Functors induced by Cauchy extension of C*-algebras

0.23 MB·

by  Kourosh Nourouzi

Functors induced by Cauchy extension of
7 C∗-algebras
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2
Kourosh Nourouzi1∗, Ali Reza2
n
a
J 1,2 Faculty of Mathematics, K. N. Toosi University of

5
Technology,

] P.O. Box 16315-1618, Tehran, Iran.
A
O January 6, 2017
.
h
t
a
m
Abstract
[
In this paper we give three functors P, [·]K and F on the category
1 of C∗- algebras. The functor P assigns to each C∗-algebra A a pre-C∗-
v algebra P(A) with completion [A]K. The functor [·]K assigns to each
8 C∗-algebraAtheCauchyextension[A]K ofAbyanon-unitalC∗-algebra
3 F(A). Some properties of these functors are also given. In particular, we
3 show that the functors [·]K and F are exact and the functor P is normal
1
exact.
0 Keywords: Pre-C∗-algebras;ExtensionsofC∗-algebras;Exactfunctors;
.
1 Cauchy extension.
0
7
1
1 Introduction
:
v
i
X Given a complex C∗-algebra A, the algebra A[[Z]] consists of all sequences
r (an)∞n=0 in A with pointwise linear operations and Cauchy product
a
((a )∞ )((b )∞ )=(c )∞ ,
n n=0 n n=0 n n=0
n
where each c = a b . It is natural to think of elements of A[[Z]] as
n k=0 k n−k
the formal powerPseries in one variable of the form ∞n=0anZn with product
P
∞ ∞ ∞
( a Zn)( b Zn)= c Zn,
n n n
nX=0 nX=0 nX=0
∗Correspondingauthor,e-mail: [email protected]
2010Mathematics SubjectClassification. 46L05,46M15
1
where c ’s are as above. One may consider the complex subalgebra
n
∞ ∞
A[Z]={ a Zn : ka k<∞},
n n
nX=0 nX=0
of A[[Z]]. It is of interest to find a C∗-algebra via A[Z] to be an extension of
A. Recall that an extension B of C by A is a short exact sequence
f g
0−→A−→B −→C −→0. (1)
of C∗-algebras (see, e.g., [1, 5, 6, 10]). For any subset Kof [−1,1] such that 0
is a limit point of K, we will define a pre-C∗-norm on A[Z]. The completion of
A[Z], denoted by [A] , is an extension of A (Proposition 7 (iii)) which will be
K
called the Cauchy extension of A.
The outline of this work is as follows. In Section 2 we introduce pre-C∗-
algebra A[Z]. In Proposition 5, it is shown that A[Z] is not a C∗-algebra.
Proposition 7 shows that the completion [A] of pre-C∗-algebra A[Z] is an
K
extension of A. We also introduce the functors P, [·] and F on the category
K
of C∗-algebras. The functor P assigns to each C∗-algebra A a pre-C∗-algebra
P(A) = A[Z]. The functor [·] assigns to each C∗-algebra A an extension
K
[A] of A by a non-unital C∗-algebra F(A), where the C∗-algebra F(A) is the
K
completion of the ideal
∞
A ={ a Zn ∈A[Z]:a =0}
1 n 0
nX=0
of A[Z]. Some properties of functors P, [·] and F are listed in Proposition 8.
K
InSection3 weshow thatthe functors [·] andF areexact. InSection4,using
K
the notion of normal exact sequence of the normed spaces introduced by Yang
[16], we provethat the functor P is normalexact. More precisely,for any short
exact sequence of C∗-algebra (1) the corresponding short exact sequence
f˜ g˜
0−→A[Z]−→B[Z]−→C[Z]−→0
is a normal exact sequence of pre-C∗-algebras. That is, B(Z)/kerg˜−→C[Z] is
an isometry. Among other results we also show that for any closed ideal I of
a C∗-algebra A, the pre-C∗-algebra I[Z] is a closed ideal of A[Z] (Proposition
8 (iii)) and the quotient A[Z]/I[Z] is a pre-C∗-algebra (Theorem 3) which is
isometric∗-isomorphicto (A/I)[Z] (Theorem4). Finally in Section5, we show
that the Cauchy extension [A] of a C∗-algebra A can be considered as a C∗-
K
subalgebra of C (K,A), the C∗-algebra of all bounded continuous functions
b
from K to A (Theorem 5 (i)). In particular, if K is compact, then [A] is
K
∗-isomorphic to C(K,A). We also give some other results in Theorem 5. A
minimax type result is given in Corollary 5.
