A note on the uniqueness of involution in locally C*-algebras

0.1 MB·English

by  Alexander A. Katz

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]