
Research Article


On Unit Element’s Norm in Some Banach Spaces 

Abdolmohammad Forouzanfar,
Sajad Khorshidvandpour
and
Mohsen Moosavi



ABSTRACT

In this study, we obtain Banach algebras which norm of their unit elements
is not one. These Banach algebras are subsets of .
Also, we present some interesting properties.





Received: January 18, 2012;
Accepted: February 29, 2012;
Published: May 18, 2012


INTRODUCTION We know normed algebras are one of the most important subjects in functional analysis. Also we know that if a normed algebra is unitary then norm of its unit is one. Most of persons who study functional analysis in a nonprofessional way think that norm of unit in any algebra should be one. We wish to present algebras in which their unit element’s norm is not one. We need to following definitions.
Definition 1: Kreyszig (1987) (Normed algebra):
An algebra over F (the real field
or the complex field C) is a linear space A over F together with a mapping (x,
y)→xy of AxA into A that satisfies the following axioms (for all x,y,zεA,
αεF):
(i) 
x (yz) = (x,y)z 
(ii) 
x(y+z) = (xy)+xz, (x+y)z = xz+yz 
(iii) 
(αx)y = α(xy) = x(αy) 
Furthermore if there exists a norm  on A such that
we have for any x,y∈ A,xy≤x
y then A is a normed algebra.
As is usual for normed linear spaces a normed algebra A is regarded as a metric space with the distance function d (x,y) = xy (x,yεA). If A is a complete metric space with defined metric, then A is called a Banach algebra.
Definition 2: Bonsall and Duncan (1973): An element
e of an algebra A is an unit element if and only if e≠0 and ex = xe = x (xεA).
Definition 3: Kreyszig (1987) (Equivalent norms):
A norm  on a vector space X is said to be equivalent
to a norm _{1} on X if there are positive numbers
a and b such that for all xεX we have ax_{1}≤x≤bx_{1}.
MAIN RESULTS
In this section we introduce some subsets of ^{k}
which are Banach algebras with defined norm. In these algebras the norm of unit
element isn’t one.
We begin with
(real numbers set). Consider
with ordinary addition and scalar product.
with defined operations is a vector space. We define the product of
as follow:
And define norm on
by:
where, c is a constant and c>1. clearly,
is a normed algebra with 1 as unit element, but 1 =
c1 = c>1.
The defined norm in Eq. 1 is equivalent with original norm
on .
The norm of 1 with original norm is one, whereas with (1) isn’t one. We
define, In general, for kεN:
Clearly, A_{k }is a subset of ^{k}
for k≥1. A_{k} ‘s with following operations are vector spaces
on :
(0,…,0, x_{k}, 0,…,0)+(0,…,0,
y_{k},0,...,0) = (0,…,0, x_{k}+y_{k},0,…,0)
α. (0, …,0, x_{k}, 0,…,0) = (0,...,0,αx_{k},0,…,0)

We define The product of A_{k} ‘s as follow:
(0,…,0, x_{k}, 0,…,0)(0,….,
0,y_{k},0,...,0) = (0,…,0, x_{k}y_{k},0,…,0) 
And define the norm on A_{k}’s by:
where, c is a constant and c>1. It is easy to verify that any A_{k}, k≥1, induces a metric by:
It is easy to show that any A_{k}, k≥1, is a Banach algebra. Since in Eq. 2 c>1 is arbitrary, thus we can obtain infinite many of Banach algebras.
Corollary 1: We can obtain infinite many of Banach algebras of ^{k}
which norm of their unit element’s norm is not one.
Note 1: According to last explanations We conclude that norm of unit element in an algebra depends on algebra norm’s. For example, We have: where, Ds(A_{k}) denotes direct sum of A_{k}’s. If we define the product in A_{k} by:
(x_{1}, x_{2}, …,x_{k}).(y_{1},y_{2},…,
y_{k}) = (x_{1}y_{1}, x_{2}y_{2},…,x_{k}y_{k}) 
And norm in Ds(A_{k}) by:
(x_{1}, x_{2},..., x_{k})
 = c max {x_{i} : 1≤i≤k} 
where, c>1 is a constant. It is easy to show that A_{k} with above definitions is a normed algebra which its unit element is (1,0,0,…….,0) and (1, 0, 0,..., 0) = c>1. Also (1, 0,0,..., 0) is the unit element in R^{k}. But its norm is one.
Note 2: Note 1 shows that we can write R^{k} as direct sum of
their subsets which any of them is a Banach algebra. Although, in the left side
of Eq. 3 the norm of (1, 0,……,0) is one where in
right side is c(>1).

REFERENCES 
1: Kreyszig, E., 1987. Introductory Functional Analysis with Application. 1st Edn., John Wiley and Sons, Canada, ISBN13: 9780471504597, pp: 4976
2: Bonsall, F.F. and J. Duncan, 1973. Complete Normed Algebras. SpringerVerlag, New York, USA., ISBN13: 9780387063867, pp: 120



