Rev.Acad.Canar.Cienc., XI (Núms. 1-2), 41-55 (1999)
REGULAR ANO SINGULAR ELEMENTS OF A COMPLETE METRIC
1-PARA-ALGEBRA ANO A METRIC 1-PARA-SEMIALGEBRA
D. K. Bhattacharya and T. Roy.
ABSTRACT : The papar introduces the notions of a metric para-algebra and a metric parasemialgebra.
lt studies regular and singular elements of an unital metric para-algebra and those of
a metric para-semialgebra with S-identity. lt also investigates the natura of radical oh metric paraalgebra
and a metric para-semialgebra.
1. INTRODUCTION : Properties of regular and singular elements of an unital Banach algebra were
studied by C.E. Rickart, 1960; G.F. Simmons, 1963 and many others. They also investigated the
natura of radical of this algebra. D. K. Bhattacharya and A. K. Maity, 1992 studied regular and
singular elements of a nonunital Banach algebra with central idempotents. Also the radical of such
algebra was studied by D. K. Bhattacharya and T. Roy, 1996.
In the present papar, it is shown that the above discussions may be carried out for an unital algebra
by dropping the property of homogeneity of the norm or even without assuming the existence of
additive inversas of elements of the algebra.
The whole discussion is limitad to four artides - the first one deals with definitions and examples of
different types of metric para-algebras and metric para-semialgebras; the second one deals with
regular and singular elements of an unital metric para-algebra; the third one studies similar elements
for a metric para-semialgebrs with S-identity; the last article investigates the natura of radical of the
above algebras.
2. SOME DEFINITIONS ANO EXAMPLES :
Definitlon 2.1. A semilinear space X over R·U{O} is an additively commutative semigroup with
identity 0,
where for all x, ye X, a, ~e R•U{O}, 1 e R•, (a+ ~)x =ax+ ~x. a(x +y)= ax +ay, 1x = x, Ox =0.
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Definition 2.2. A semi algebra X over R•U{O} is a semilinear space over R•U{O} where there is an
additional composition o : XxX -t X, called multiplication such that
(i) x o(y oz)= (x oy) oz
(ii) x o(y+z)=xoy+xoz ' (x+y)oz=xoz+yoz
(iii) 9 X= 9 = X 9
(iv) a( X o y) = ( ll X) o y = X o ( ll y) ' '<:/ X ' y ' z E X ' ll E R·u{ o} .
Definition 2.3. Let X be an unital semialgebra. The multiplicativa identity 1 of X is called a S-identity,
if for each r e X, there exists a unique re e X such that r + re = I .
Definition 2.4. A paranorm on a linear space X over a field F is a function
p : X--t R·u{o} such that
(i) p(X) = O iff X = 9
(ii) p(x+y) ~ p(x) + p(y)
(iii) p(-x) = p(x)
(iv) IA., - A.1--t O, p( x, - x)--t O=} p(A.,x, - A>c) -t O as n -too
where {xJ e X, x, y e X, <A.Je F, A. e F,. 9 is the additive identity of X.
(X, p) is called a paranormed space ovar F.
Remark 2.1. Every linear metric space/metric linear space (X. p) is a paranormed space (X, p) where
p( x) = p( x , 9) , x e X [9 being the additive identity of X].
Examples of paranormed spaces
Example 2.1. l_(r) = [x={x,J, X,. E R : Sf lx..I'' <00]
where {rk} is a bounded sequence of reals o < rk ~ H, H = Sup r., inf rk > O. 1_ (r) is a paranormed
k
space over R under componentwise operations, where paranorm p is given by
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p(x) = ~ lx.l~JM , M = max (1, H).
k
Example2.2. l{r) =[x={x.}. x. e R; tlx.I'' existsJ.where {r,} isdefinedasinexample2.1.
l(r) is paranormed space over R under componentwise operations with paranorm
p(x) = ( ~ lx.I'' r
Example2.3. x=[x={x.} . x, E R. S~Plx.I < 00].
X is a paranormed space over R under componentwise operations, with paranorm
p(x)= Sup K .
• i +lx.I
Remark 2.2. Every normed linear space is a paranormed space, but the converse is not true.
Remark 2.3. A linear space endowed with an invariant metric may not induce a paranorm.