2
2 Cauchy extension of C∗- algebras
LetAbeacomplexBanachalgebraandA[[Z]]bethecomplexalgebraconsisting
of all formal power series in A. If A has a unit, then an element F = F(Z) =
∞ a Zn ∈A[[Z]] is invertible if andonly if a is aninvertible element in A.
n=0 n 0
PIn particular, 1+Z2 is invertible in A[[Z]] and we have
∞ ∞
(1+Z2)( (−1)nZ2n)=( (−1)nZ2n)(1+Z2)=1. (2)
nX=0 nX=0
The subalgebra
∞ ∞
A[Z]={ a Zn ∈A[[Z]]: ka k<∞}
n n
nX=0 nX=0
can be equipped with a norm as
∞
kFk= ka k, (3)
n
nX=0
for all F(Z)= ∞ a Zn ∈A[Z].
n=0 n
P
Proposition 1. Let A be a Banach algebra. Then A[Z] with the norm given
in (3) is a Banach algebra.
Proof. To show that A[Z] is a Banach algebra, let (F ) = ( ∞ a Zn) be a
k n=0 kn
sequence in A[Z] such that ∞k=0kFkk<∞. Then P
P
∞ ∞ ∞ ∞
ka k= ka k<∞.
kn kn
Xk=0nX=0 nX=0Xk=0
Let c = ∞ a and F = ∞ c Zn. Then F ∈A[Z]. Let ε>0 be given.
n k=0 kn n=0 n
There exisPts a positive integerPN suchthat ∞k=N+1 ∞n=0kaknk<ε. We have
P P
k N F −Fk = k ∞ ( ∞ a )Znk
k=0 k n=0 k=N+1 kn
P = P∞ k P∞ a k
n=0 k=N+1 kn
≤ P∞ P∞ ka k
n=0 k=N+1 kn
= P∞ P ∞ ka k
k=N+1 n=0 kn
P P
< ε.
This completes the proof.
Proposition 2. Let A be a Banach algebra. If F(Z) = ∞ a Zn ∈ A[Z],
n=0 n
then N a Zn →F(Z) as N →∞, P
n=0 n
P
3
Proof. Since
N ∞ ∞
kF(Z)− a Znk=k a Znk= ka k,
n n n
nX=0 n=XN+1 n=XN+1
we get the desired limit.
Now one can consider any element F(Z)= ∞ a Zn ∈A[Z] as a conver-
n=0 n
gent series in A[Z]. P
If A is a C∗-algebra, we can define an involution ∗ in A[Z] by F∗(Z) =
∞ a∗Zn for any F(Z) ∈ A[Z]. In this case, A[Z] equipped with this invo-
n=0 n
lPution and the norm given in (3) is a ∗-Banach algebra.
Proposition 3. Let A be a C∗-algebra. There is no norm on involutive algebra
(A[Z],∗)which makes it a C∗-algebra. In particular, (A[Z],∗)equipped with the
norm given in (3) is not a C∗-algebra.
Proof. We suppose on the contrary that there exists a norm k ·k such that
(A[Z],∗,k·k) is a C∗-algebra. Suppose that A is unital. By (2) the element
1 + Z2 is not invertible in A[Z]. This implies that −1 ∈ σ(Z2) which is a
contradiction. Nowlet A be non-unitalanda∈A be self-adjointwith kak>1.
Applying (2) for aZ we getthat 1+a2Z2 is notinvertible in (A⊕C)(Z). That
is, −1∈σ(a2Z2) which is again a contradiction.
For a C∗-algebra (A,k·k) if F(Z) = ∞ a Zn ∈ A[Z] and −1 ≤ t ≤ 1
n=0 n
then P
∞ ∞
ka tnk≤ ka k<∞.
n n
nX=0 nX=0
Hence F(t)= ∞ a tn is norm-convergentin A.
n=0 n
For any F(PZ),G(Z)∈A[Z] and λ∈C,t∈[−1,1] we have
(λF(Z))(t)=λF(t), (4)
(F(Z)+G(Z))(t)=F(t)+G(t), (5)
(F(Z)G(Z))(t)=F(t)G(t). (6)
Note that the equalities (4) and (5) are clear and the proof of (6) is similar to
that of complex case (see [15, p. 74]).
Thefollowingpropositionhasastraightforwardproofwhichisomittedhere.
Proposition 4. Suppose that K is a subset of [−1,1] such that 0 is a limit
point of K and (a )∞ is a sequence in C∗-algebra A. If
n n=0
(i) F(Z)= ∞ a Zn such that ∞ ka k<∞;
n=0 n n=0 n
P P
(ii) F(t)=0 for any t∈K,
then a =0 for all n.
n
4
Hereafter, throughout the paper K will denote a subset of [−1,1] such that
0 is a limit point of it.
