3) References [1] Arens, R., A generalization of normed rings. (English) Pacific J. Math., Vol. 2, 1952, pp. 455–471. [2] Bohnenblust, H.F.; Karlin, S.,Geometrical properties of the unit sphere of Banach alge- bras. (English)Ann.ofMath.(2)No.62,1955,pp.217–229. [3] Fragoulopoulou,M.,Topological algebraswithinvolution.(English)North-HollandMathe- maticsStudies,Vol.200,ElsevierScienceB.V.,Amsterdam,2005,495pp. [4] Inoue,A.,Locally C∗-algebra.(English),Mem.Fac.Sci.KyushuUniv.Ser.A,Vol.25,1971, pp.197–235. [5] Kelley,J.L.,Generaltopology.(English)Reprintofthe1955edition[VanNostrand,Toronto, Ont.]. Graduate Texts inMathematics, No. 27. Springer-Verlag, New York-Berlin, 1975, 298 pp [6] Michael, E.A., Locally multiplicatively-convex topological algebras. (English) Mem. Amer. Math.Soc.,No.11,1952,79pp. [7] Rickart,C.E.,GeneraltheoryofBanachalgebras.(English)TheUniversitySeriesinHigher MathematicsD.vanNostrandCo.,Inc.,Princeton,N.J.-Toronto-London-NewYork,1960,394 pp. [8] Tr`eves, F., Topological vector spaces: Distributions and Kernels. (English), New York- London: AcademicPress.,1967, 565pp. Dr. Alexander A. Katz, Department of Mathematics and Computer Science, St. John’sCollegeofLiberalArtsandSciences,St. John’sUniversity,300HowardAvenue, DaSilvaAcademic Center 314,StatenIsland, NY10301,USA E-mail address: [email protected]