Examples 2.1. 2.2. and 2.3. are all linear spaces with translation invariant metrics
p(x,y) = s~plx, - y,j''", p(x,y) = (~ lx.-Y.l}JM and
p( X. Y) = sup 1 1¡ - Y' 1 I respectively, where all the me tries p induce p given by p(x) = p( x, 0) .
k +X. -y.
But if in Example 2 .1 . inf r, > O then corresponding metric p , although translation invariant, can
not induce the paranorm p.
Definition 2,5. Let X be a linear space over F. A function p : X -+ WU{O} is called a l'"paranorm
on X, if for x, y e X , a e F
(i) p(x) =O iff x = 0 (ii) p(x+ y) :!> p(x)+p(y) (iii) p(-x) = p(x)
(iv) p( ax) :!> la! P(x) , la! ;;: 1
la! P(x) < P(a x) < P(x), la!< 1 .
EXAMPLES OF 1-PARANORMED SPACES.
The spaces l_(r) (example 2.1.) and X (example 2.3.) are 1-paranormeó spaces.
Remark2.4. A normed linear space is not a 1-paranonned space and a 1-paranormed space is
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not a normed linear space.
Remark 2.5. A paranormed space may not be 1-paranormed space. In fact, l(r) (example 2.2) is not
a 1-paranormed space.
Remark 2.6. Every 1-paranormed space (X, p) is a paranormed space.
Let {x.} e X and x e X such that P(x. - x)-+ O as n ....+ oo;
let {a.} e F and a e F be such that la. - aj ....+ O as n ....+ oo
Now P(a.x. - ax) s p [a.(x. - x)) + p [(a. - a)x)
lf la.I ~ 1 , then p [a.(x.-x)) s la.I p(x.-x); also as la. -al < l. so
P[(a. - a)x) < P(x) , hence P[(a.-a)x) ~ la. - a¡.(1.P(x) forsome~>1 .
Hence p(a.x. -ax)~ O as n~ .,;..
lf la.I < 1 , then p [a.(x. -x)) < p (x. - x) and
p [(a.-a)x) ~ la. - al ~P(x) , ~>l.
So, p( a.x. - a x) ~ O as n ~ 00• Hence X is a paranormed space.
Remark 2. 7. Normed linear spaces and 1-paranormed spaces form two distinct subsets of the set of
all paranormed spaces.
Definition 2.6. A semiparanorm on a semilinear space X over R·U{o} is a function p : X ~ R•U{o}
such that for all x, y e X
(i) P(x)=O iff x=9 (ii) P(x+y) ~ P(x)+P(y) .
(iii)Forevery {x.}cX and x e X forwhich IP(x.)-P(x~ ~Oasn~ oo
and for every p ... } e R•U{o} , A. e WU{O} for which IA.. -Aj~ O as n ~ oo,
it is implied that j P(A..x. ) - p(A. x) j ~ O as n ~ oo.
(X,p) is called a semiparanormed space over R·U{o} .
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Remark 2.8. A semiparanorm on a semilinear space X can never induce an invariant metric on X,
as for x, y e X, x-y is undefined. However, a semilinear space X endowed with an invariant metric
p may or may not induce a semiparanorm p on X.
A semilinear space X equipped with an invariant metric p is called a metric semilinear space if p
induces a semiparanorm p on X. In this case, it is also callad a semiparanormed space and is
denoted by (X, p).
EXAMPLES QF SEMIPABANORMED SPACES.
Example2.4. l:(r) = [x={x.}, x. >0: s~p(x.)" < oo]
is a semiparanormed space over WU{O} under componentwise operations with semiparanorm
P(x) = s~p (x.)'"' , inf r. >O, where {r.} is defined as in Vr).
lt is nota semiparanormed space if inf ~>O.
Example2.5. r(r)= [x= {X.}rX. >O: ~(x.)" <oo]
is a paranormed space over WU{O} under componentwise operations with semiparanorm
P(x) = ( ~(x.)" r
Definition 2. 7. Let X be an algebra over F with invariant metric p . lt is called a metric paraalgebra
over F if it is a paranormed space with paranorm P( x) = p( x, 9) , where for all x, y, z, e X,
P(xy, xz) :!> p(x, 9) p(y,z); p(xy, zy) :!> p(x,z) p(y,9).