Proposition 5. The following statements hold:
(i) The functional k·k defined by
K
∞
kFk = supk a tnk,
K n
t∈K nX=0
for all F =F(Z)= ∞ a Zn ∈A[Z], is a norm;
n=0 n
P
(ii) (A[Z],∗,k·k ) is a pre-C∗-algebra but not a C∗-algebra;
K
(iii) kFk ≤kFk for all F ∈A[Z];
K
(iv) If F(Z)= ∞ a Zn, then N a Zn →F(Z) as N →∞ in k·k .
n=0 n n=0 n K
P P
Proof. (i) From (4), (5), (6) and Proposition 4 it is easily seen that k·k is a
K
norm. (ii) By the definition of k·k we have the identity kF∗Fk = kFk2 .
K K K
Therefore (A[Z],∗,k·k ) is a pre-C∗-algebra which by Proposition 3 is not a
K
C∗-algebra. (iii) By the definition of k·k is clear. (iv) The proof follows from
K
Proposition 2 and Part (ii).
We will call the completion [A] of pre-C∗-algebra (A,∗,k · k ) the K-
K K
Cauchy or simply the Cauchy extension of A. It is clear that [A] is a C∗-
K
algebra.
Proposition 6. Let A be a C∗-algebra. The following hold:
(i) If I is an ideal of A[Z], then the completion Iˆ of (I,k·k ) is a closed ideal
K
of [A] ;
K
(ii) If I is a closed ideal of A, then [I] is a closed ideal of [A] .
K K
Proof. (i) Let I be an ideal of A[Z]. Then the completion Iˆ of (I,k·k ) is
K
a closed ideal of [A] . Choose any element F ∈ Iˆ and G ∈ [A] . Let (F )
K K n
and (G ) be two sequences in I and A[Z] respectively converging to F ∈ Iˆ
k
and G ∈ [A] . For any k,n ≥ 1 we have F G ,G F ∈ I. This implies that
K n k k n
FG ,G F ∈Iˆ, for all k≥1 and so FG,GF ∈Iˆ. That is Iˆ is a closed ideal of
K k
[A] .
K
(ii) Consider F ∈I[Z] and G∈A[Z]. It is clear that FG,GF ∈I[Z], i.e., I[Z]
\
is an ideal of A[Z]. Now Part (i) implies that (I[Z])=[I] is a closed ideal of
K
[A] .
K
5
For a C∗-algebra A define
∞
A ={F(Z)= a Zn ∈A[Z]:a =0 for n>0},
0 n n
nX=0
∞
A ={F(Z)= a Zn ∈A[Z]:a =0}.
1 n 0
nX=0
Denote the completion of A by Aˆ . It is clear that A is an ideal of A[Z]
1 1 1
and by Proposition 6, Aˆ is a closed ideal of [A] . Hence if A =6 0, then [A]
1 K K
has a proper closed ideal Aˆ . Consequently no simple C∗-algebra is a Cauchy
1
extensionof someC∗-algebra. It is worthmentioning that there is no ideal I of
A such that I[Z]=A . Since A is naturally ∗-isomorphic to A we always use
1 0
A instead of A as a subalgebra of A[Z].
0
Suppose that A,B,E are C∗-algebras such that B is an ideal of E. It is said
to be E an extension of A by B if there is a short exact sequence
i p
0−→B −→E −→A−→0,
where i(B) = kerp and i,p are injective and surjective ∗-homomorphisms re-
spectively (see, e.g.,[1]).
Proposition 7. Let A be a C∗-algebra. The following statements hold:
(i) Every element F of [A] has a unique representation F = a+G, where
K
a∈A and G∈Aˆ ;
1
(ii) kak =kak≤ka+Gk , for all a∈A and G∈Aˆ ;
K K 1
(iii) [A] is an extension of A by Aˆ ;
K 1
(iv) Aˆ is not unital as a C∗-subalgebra of [A] .
