Advances in Pure Mathematics, 2013, 3, 660-665
Published Online November 2013 (http://www.scirp.org/journal/apm)
http://dx.doi.org/10.4236/apm.2013.38088
Open Access APM
A Real p-Homogeneous Seminorm with Square
Property Is Submultiplicative
Mohammed El Azhari
Department of Mathematics, Ecole Normale Supérieure, Rabat, Morocco
Received October 10, 2013; revised November 10, 2013; accepted November 17, 2013
Copyright © 2013 Mohammed El Azhari. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
We give a functional representation theorem for a class of real p-Banach algebras. This theorem is used to show that
every p-homogeneous seminorm with square property on a real associative algebra is submultiplicative.
Keywords: Functional Representation; p-Homogeneous Seminorm; Square Property; Submultiplicative
1. Introduction
J. Arhippainen [1] has obtained the following result:
Theorem 1 of [1]. Let q be a p-homogeneous semi-
norm with square property on a complex associative
algebra A. The n
1) Ker(q) is an ideal of A;
2) The quotient algebra A/Ker(q) is commutative;
3) q is submultiplicative;
4)
1
p
q is a submultiplicative seminorm on A.
This result is a positive answer to a problem posed in
[2] and considered in [3-5]. The proofs of (3) and (4)
depend on (2) which is obtained by using a locally
bounded version of the Hirschfeld-Zelazko Theorem [1,
Lemma 1]. This method can not be used in a real algebra;
if q is the usual norm defined on the real algebra H of
quaternions, Ker and H/Ker is non-
commutative, then the assertion (2) does not hold in the
real case.
 
0q

Hq
The purpose of this paper is to provide a real algebra
analogue of the above Arhippainen Theorem, and this im-
proves the result in [6]. Our method is based on a func-
tional representation theorem which we will establish; it
is an extens ion of the Abel-Jar osz Theore m [7 , Theore m 1]
to real p-Banach algebras. We also give a functional rep-
resentation theorem for a class of complex p-Banach a lg e-
bras. As a consequence, we obtain the main result in [8].
2. Preliminaries
Let A be an associative algebra over the field K = R or C.
Let


0,1p, a map

.: 0,A

is a p-homo-
geneous seminorm if for in A and
,ab

in K,
aba b

 and p a  a. Moreover, if
0a

imply that 0a

, . is called a p-homo-
geneous norm. A 1-homogeneous seminorm (resp.norm)
is called a seminorm (resp.norm). . is submultipli-
cative if abab for all in A. ,ab . has the 
square property if 2
2
aa for all a. If A. is a
submultiplicative p-homogeneous norm on A, then


,.A is called a p-normed algebra, we denote by M(A)
the set of all nonzero continuous multiplicative linear
functionals on A. A complete p-normed algebra is called
a p-Banach algebra. A uniform p-normed algebra is a
p-normed algebra


,.A such that 2
2
aa for all
aA

. Let A be a complex algebra with unit e, the
spectrum of an element aA

is defined by




1
,Sp aCeaA 


where 1
A

is the set of all invertible elements of A. Let
A be a real algebra with unit e, the spectrum of aA

is
defined by
 


221
,SpasitCaseteA  .
Let A be an algebra, the spectral radius of an element
aA

is defined by



 
sup ,ra Spa  . Let


,.A be a p-normed algebra, the limit 1
lim n
p
n
na 
exists for each aA

, and if A is complete, we have

M. EL AZHARI 661

1
lim n
p
n
n
ra a

 for all . A -algebra is a aA
complex algebra with a mapping :,
A
Aa a

 ,
such that, for in A and
,ab ,C




 
,,
,.
aaabab
aaabba 








The map is called an involution on A. An element
is said to be hermitian if . The set of all
hermitian elements of A is denoted by H(A).

aAaa

3. A Functional Representation Theorem for
a Class of Real p-Banach Algebras
We will need the following result due to B. Aupetit and J.
Zemanek ([9,10]) , th eir alg ebr aic approach works for real
p-Banach algebras.
Theorem 3.1. Let
A
be a real p-Banach algebra with
unit. If there is a positive constant

such that
for all in
 
rab rarb  ,ab
A
, then for every
irreducible representation

of
A
on a real linear
space , the algebra
E


A
 is isomorphic (algebrai-
cally) to its commutant in the algebra of all
linear transformations on .