Definition 2.8. let X be an algebra over F. lt is called a paranormed algebra over F if lt is a
paranormed space with paranorm p where
P(xy) :!> P(x) P(y), '\::/ x, ye X.
Remark 2.9. Every metric paraalgebra is a paranormed algebra.
EXAMPLES OF METRIC PARAALGEBRAS
Example 2.1 . and example 2.2. are metric paraalgebras under an additional composition of
multiplication defined componentwise.
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Definition 2.9. Let X be an algebra ovar F with invariant metric p . lt is callad a parametric paraalgebra
ora paranormed paraalgebra ovar F if it is a paranormed space with paranorm P(x) = p(x, 9)
where the operation of miltiplication is continuous with respect to the paranorm p, i.e.
(i) { x,,} e X, x e X , P(x,, -x)-+ O~ P(x,,y-xy)-+ O as n-+ oo, for each y e X.
(ii) {y.} e X, y e X., p(y0 -y)-+O~P(xy. -xy)-+0 as n-+ 00 for each x e X
Remark 2.1 O. A paranormed algebra is always a paranormed paraalgebra but the converse is not
true. Example 2.3. under componentwise operation of multiplication is a paranormed paraalgrbra
which is not a paranormed algebra.
lndeed, let {f,}={x,,,}cX and f={x,,} belongtoXsuchthat P(f,-f)-+0 asi-+oo .
Then sup ll -x.j ¡ -+O as i-+ oo
n l+x.,-x,,
Thus jx,,, - x,,j-+ O i.e. jx.,y. -X.Y.!-+ a as i-+ oo, for each bounded {Y.} e X.
Hence if g ={Y.} , then P(t, g- fg)-+ o as i-+ oo, for each ge X .
lt may be similarly shown that if 9; = {g,,} e X , g = {9.} e X be such that
P(g, -g)-+ O as i-+ 00 , then P(fg, -fg)-+ O as i-+ oo for each f e X. So X is a paranormed paraalgebra.
Now we show that P(fg) s p(f) P(g) does not hold for all f, g e X.
In fact, for
f={x.} , p(ff)=s~p [x.' /(1+x.' )]
~~ [ lx.I / (1+lx.I )] ~ [ lx.I / (1+lx.I )]
= p(f) p(f)
Hence X is not a paranormed algebra.
Remark 2.11. Definitions and examples of 1-paranormed paraalgebra, 1-paranormed algebra may
be similarty given. Moreover, all these definitions and examples may be given in the setting of
semilinear space as well.
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3. REGULAR ANO SINGULAR ELEMENTS OF AN UNITAL METRIC PARA-ALGEBRA.
Theorem 3.1. Let ·(X, P) be a complete metric paraalgebra with identity 1 . Then each r e X for
which P(r , 1) < 1 is regular 1i!nd its inversa s is given by s = 1 + Í,(I - r)"
n=I
Theorem 3.2. Let G and S denote respectively the set of regular and singular elements of a
complete metric paraalgebra (X, P ). Then G is an open set and S is a ciosed set.
Theorem 3.3. The mapping r-+ r-1 of G into G is continuous. As an application of theorem 3.2.
the following theorem follows :
Theorem 3.4. Let R be the radical of a complete metric paraalgebra (X. P) where R is taken as
the intersection of all maximal left ideals of X. Then X/R is a complete semisimple metric paraalgebra.
We simply state the theorems 3.1. - 3.4. , as the proofs are parallel in case of a Banach algebra.
We now define a topological divisor of zero in a metric 1-paraalgebra and obtain one of its important
properties.
Definition 3.1. An element z of a metric 1-paraalgebra (X, P) callad a topological divisor of zero,
if there exists a sequence { z.} e X such that p( z.) -+ O as n -+ oo but either zz. or z.z tends to
zero as n -+ oo .
Theorem 3.5. Let Z denote the set of all topological divisors of zero of a metric 1-paraalgebra
(X , P) , then Z !;; S ; also sorne boundary points of S form a subset of Z.
Proof. Z ¡;;; S is clear, because if z e Z but z E S then z-1 exists. As z e Z , so there exists { z.} e X
such that p(z.) -+ O as n-+ 00 but zz.-+ O or z.z-+ O as n-+ oo .
