jordan-arens irregularity

Jordan-Arens Irregularity

Chris Auger

A thesis submitted to the Faculty of Graduate Studies

Carleton University

in partial fulfillment of the requirements

for the degree of Master of Science in the

Ottawa-Carleton Institute for Graduate Studies and Research

in Mathematics and Statistics

©2007 Chris Auger

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Abstract

The topological centres of the convolution algebra of nuclear operators has been

studied recently prim arily by Neufang. This thesis is primarily composed of three

sections. The first section focuses on calculating the two topological centres of this

algebra for discrete of groups. The second section uses this result to discuss the

non-hereditary property of strong Arens irregularity to subalgebras. This is done by

showing the existence of a strongly Arens irregular Banach algebra which admits a

subalgebra that is not strongly Arens irregular. Finally, the last section introduces

the notion of a Jordan topological centre of a Banach algebra and successfully shows

that the convolution algebras of nuclear operators and l \ (G) are both Jordan strongly

Arens irregular.

ii


Acknowledgements

I would like to take this opportunity to thank my thesis supervisor Matthias Ne-

ufang. His guidance and patience throughout has been inspiring and was crucial to

the completion of this thesis.

iii


Contents

1 Introduction 1

2 Prelim inaries 3

2.1 Banach algebras and m o d u le s ................................................................... 3

2.2 Arens products and topological ce n tre s ................................................... 5

2.3 Nuclear operators . . ................................................................................ 10

3 The topological centres o f the convolution algebra of nuclear opera

tors 14

3.1 The convolution p ro d u c t............................................................................. 14

3.2 The first topological centre of A f(Lp( G ) ) ................................................ 20

3.3 The second topological centre of J\f(Lp( G ) ) ............................................. 21

4 Strong Arens irregularity is not hereditary 31

4.1 Strongly Arens irregular Banach algebras w ith an Arens regular sub

algebra 31

5 Jordan Topological Centres 38

5.1 General results for Banach a lgebras.......................................................... 38

5.2 J o rd a n to p o lo g ic a l c e n tre of (G ) ........................................................................... 42

5.3 Jordan topological centre of the convolution algebra of nuclear operators 52

5.4 Conclusion and main re s u lts ....................................................................... 53

iv


Chapter 1

Introduction

The Banach spaces LP(G), w ith 1 < p < oo and G a locally compact group, have

been studied extensively as well as the Banach space B (Lp (G)), the space of bounded

linear operators on LP(G). The projective tensor product LP(G) ®7 L q{G) (w ith

1 < p, q < oo being conjugate indices) can be identified w ith the space Af (Lp (G ))

of nuclear operators on Lp (G) and, when p = 2, w ith H = L 2(G) a H ilbert space,

A f(L 2(G)) coincides w ith T(Tt), the space of all trace class operators. In [18] M.

Neufang introduces a new product on Af (Lp (G)) and discusses many of the relations

between L 1 (G) and T (H) suggesting that T (H) w ith this product is a noncommu-

tative version of the group algebra L i (G). Thus we may view T ( t2 (G)) w ith the

usual composition as a noncommutative version of l \ (G) w ith the pointwise product

and T (L2 (G)) w ith this new product as a noncommutative version of L \ (G) w ith

the convolution product. I t is w ith this view that we denote this new product by

* (the usual symbol for convolution) and describe it as a convolution type product

on A f (Lp (G)). We would like to point out that this algebra has been studied subse

quently by others including Aristov in [1] and Pirkovskii in [22] where in these papers

they look at b ipro jectivity and biflatness for (J\f (L2 (G )) , *). Their results are similar

to those of (L i (G ) , *) which strengthens the analogy between the two convolution

algebras.

In [18] Neufang also discusses briefly the relation of the topological centres of

L \ (G) and A f (Lp (G)) w ith the different products. This relation is as follows: both

1


( t i (G) , •), where • is the pointwise product, and (Af (Lp (G ), o)) are Arens regular;

also for |G| < oo both (L \ (G) , *) and (Af (Lp (G) ) , *) are (triv ia lly) Arens regular.