LE
E
Let
A
be a real p-Banach algebra with unit such that

1
p
amra for some positive constant and all m
aA. Let

X
A be the set of all nonzero multi-
plicative linear functionals from A into the noncom-
mutative algebra
H
of quaternions. For aA

, we
consider the map
 


:,
J
aXAHJaxxa for
all

x
XA. We endow

X
A

with the weakest
topology such that all the functions ,
J
aa A

, are
continuous. The map
 


:,,
J
ACXAH Jaa,
is a homomorphism from
A
into the real algebra of all
continuous function s from


X
A into
H
.
Theorem 3.2. If

is an irreducible representation of
A
, then

A
 is isomorphic to , or RC
H
.
Proof. Let and , we have
,ab A1n

nn n
aba b,
then

111
npn
p
p
aba b.
Letting , we obtain n
 


2
rab mrarb. Let

be an irreducible representation of
A
on a real
linear space . By Theorem 3.1,
E


A

is isomorphic
to its commutant Q in the algebra


LE
y of all linear transformations on . Let 0 be a fixed nonzero
element in . For E
E
y
E, we consider
 
0
inf, and
E
yaaAay  
By the same proof as in [11, Lemma 6.5], .
E
is a
p-norm on and Q is a real div ision p-normed algebra
of continuous linear operators on . By [12], Q is
isomorphic to , or
EE
RC
H
.
Proposition 3.3.
A
is semisimple and


X
A is a
nonempty set which separates the elements of A.
Proof. By the condition

1
p
amra for all aA

,
we deduce that
A
is semisimple. Let be a nonzero
element in a
A
, since
A
is semisimple, there is an irre-
ducible representation

of
A
such that


0a 

.
By Theorem 3.2, there is

:
A
H
To
 an isomor-
phism (into). We consider the map ,:TAH

 
is a multiplicative linear functional. Moreover,




0Ta a 


since


0a 

and

is injective.
Proposition 3. 4.
1)

1
p
x
aa for all and aA


;
x
XA
2) An element is invertible in A if and only if a


J
a is invertible in



,;CXA H
3)






aaASp aSpJ for all .
Proof. (1): Since
H
is a real uniform Banach algebra
under the usual norm
   
1
.,
p
HA
x
arxa raa
for all aA

and


.
x
XA
(2): The direct implication is obvious. Conversely, let

be an irreducible representation of
A
. By Theorem
3.2, there is


:
A
H   an isomorphism (into).
Since


A

oX   and

J
a is invertible,



 
0

J
ao a   
,
then


0.a 

Consequently, a is invertible.
(3):


s
it Spa

iff

221
aseteA


Iff


e


1
,CXAH


22
Jaset by (2)
Iff
     
21
2,JasJetJeCXA H


Iff




.
s
itSp Ja
Proposition 3.5.


X
A is a Hausdorff compact
space.
Proof. Let x1, x2 in


1
,2
X
Ax x, there is an element
aA

such that




12
x
axa, i.e.



12
J
ax Jax, so


X
A is Hausdorff. Let and aA
1
,p
a
KqHqa


 



,

y.
a
K
is compact in
H
. Let
K
be the topological
product of a
K
for all is compact by the
Tychonoff Theorem. By Proposition 3.4(1),
,KaA


X
A is a
subset of
K
. It is easy to see that the topology of
Open Access APM

M. EL AZHARI 662

X
A is the relative topology from K and that


X
A
is closed in . Then
K

X
A

:,
is compact.
Theorem 3.6. The m a p
 
,,
J
AC HaJaXA
is an isomorphism (into) such that

11
1
p
p
s
maaa
J
for all , where
aA.
s
is the supnorm on



,A H

CX .
If , we have 1m

1
p
s
aJa for all .aA
Proof. By Proposition 3.3,
J
is an injective homo-
morphism. Let by Proposition 3.4(3), ,aA
 


s
ra rJaJa
since is a real uniform Banach algebra


A

,CXH
under the supnorm .
s
. Moreover,

1
p
s
J
aa by
Proposition 3.4(1). Then

11
1.
p
p
s
mara Jaa 

As an application, we obtain an extension of the Kul-
karni Theorem [13, Theorem 1] to real p-Banach alge-
bras.
Theorem 3.7. Let be an element in a
A
such that
then belongs to the center of