So p(z,,) = ~(z-1 z)z,, ) = P(r'(zz,,)) s P(z-1) p(zz,,) ~O as n ~ "°·
This is a contradiction. So z e S. Now to prove the other part, we take z e bd S . As S is ciosed,
so there exists {r.} e G such that P (r. - z)-+ O as n-+ oo. Also the sequence { ~r. -i)} is
unbounded, for otherwise, if p(r.-1
) < 00 then using p(r. -z)-+ O as n-+ oo. the inequality
P(r.-1z- 1) = p (r. -1(z - r.) ) ~ P(r.-1) P(z-r.), it may be shown that P(r. -1z- 1) < l. So r.-1z e G .
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Hence z = rn(rn 'z) E G . This is a contradiction. Therefore p( rn 1
) ~ 'f• as n ~ ·~ .
Let z" = (~;~',J
Sop(z.) ={~;"',)).As p{rn')~ooasn~ oo ,
so 3 N(> O) such that p(rn ') > M , M being large at pleasure, 'f n ~ N .
So, l < _..!._ < l. Using ¡~1 p (x) < p (ax)· < p (x) , if 1~1 < 1 and taking x = r"',
P(r. _,) M u¡ "1
We get P(r~-• ) p (r.-') < p (zn) < P(r" ·'). 'f n ~ N.
i.e., 1<p(z")<p(r" 1) , 'in~ N.
So, p(z")~O as n ~ oo. Moreover, it is concluded that {z,,} may not always tend to infinity as n ~ °"·
In case, p(z,,)~ oo as n ~ oo, we show that p(zz")~O as n ~ oo.
In fact, p(zz") = p(z - r" + r", z") ~ p{(z - r")z,,} + p(r"z") ~ p(z - r") p(z.) + p(r" z.).
- 1
Now, p(r.z.) = p(r". ;;" -•» = p (a"I) where ª" = p(r:-.) ~O as n ~ oo.
As pis paranorm so p(a" I) ~ p(OI) =O
So, p(zz")~O as n ~ oo but p(z")~O as n ~ oo, so z becomes a topological divisor of zero. As z"
depends on r"-1 and hence on r" andas r", in its term, is chosen corresponding to the boundary point
z, so it may be concluded that there are some points on the boundary of the set of singular elements
such that they are topological divisors of zero.
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4. REGULAR ANO SINGULAR ELEMENTS Qf A COMPLETE METRIC 1- PARA-SEMIALGEBRA
4.1 Propertles of reaular and sinaular elements.
Theorem4.1.1. Let (X,p) be a complete metric para-semialgebra wit'1 I as its S-identity. Then
each r e X, for which p(I, r) < 1, is regular. Further, the inverse of r is given by r' = I + L t"
where r + t = I .
By given conditions p(t) = p(t, 0) = p(r + t, r) = p(I, r) < 1. As p(t") :o; (p(t))" < 1, so partial
sums of L t" from a Cauchy sequence in X. As X is complete, so L t" converges to sorne
1 1
element of X. We denote it by L t" . Now
1
=p (! r t" + ft" , t+ ft")
1 2 1
= p (fr t" + f t" , f t")
1 2 1
- - - - - Again I = r + t => t" = r t" + t"•1• So L t" = L r t" + L t"'' = L r t" + L t"
. 1 1 1 1 2
Hence p (r r' , 1) = p ( ~ t" , ~ t") =O. So r r' = I . lt may be similarly shown that r' r = I . Thus r' is
the inverse of r. This proves the theorem.
Theorem 4.1.2. Let (X,p) be defined as in theorem 4.1.1. Then the set of regular elements Gis
open and the set of singular elements S is closed.
fr22! : We only prove that G is open. Let r e G be arbitrary and let r' be its inversa. We show that
the open ball with centre r and radius 1/p(r') líes in G. Let y e X such that p(y, r) < 1/p(r' ).
We prove that y e G.
Now p(yr',I) = P(yr',r r' ) $p(y,r) p(r',0)$(1/p(r')) P(r')=L
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Hence y r' e G. As composed of two regular elements is a regular elerrient,
so (y r' )r = y(r r' ) = y e G. Thus G is open and hence S is ciosed.
Theorem 4.1.3. Let (X, p) be defined as in theorem 4.1.1., then the mapping r~ r _, of G into G is
continuous.