Now, the topological centre of (L i (G) , *) has been studied prim arily by Lau and

Losert in [13] and more recently using a different approach by Neufang in [19] where

it is shown that for any locally compact group G, (L\ (G) , *) is left and right strongly

Arens irregular. In [15] Neufang shows that (Af (Lp (G) ) , *) is left strongly Arens

irregular but not right strongly Arens irregular. In [5], Dales and Lau develop a

construction of a Banach algebra that is left but not right strongly Arens irregular

where they mention that the convolution algebra of nuclear operators developed by

Neufang is an earlier example of an algebra having this property.

In this thesis we shall give the necessary definitions and some previous results

about topological centres in section 2 that w ill be used throughout. In section 3 we

w ill provide the details for showing that for a locally compact, noncompact group G,

(Af (Lp (G ) , *)) is left strongly Arens irregular but not right strongly Arens irregular

as is done in [15]. In section 4 we w ill construct, by using a result by Dales and Lau [5],

a Banach algebra that is strongly Arens irregular while adm itting a subalgebra that

is Arens regular and not strongly Arens irregular, which answers in the affirmative a

question (not published) raised by Neufang. In section 5 we look at the Jordan algebra

of a given Banach algebra. By forcing a Banach algebra to be commutative, via the

Jordan product, we look at these effects in the bidual level. The motivation for this

comes from the convolution algebra of the nuclear operators. As was stated above, in

[19], Neufang shows that this algebra is left but not right strongly Arens irregular and

we look at how this affects the continuity of the Jordan product extended to the bidual

level. We introduce the notion of the Jordan topological centre of a Banach algebra

and determine the Jordan topological centre of (L i (G ) , *) and (Af (Lp (G ) ) , *) for

two different classes of groups. The technique that we use to determine the Jordan

topological centre for these algebras is of interest on its own which is a strengthening

of the factorization result developed in [18] by Neufang.

2


Chapter 2

Prelim inaries

In this section we give some of the definitions required throughout this thesis. Other

definitions w ill be stated as needed. As well, we provide a few basic well known

results that w ill be needed later. Although we provide these definitions and basic

results, some background knowledge of these areas are assumed. For a more thorough

introduction to Banach algebras we refer the reader to [4] and [21], and for measure

theory to [3] and [11].

2.1 Banach algebras and modules

Definition 2.1.1 An algebra over F is a vector space A over a field F such that

there is a multiplication between elements that is distributive and associative, i.e.

Vx, y,z G A, we have

i ) x (y + z) = xy + xz

i i ) (x + y) z = xz + yz

Hi) (xy) z = x (yz)

and is linked to scalar multiplication in an obvious way, i.e. Va € F, we have

(ax) y = a (xy) — x (a y ) .

N ote : For this thesis, when the term algebra is used, it refers to a algebra over

C.

3


D e fin it io n 2 .1.2 A Banach space is a complete normed space.

Definition 2.1.3 A Banach algebra is an algebra A, together with a norm || ||,

such that (A, || ]|J is a Banach space and\/x ,y E A, \\xy\\ < ||x||||?/||.

Note: We can take as a definition of a Banach algebra to be a Banach space A,

w ith norm || ||, along w ith an algebra structure where 3K > 0 such that Vx,y E A,

\\xy\\ < i f 11x|| ||?/1| since by putting ||| ||| — K || || we get the desired inequality. There

is no problem in doing this since scaling a norm does not change any of its topological

properties.

Definition 2.1.4 Let A be an algebra. A left A-m odule is a vector space X such

that there is an action of A on X (i.e. a map A x X ► X ) which is denoted by a - x

fo r all a E A and x E X which satisfies the following:

i ) (a + b) ■ x = a • x + b • x, Va, b E A, Vx E X ,

ii) (ab) ■ x = a ■ (b ■ x ) , Va, b E A, Vx E X ,

Hi) a • (x + y) = a • x + a • y, Va E A, Vx, y E X ,

and i f A has a unit, then

iv) 1 ■ x = x, Vx E X .

There is an analogous definition for a right A-module in which the multiplication

of the elements between A and X are reversed, i.e. the action of A on X is denoted

by x ■ a for all a E A and for all x E X .