1 K
Proof. (i) Let (F ) be a Cauchy sequence in (A[Z],k·k ), where
k K
∞
F = a Zn ∈A[Z].
k kn
nX=0
Let ε>0 be given. Then kFk−Fk′kK <ε for sufficiently large k,k′ . Suppose
that (t ) is a sequence in K such that t −→0 as m−→∞. By the definition
m m
of k·k we have
K
∞ ∞
kak0−ak′0k= lim k (akn−ak′n)tnmk≤ supk (akn−ak′n)tnk
m→∞ nX=0 t∈K nX=0
=kFk−Fk′kK <ε,
6
for sufficiently large k,k′. Furthermore
∞
supk (akn−ak′n)tnk<2ε.
t∈K nX=1
Therefore the sequences (a ) and ( ∞ a Zn) are Cauchy in A and A ,
k0 n=1 kn 1
respectively. For F ∈ [A]K, let F =Plimk→∞Fk, where Fk = ∞n=0aknZn ∈
A[Z]. Then F =a+G, where a −→a∈A and ∞ a ZnP−→G∈Aˆ as
k0 n=1 kn 1
k → ∞. Since Aˆ ∩A = 0, then this representatioPn is unique. Hence [A] is
1 K
the internal direct sum of subspaces A and Aˆ , i.e., [A] =A⊕Aˆ .
1 K 1
(ii) Note that if a∈A and G=lim ∞ a Zn ∈Aˆ , then
k→∞ n=1 kn 1
P
∞ ∞
ka+ a Znk = supka+ a tnk,
kn K kn
nX=1 t∈K nX=1
for all k ≥ 1. A similar method to that used in Part (i) implies that kak ≤
ka+ ∞ a Znk, for all k ≥ 1. Therefore kak ≤ ka+Gk, for all a ∈ A and
n=1 kn
G∈APˆ .
1
(iii) Define p :[A] −→A by p (a+G)=a, for all a∈A and G∈Aˆ . It is
A K A 1
easilyseenthatp isasurjective∗-homomorphismandkerp =Aˆ . Therefore
A A 1
we have the short exact sequence
0→Aˆ ֒→i [A] →pA A→0. (7)
1 K
This shows that [A] is an extension of A by Aˆ .
K 1
(iv)SupposeonthecontrarythatAˆ isunitalwithunitU(Z). SinceaZU(Z)=
1
aZ foralla∈A,wehavetaU(t)=taforanyt∈K anda∈A. Thisimpliesthat
aU(t)=a for all t6=0 and therefore lim U(t)6=0, which is a contradiction.
t→0
Remark 1. Each ∗-homomorphism f : A −→ B of C∗-algebras induces a ∗-
homomorphism f˜:A[Z]−→B[Z] between pre-C∗-algebras A[Z] and B[Z] by
∞ ∞
f˜( a Zn)= f(a )Zn, (8)
n n
nX=0 nX=0
where ∞ a Zn ∈A[Z].
n=0 n
P
Remark 2. If we define P(A) = A[Z] for any C∗-algebra A and P(f) = f˜,
for any ∗-homomorphism f : A −→ B of C∗-algebras, then P is a functor
from the category of C∗-algebras to the category of pre-C∗-algebras. Each ∗-
homomorphism f˜: A[Z] −→ B[Z] defined by (8) induces a ∗-homomorphism
fˆ: [A] −→ [B] . It is easy to see that [·] is a functor from the category
K K K
of C∗-algebras to itself as [f] = fˆ. Now defining F(A) = Aˆ and F(A −f→
K 1
B)=fˆ| :Aˆ −→Bˆ , for C∗-algebras A, B and ∗-homomorphism f, we get a
functorAoˆ1n the1catego1ry of C∗-algebras which assigns, by Proposition 7 (iv), to
any C∗-algebra a non-unital C∗-algebra.
7
By A∼=B we mean that the C∗-algebras A and B are ∗-isomorphic.
Proposition 8. Let f :A−→B be a ∗-homomorphism of C∗-algebras. Then
(i) f˜is a contraction;
(ii) f˜and fˆare isometries provided that f is an isometry;
(iii) f˜is surjective provided that f is surjective;
(iv) If f is a ∗-isomorphism, then both f˜and fˆare ∗-isomorphisms;
(v) kerf˜=(kerf)[Z];
(vi) Imf˜=(Imf)[Z];
(vii) If I is a closed ideal of A, then I[Z] is a closed ideal of (A[Z],k·k ). In
K
particular,
p′
0−→I[Z]֒→A[Z]−→A[Z]/I[Z]−→0
and
p˜
0−→I[Z]֒→A[Z]−→(A/I)[Z]−→0
are short exact sequences;
(viii) [A⊕B]K ∼=[A]K ⊕[B]K;
(ix) (A\⊕B) ∼=Aˆ ⊕Bˆ .
1 1 1
Proof. The proof of (iv) follows from (ii) and (iii). The proofs of (v) and (vi)
are straightforwardand the proof of (ix) is similar to Part (viii). We prove the
others.