Sp a,Ra.
A
Proof. By Theorem 3.6,
 
:,

J
ACX
A

Sp aR
AH is an
isomorphism (into). Let a with Let .

x
XA and

x
ast where
s
R and
12 3
.ttitjtk 
Suppose that We have
0.t
 

22
2 222
123
x
as ttttt,
Then


220xa st.
Consequently
  
s
i tSpxaSpa
with 0,t a contradiction. Then




,
J
aCXAR
and
 


J
aJb JbJa
for all in b,
A
i.e. for all b in

0Jab ba
A
.
Since
J
is injective, for all b in 0 ab ba .
A
4. A Functional Representation Theorem for
a Class of Complex p-Banach Algebras
Let . be a submultiplicative p-homogeneous se-
minorm on a complex algebra
A
. For ,aAa is de-
fined as follows:
1
inf n
1
p
i
aa
i

A
,
where the infimum is taken over all decompositions of
satisfying the condition 1
i, 1
an
i
aa,, .
n
aa


By [14, Theorem 1], . is a submultiplicative seminorm
on ,
A
it is called the support seminorm of .. Also, it
is shown [14] the following result:
Theorem 2 of [14]. Let
A
be a complex algebra, .
a submultiplicative p-homogeneous seminorm on
A
,
and . the support seminorm of .. Then
11
lim lim
nn
p
nn
nn
aa
 

for all .aA

In the proof of this theorem, Xia Dao-Xing uses the
following inequality: If 1m
aa a

 and , then 1n
1
11
1
!
!!
m
m
p
nm
nm
n
aaa




 




.
If the algebra is commutative,

1
1
1
1
1
!
!! m
m
n
nm
m
nm
aaa
naa




 





,
then
1
11
1
!.
!!
m
m
p
nm
nm
n
aa
 


 




a
This inequality is not justified in the noncommutative
case; if the algebra is noncommutative, we only have
1
11
1
!
!!
m
m
nm
nm
n
aaa




 

.
For the sequel, we will use Theorem 2 of [14] in the
commutative case.
Theorem 4.1. Let


,.A be a complex p-normed
algebra such that 22 ama for some positive con-
stant and all .maA

Then
1
1p
p
aa ma and
2
22
p
ama for all aA

, where . is the support
seminorm of ..
Proof. The completion of B

,.A

is a p-Banach
algebra such that 2 bmb2 for all b, it is com-
mutative by [1, Lemma 1], so B
A
is commutative. By
induction, 2
122n
n n
am a



 for all a and , A1n
then 1
lim nn n
ama

 for all By the com- .aA
mutative version of [14, Theorem 2], we have
Open Access APM

M. EL AZHARI 663
11
1
11
1
lim
lim
n p
p
n
pn
n pp
n
n
aa ma
mam



a
for all . From the above inequalities, aA

2
1
2
222
p
p
p
aa mama .
Corollary 4.2. Let


,.A be a complex uniform
p-normed algebra. Then 1
p
aa for all .aA
Theorem 4.3. Let


,.A be a complex p-Banach
algebra with unit such that 22
ama for some posi-
tive constant and all . Then the Gelfand map
is an isomorphism (into) such that
m
 
MaA

:GA CA


21
11
pp
p
p
s
ma maGaaa

 
for all where ,aA.
s
is the supnorm on




.CMA
Proof. A is commutative by [1, Le mma 1]. By Theore m
4.1,
1
1p
p
aa ma for all , then aA

,.A

is a
complex commutative Banach algebra with unit. Clearly

 
,. ,.