Definition 2.1.5 An A-bimodule X is both a left A-module and a right A-module

such that

(a • x) • b = a • (x • b) Va, b E A Vx E X.

Definition 2.1.6 Let Abe a Banach space. A Banach A-bimodule X is a Banach

space X such that X is an A-bimodule, and the operation

A x X x A —> X , (a,x,b) i—> a • x • b

is jo in tly continuous, where A x X x A carries the product topology.

4


Example: Let A be a Banach algebra. Then

A* = { / : A —» C | / is linear and continuous} ,

the topological dual of A, is a Banach A-bimodule in the following sense: for a ,b E A

and / G A*

(a ■ f , b ) = ( / , ba)

and

{ / - a ,b ) = ( / , ab)

where ( , ) is the dual pairing of A* and A (i.e. ( / , a) f ( a ) ) .

One can define similar actions of A** on A* making A* a Banach A**-bimodule

and we explore this in the next section.

2.2 Arens products and topological centres

Let (A , *) be a Banach algebra. The Arens products are defined on the bidual A** of

A via module actions of A** and A on A*.

Definition 2.2.1 The firs t (or left) A rens product, denoted by ©, is defined as

follows. Let m ,n G A**, h e A*, f , g G A, then m Q n G A**, n © h G A* and

h O f G A* are given by

(m © n, h) := {m, n Q h )

(n Q h , / ) := (n , h ® f )

(h © / , g) := ( h , f * g ) .

Definition 2.2.2 The second (or right) A rens product, denoted by o, is defined

as follows. Let m, n G A**, h € A*, f , g G A, then m o n g A**, h o rn G A* and

f o h G A* are given by

(m o n, h) := (n , h o rn )

( h o m , f ) := ( m j oh)

( f o h ,g ) := (h , g * f }.

5


In order to show that these two products on A** are extensions of the original

product on A, let m ,n E A and f E A*. When we write (m, / ) we mean (rh , / ) where

the map m t—» m is the canonical embedding of A into A**. Then

(m Q n , f ) = (m, n Q f )

= (n © / , m)

= { n , f Q m)

= ( f © m, n)

= ( / , m * n)

= (m * n , f )

and

(m on, f ) = {n, f om )

= ( f om , n)

= {m ,n o / }

= (no f , m)

= ( f , m * n)

= (m * n , f )

D efinition 2.2.3 Let X be a Banach space. We say that a net {4>a)aei C X * con

verges w * to some element (j) i f fo r every f E X we have that {<fa , f ) — > (<f>, / ) .

I t is clear by the definitions of the Arens products that the maps n i-» n © m and

n h-> m o n a rc w* — w* co n tin u o u s fo r ev e ry m G A**, a n d so wc define th e f irs t a n d

second topological centres as follows.

D efinition 2.2.4 The f irs t topological centre of A**, denoted by Z \ (A**), is the

set

Z \ (A**) = {m E A** | the map n i—> m 0 n is w* — w* continuous}.

6


D e fin it io n 2.2.5 The second topolog ical centre of A**, denoted by Z f(A ** ) , is

the set

Z? (A**) — {m E A** | the map n n n o m is w* — w* continuous}.

Using the same techniques employed to show that the Arens products are exten

sions of the original product, we w ill now show that A is always contained in each of

the centres.

P ro p o s itio n 2.2.1 Let A be a Banach algebra. Then A C Z ] (A**) f l Z f (A **).

P ro o f Let m E A and let {na)aeI be a net in A** converging w* to 0. We have to

show that m Q n a — > 0 (w*). For every f E A* we have

(m © nQ, / ) = {m ,na © / )

= (na 0 / , m)

(na, f (Dm) — ► 0.

Similarly,

(na om , f ) = ( m , f o n a)

= ( / o n a,m)

= (na, m o f ) — > 0 .

This shows that m e Z \ (A**) f l Z t2 (A**). |

There is an alternative way of describing the first and second topological centresyj*

of A**, and we w ill show this next by using Theorem 8.6 in [2] which gives A = A**

as Banach spaces.

P ro p o s itio n 2 .2.2 Let A be a Banach algebra. Then

i)Z l (A**) = {m E A** | m Q n = m o n Vn E A**}, and

i i )Z f (A**) = {m E A** \ n Q m = n o m Vn E A**}.