(i) suppose that F(Z)= ∞ a Zn ∈A[Z]. Then
n=0 n
P
kf˜(F)k = k ∞ f(a )Znk
K n=0 n K
= suPp k ∞ f(a )tnk
t∈K n=0 n
= sup kfP( ∞ a tn)k
t∈K n=0 n
≤ sup k P∞ a tnk
t∈K n=0 n
P
= kFk .
K
(ii) If f is an isometry, then the proof of (i) shows that kf˜(F)k =kFk , for
K K
all F ∈A[Z]. That is f˜and consequently fˆis an isometry.
8
(iii) Let f be surjective and G = ∞ b Zn ∈ B[Z]. For any integer n ≥ 0,
n=0 n
there exists an ∈ A such thaPt bn = f(an). For any integer n ≥ 0 there
exists a′ ∈kerf such that
n
ka +a′ k≤ka +kerfk+2−n. (9)
n n n
Since A/kerf ∼=B, we have
ka +kerfk=kf(a )k=kb k. (10)
n n n
Define a′′ = a +a′ , for all n ≥ 0. Now we see from (9) and (10) that
n n n
F(Z)= ∞ a′′Zn ∈A[Z]andf(a′′)=b ,foreachn≥0,andtherefore
n=0 n n n
f˜(F)=GP.
p˜
(vii) Exactnessoffirstdiagramisclear. Part(iii)showsthatA[Z]→(A/I)[Z]
p
induces by the projection A−→A/I is surjective. By (v) kerp˜=I[Z] is
a closed ideal of A[Z]. This completes the proof.
(viii) It is easily seen that
T :A[Z]⊕B[Z]−→(A⊕B)[Z]
defined by
∞ ∞ ∞
T( a Zn, b Zn)= (a ,b )Zn,
n n n n
nX=0 nX=0 nX=0
for all ∞ a Zn ∈A[Z] and ∞ b Zn ∈B[Z], is a ∗-isomorphism.
n=0 n n=0 n
P P
3 Exactness of the functor [·]
K
In this section we show that [·] is an exact functor. We first recall some
K
definitions of the category theory [11].
f
Recall that a map X −→Y in a category C is called an epimorphism if for
g h
all maps Y −→ Z and Y −→ Z in C with g ◦f = h◦f, we have g = h. In
the category of C∗-algebras, a ∗-homomorphism f : A → B is an epimorphism
if and only if it is surjective [13].
f
Suppose that X −→ Y is a map in a category C with zero object. A map
Z −→j X is a kernel of f if f ◦j =0 and for any map Z′ −g→X in C such that
f◦g =0,there exists aunique mapZ′ −h→Z suchthatj◦h=g. Forexample,
if A −f→B is a ∗-homomorphism of C∗-algebras, then the inclusion kerf ֒→ A
is a kernel of f.
Theorem 1. The functor [·] is exact.
K
9
Proof. Suppose that
f g
0−→A−→B −→C −→0
is a short exact sequence of C∗-algebras. We must show that
fˆ gˆ
0−→[A] −→[B] −→[C] −→0 (11)
K K K
is a short exact sequence of C∗-algebras. We first show that if f : A → B is
a surjective ∗-homomorphism of C∗-algebras, then fˆ: [A] → [B] is also a
K K
surjective ∗-homomorphism of C∗-algebras. To do this suppose that [B] →h C
K
and [B] →g C are ∗-homomorphism of C∗-algebras such that h◦fˆ = g ◦fˆ.
K
From Proposition 8 (iii), we have fˆ(A[Z]) = B[Z]. So for any G(Z) ∈ B[Z]
there exists an element F(Z)∈A[Z] such that
h(G(Z)) = h(fˆ(F(Z)))
= (g◦fˆ)(F(Z))
= g(G(Z)).
Thisimpliesthath| =g| andthereforeg =h. Hencefˆisanepimorphism
B[Z] B[Z]
and consequently is surjective by [13].
Now we show that if A −f→ B is a ∗-homomorphism of C∗-algebras, then
kerfˆ=[kerf] . To prove this, suppose that C −g→[A] is a ∗-homomorphism
K K
ofC∗-algebrassuchthatfˆ◦g =0. Ifgˆ =pˆ ◦gˆandC ֒→iC [C] isthe injection,
C A K
j
then gˆ ◦ i = g. Since kerf ֒→ A is a kernel of f, there exists a unique
C C
h
∗-homomorphismC −→kerf such that the diagram
f
kerf A B
g
h C
C
is commutative. Since [·] is a functor we get the commutative diagram
K
j fˆ
[kerf] [A] [B]
K K K
hˆ gˆC
[C]
K
10