M
AMAMA is a nonempty compact
space. As in the proof of Theorem 4.1, we have
 


11
1
11
lim
sup ,
.
n pn
n
p
pp
s
am a
mfafM
mGa ma




A
Let from the above inequalities, ,aA

21
1
pp 1
p
p
s
ma maGaaa

 .
Corollary 4.4. Let


,.A be a complex uniform
p-Banach algebra with unit. Then the Gelfand map
is an isomorphism (into) such that


:GA CMA


1
p
s
aa Ga
for all . aA
Theorem 4.5. Let


,.A be a complex p-normed
-algebra with unit such that
1) 22
ama for some positive constant and
all m
;aA
2) Every element in

H
A has a real spectrum in the
completion of .
B
A
Then the involution is continuous on

A
and the
Gelfand map



:GB MB

C is a -isomorphism
such that

211
p
p
p
s
mb Gbb
 for all in b.B
Proof. By Theorem 4.3, it remains to show that the
involution * is continuous on
A
,


Gb Gb

 for
all ,bB

and G is surjective. Let

,hHA






,
B
Sphf hfM BR


by (2). Let ,aA

we have with
1
ahih
212
,hh


H
A. Let


,
f
MB



 


121 2
12 12
fafh ihfhifh


f
hifhfh ihfa

 
 
since


1
f
h and


2
f
h are real. Then


GaGa


for all .aA

By Theorem 4.3,

 
21
1
pp
s
p
s
s
ma Ga
GaGaa




for all ,aA

then 2 ama
 for all .aA

Con-
sequently, the involution

is continuous on A and can
be extended to a continuous involution on which we
will also denote by B
.

Let there exists a
sequence ,bB


nn
a in A such that n. Since the in-
volution on B and the Gelfand map
ab




CMB:GB
are continuous, we have


n
Ga Gb




and
 
,
n
Ga Gb


then



.Gb Gb

By the Stone-Weierstrass Theorem, we deduce that
is surjective.
GAs a consequence, we obtain the main result in [8].
Corollary 4.6. Let
A
be a complex uniform p-
normed

-algebra with unit such that every element in


H
A has a real spectrum in the completion of B.
A
then is a commutative -algebra. B C
5. The Main Result
Theorem 5.1. Let
A
be a real associative algebra.
Every p-homogeneous seminorm with square pro- q
perty on
A
is submultiplicative and
1
p
q is a sub-
multiplicative seminor m on .
A
Proof. By [1], there exists a positive con stant such
that m






qabmqaqb for all . ,abA


Kerq is
an ideal of ,
A
the norm . on the quotient algebra


Ker
A
q defined by

qKeraqa is a p-norm
with square property . Define


Ker Keraqmaq
for all .aA

Let ,,ab A

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M. EL AZHARI 664
 
 
 
2
Ker Ker
Ker Ker
KerKer ,
ab qmab q
ma qb q
aqbq

 
 
then


Ker, .Aq

is a real p-normed algebra. Let
,aA
 




22
2
2
1
2
1
Ker Ker
Ker
Ker
Ker
aqmaq
ma q
mmaq
ma q






i.e.
 
22
KerKer .aqma q
The completion of B


Ker, .Aq

satisfies also
the property 22 bmb for all and by induc- ,bB
tion 2
12 2
nn
n
bm b


 for all and then bB1,n

p
bmrb for all We consider two cases: . bB
B is unital: By section 3,

X
B
:
is a nonempty com-
pact space and the map
 
,
J
BCXBH is an iso-
morphism (into). By Proposition 3.4(3),





rb rJb
for all . Let bB,
bB
   
p
p
p
s
bmrb mrJbmJb
since is a real uniform Banach algebra
under the supnorm


,CXB H

.
s
. Then

1
p
s
bmb Jb


for all


KerbA q, so . is submultiplicative and
1
.
p
is a submultiplicative norm. Consequently, is q
submultiplicative and
1
p
q is a submultiplicative semi-
norm.
B is not unital: Let 1 be the algebra obtained from
by adjoining the unit. By the same proof of [15,
Lemma 2] which works for real p-Banach algebras, there
exists a p-norm on such that
B
B
B
N1
1) is a real p-Banach algebra with unit;

1,BN

2)
 