Proof: Let r, r0 e G such that p(r, ro> < (1/2) p(r0-1) ; then
p (r0 · • r, l)=p (r0• 1 r, r~' r0)~P (r0- ') p (r, r0 )<1/2 . Sor;' re G.
- 1 ~
Now r·• r0 = (r0 _, r) = l + L t" where r0 _, r + t = I . Hence
1
P (t)=p (t, 9)=p (r0 · 1 r+ t, r0 · 1 r)=p (1, r0• 1 r)<l/2 .
= P (r0 _,) p ( 1 + ~ t" , 1)
= P (r0- ') P (~(t"))~P (r0- ') (~p (t))"
= p (r0 -
1
) [p (t)/ (1-p (t))]. [asp(t) < 1]
< 2 p (r0- ') p (t), [as p(t) < 1/2]
~p (r0 - 1) p (r0 , r) .
So p (r-•, r0-')< 2p (r/) p (r0 , r) p (r0-')= 2 (P (r0- ') r p (r, r0 )
= k p (r , r0 ) , k >O .
Hence r ~ r ·1 of G into G is a continuous map.
4.2 RADICAL OF A SEMIALGEBRA WITH S-IDENTITY
In this articie, it is shown that the usual conceps of radical of an algebra fails for a semialgebra. In
fact, it is nolonger a two sided ideal and hence the corresponding quotient algebra with respect to
this radical can not be obtained. However, left and right radicals for the semialgebra may be defined
and their characterizations may be obtained.
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Definition 4.2.1. Let. X be a sernialgebra over a field F. A left radical R, of X is defined as the
intersection of all rnaxirnal left ideals. A right radical R, is defined as the intersection of all rnaximal
right ideals,
Definition 4.2.2. A sernialgebra is callad sernisirnple if each of R, and R, is a zero ideal of the
sernialgebra.
Theorem 4.2. 1. Let R, be the left radical of a sernialgebra with 1 as its S-identity. Then R1 consists of
precisely those elernents r of X such that each r° E X for which x r + r° = 1, for sorne x E X is left
regular, without being right regular.
Proof : We first prove the following lernrnas :
Lemma 1. lf r E R1 and if r + r° = 1, then r° is left regular. lf possible let r° be not left regular. Then
[x(r°) : x E X] is a proper left ideal of X containing r°. We now irnbed [x(r°) : x E X] in a rnaxirnal left
ideal M of X. Obviously r, r° E M. So r+r° = r is irnpossible, as M is proper. Hence r° is left regular.
Lemma 2. lf r E R1 and if xr + r° = 1 for sorne x E X, then each such r° E X is left regular but r° is not
right regular.
f.!!!!!! : Since r E R1 => xr E RI' so by Lernrna 1, r° is left regular. lf possible let r° be right regular also.
Then there exists a unique s E X such that r° s = l. Now r° + xr = 1 => 1 + (xr)s = s. As s is unique, so
for given r and x, (xr)s is also unique. Hence 1 + (xr)s = s means that the cancellation law for addition
holds in X. But this is irnpossible for X. So r° is not right regular.
L.emma 3. Let r E X. Let r° E X be left regular and let r° be not right regular.
lfxr+r°=I forsornexE XthenrE Rr
Proof : lf possible let r E R1• Then r does not belong to sorne rnaxirnal left ideal M of X. So
{rn+xr : rn e M, x e X} is a left ideal containing both M and r. So {rn + xr} is irnproper. Hence xr+rn=I
for sorne rn e M and sorne x E X. Therefore by Lernrna 2, rn is left regular and rn is not right regular.
But this irnplies that rn can not belong to a proper left ideal. Hence we get a contradiction as rn e M.
SoreRr
Proof of the rnain theorern follows directly frorn Lernrna 2 and Lernrna 3.
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Theorem 4.2.2. Let R, be the right radical of a semialgebra X with I as the S-identity. Then R,
consists of precisely those elements r e X such that each rxi e X for which rx + rxi = I for sorne
x e X is right regular but rxi is not left regular.
Proof follows as in Theorem 4.2.1 .
Theorem 4.2.3. R(' R, = { 0} where R, and R, are defined as above.
Proof : Let r :F- 0 e Rr Then each r° e X such that xr + r° = I , for sorne x e X is left regular (not right
regular). So there exists a unique s e X such that s r° = I . Now let rxi e X be such that rx + r'Xl= I,
for the same r and x taken above. We show that rxi can not be right regular. lf possible let rxi be right
regular. Then there exists a unique s' e X such that
rxi s' = I . Now rx + rxi = I => (rsx) rx + (rsx)r'Xl = rsx
=> r(sxr)x + (rsx)r'Xl = rsx
=> r(sxr)x + rx + (rsx)r'Xl = rsx + rx
=> rsx + (rsx)r°° = rsx + rx [xr + r° = 1 => sxr + I = s]
=> (rsx) s' + rsx = (rsx) s' + rx s' .
As cancellation law does not hold in x, so rsx *" rx s . Naturally, any s' can act as the right
inversa of ,.00 provided rsx *" rxs' . Thus s' fails to be unique. This is a contradiction. Hence
r e: R, So R/°" R, = { 0 }.
4.3. Radical of a metric para-semialgebra with S-identity
Theorem 4.3.1. The left radical R1 of a complete metric para-semialgebra (X, p) with S-identity is a
proper closed left ideal of (X, p).
Proof: We show that Lis closed. lf Lis not closed, then Le [ (closure of L). As Lis proper, so
L e S (the set of all singular elements). As S is closed, so [ e S. Hence [ is also proper. This
contradicts maximality of L, as L. e [ . So L is closed. Thus R, is the intersection of closed ideals.
Hence R, is closed.
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Theorem 4.3.2. The right ideal R, of a complete metric para-semialgebra with S-identity is a proper
closed right ideal.
Proof is parallel to Theorem 4.3.1.
4.4. Topological divisor of zero in a metric 1-para-semialgebra
Theorem 4.4.1. Let (X, p) be a complete metric 1-parasemialgebra with I as the S indentity Let Z
and S denote respectively the set of all topological divisors of zero and that of all singular elements
of X. Then Z !.;;; S. also sorne boundary points of S may form a subset of Z.
Proof : Z !.;;; S is obvious. To prove the second part, we take z e bd S. We show that
z E Z. Now S being closed, there exists {r J e G such that p(rn, z) ~O as n ~ °"· We can prove that
p(rn·1) ~ oo as n ~ oo .
lt may be shown that p (rn-1Z, 1)<1. So rn -l Z e G . Hence z = r"(r"-1z)E G . This is a
contradiction.
Therefore p (r" -l) ~ 00 as n ~ oo .
We now write z = r" + rnº· Then
p (rn º) = p (rn o' e) = p (rn +r.°, rn) = p (z ' rn) ~o as n ~ 00 .
Next let
- 1 rn
Zn=p(rn-1)
. Then as in theorem 3.5, it may be shown that
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where n"
As a , __. O where n -+ º'" so P( a, 1)-+ Pl O 1) = O as n -+ oo . Also p(r, 0 )-+ O as n -+ 00 , if
PI_ z,,) ~ "" Then P(. z z,,) ~O as n-+ 00 • Hence z z,, -+0 as n--too. Thus for those z e bd S , for
which corresponding z, ~ oo ,
we see that z e Z . This completes the proof.
CONCLUDING REMARKS :
1. Regular and singular elements (in general) can be studied on metric paraalgebra (paranormed
algebra) and on metric para-semialgebra. Even characterization of radicals can be obtained in such
cases.
2. Special type of singular elements (topological divisors of zero) can be studied only on a metric
1-paraalgebra and on a metric 1-para-semialgebra. For a Banach algebra, the definition of topological
divisors of zero is more restricted compaired to that given for a metric 1-para-algebra and 1-parasemialgebra.
For a Banach algebra topological divisors of zero are always permanent singular
elements, but no such definite conclusion can be drawn for each singular element of a 1-paranormed
algebra and a 1-paranormed semialgebra.
REFERENCES
1. Bhattacharya, D.K. and Maity, A.K : Invertible elements of a Banach algebra without identityANALELE
STINTIFICE ALE UNIVERSITATll 'ALI. CUZA'
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New Delhi, Bombay, Calcutta (India) 1991.
54
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
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Department of Pure Mathematlcs
Unlverslty of Calcutta
35, Ballygunge Circular Road
Calcutta - 700 019
INDIA.
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© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017