1
13
pB
Nbmr b for all ;
1
bB
3) and
N. are equivalent o n .B
By section 3,

1
X
B
1
:
is a nonempty compact space
and the map


1

,
J
BCXBH
bB is an isomorphism
(into). Let ,
 
1
p
p
BB
bmrbmrb
by (3)


p
mr J b by Proposition 3.4(3)

p
s
mJb by the square property of the supnorm.
Then

1
p
s
bmb Jb

 for all


Ker ,bA q
so . is submultiplicative and 1 .
p
is a submultipli-
cative norm. Consequently, is submultiplicative and q
1
p
q is a submultiplicative seminorm.
REFERENCES
[1] J. Arhippainen, “On Locally Pseudoconvex Square Alge-
bras,” Publicacions Matematiques, Vol. 39, No. 1, 1995,
pp. 89-93.
http://dx.doi.org/10.5565/PUBLMAT_39195_06
[2] S. J. Bhatt and D. J. Karia, “Uniqueness of the Uniform
Norm with an Application to Topological Algebras,” Pro-
ceedings of the American Mathematical Society, Vol. 116,
1992, pp. 499-504.
http://dx.doi.org/10.1090/S0002-9939-1992-1097335-4
[3] S. J. Bhatt, “A Seminorm with Square Property on a Ba-
nach Algebra Is Submultiplicative,” Proceedings of the
American Mathematical Society, Vol. 117, 1993, pp. 435-
438.
http://dx.doi.org/10.1090/S0002-9939-1993-1128724-8
[4] H. V. Dedania, “A Seminorm with Square Property Is
Automatically Submultiplicative,” Proceedings of the In-
dian Academy of Science, Vol. 108, 1998, pp. 51-53.
[5] Z. Sebestyen, “A Seminorm with Square Property on a
Complex Associative Algebra Is Submultiplicative,” Pro-
ceedings of the American Mathematical Society, Vol. 130,
2001, pp. 1993-1996.
http://dx.doi.org/10.1090/S0002-9939-01-06278-5
[6] M. El Azhari, “A Real Seminorm with Square Property Is
Submultiplicative,” Indian Journal of Pure and Applied
Mathematics, Vol. 43, No. 4, 2012, pp. 303-307.
http://dx.doi.org/10.1007/s13226-012-0018-z
[7] M. Abel and K. Jarosz, “Noncommutative Uniform Al-
gebras,” Studia Mathematica, Vol. 162, 2004, pp. 213-
218. http://dx.doi.org/10.4064/sm162-3-2
[8] A. El Kinani, “On Uniform Hermitian p-Normed Alge-
bras,” Turkish Journal of Mathematics, Vol. 30, 2006, pp.
221-231.
[9] B. Aupetit and J. Ze manek, “On the Real Spectral Radius
in Real Banach Algebras,” Bulletin de l’Académie Polo-
naise des Sciences Série des Sciences Math., Astr. et
Phys., Vol. 26, 1978, pp. 969-973.
[10] J. Zemanek, “Properties of the Spectral Radius in Banach
Algebras,” Banach Center Publications, Vol. 8, 1982, pp.
579-595.
[11] A. M. Sinclair, “Automatic Continuity of Linear Opera-
tors,” Cambridge University Press, Cambridge, 1976.
http://dx.doi.org/10.1017/CBO9780511662355
[12] Ph. Turpin, “Sur une Classe D’algèbres Topologiques,” C.
R. Acad. Sc. Paris, Série A, Vol. 263, 1966, pp. 436-439.
Open Access APM

M. EL AZHARI
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665
[15] R. A. Hirschfeld and W. Zelazko, “On Spectral Norm
Banach Algebras,” Bulletin de l’Académie Polonaise des
Sciences Série des Sciences Math., Astr. et Phys., Vol. 16,
1968, pp. 195-199.
[13] S. H. Kulkarni, “Representations of a Class of Real

B-Algebras as Algebras of Quaternion-Valued Func-
tions,” Proceedings of the American Mathematical Soci-
ety, Vol. 116, 1992, pp. 61-66.
[14] D.-X. Xia, “On Locally Bounded Topological Algebras,”
Acta Mathematica Sinica, Vol. 14, 1964, pp. 261-276.