7


P ro o f Let m G Z \ (A**) and n G A**. Let (m a)aeI and (np)/3eJ be nets in A such

that ma —»• m and np —► n (w*). Then

m o w = w* — lirng(m o rip)

= w* — lim Jg(iy* — lim a(ma o np))

— w* — lim Jg(«;* — lim a(ma 0 np))

— w* — \imp(m © np)

= m © n.

Similarly, assuming now that m € Z( (A**),

n o m = w* — limp(np o m)

— w* — lirng(u>* — lim a(np * ma))

= w* — lim p(w* — lim a{np © m a))

= w* — lirn0(np © m)

= n@ m

The other directions are clear. |

C o ro lla ry 1 Both Z^(A**) and Z?(A**) are subalgebras of A** on which © and o

coincide.

D e fin it io n 2.2.6 Let A be a Banach algebra. Then A is called le f t ( r ig h t ) s tro n g ly

A re n s ir regu la r, abbreviated LSA I (RSAI), i f Z) (A**) = A (Z( (A**) = A). The

algebra A is called s tron g ly A re n s i r re g u la r i f Z} (A**) = Z ( (A**) = A and called

A re n s regu la r i f Z) (A**) = Z( (A**) = A**.

We w ill now prove that L \ (G) is strongly Arens irregular, a result that w ill be

used frequently in this thesis. This result has been shown by Lau and Losert in [13]

and by Neufang in [18] and [19] and we give a general proof for any Banach algebra

that has two fundamental properties which w ill now be defined (cf. [17]).

D e fin it io n 2.2.7 ([17]) Let X be a Banach space and k a cardinal number.

i)A functional f G X** is called w* — k continuous i fV nets (xa)aGl C B a ll(X *) with

8

(np G A C Zf(A**))

(ma,np G A \/a,(3)

(m G Z[(A**))

orn G Z t2 (A**))

(ma,np G A Va:,/?)

(np G A C Z[(A**))


< |L| < « such that xa — > 0 (w*), we have ( f , x a) — > 0.

i i) We say that X has M azu r’s property o f level n i f every functional in X** which

is w* — k — continuous, actually belongs to X .

D e fin it io n 2.2.8 ([17]) Let A be a Banach algebra and k a cardinal number. We

say that A* has the left A**-factoriza tion property o f level k i f

V (/ii) ig/ C B a ll (A * ) , with |/| = k, 3 ( ^ ) . €/ C B all (A **) 3h e A*

such that hi = if i Q h fo r all i £ I .

Similarly we say that A* has the right A** -factorization property o f level k,

i f

V (hi)ieI C B a ll (A*) , | / | = «, 3 C0i)ie / C Ball (A**) 3h e A*

such that hi = ho 'if i fo r all i £ I .

Theorem 2.2.1 ([17]) Let A be a Banach algebra with the Mazur property of level

k > Hq- I f A* has the left A**-factorization property of level k (resp. right A**-

factorization property of level k) then A is LSAI (resp. RSAI).

P roof Let m £ Z [ (A**) and take a net (ha)aeI C Ball(A*) w ith |/| < k which

converges w* to 0. Since A has the Mazur property of level k . we only need to

show that (m, ha) converges to 0. I t is sufficient to show that every convergent

subnet ((m, ha))aGl converges to 0. Let ((m , ha0))^ be a convergent subnet. As

A* has the left A**-factorization property of level k we have that ha = i [a © h w ith

( 'fo)OJ=j C Ball (A**) and h £ A*. By Alaoglu’s Theorem, there exists a ltd-convergent

subnet • Let E — w* — lim E Ball (A**). Now, using the fact that ha

converges w* to 0, we have for all a E A

(E © h,a) = (E , h 0 a) = lim , h Q a ^ — lim 0 h, = lim ( h a^ ,a ^ > = 0.

9


Thus E Q h — 0. So, as m e Z[ (A **):

lim (to, ha0 ) = lim ^to,

= lim (m , 4>a^ © h

= lim ( m © ij>afh,7 \

= (to © E, h)

= (to, E Q h)

= 0.

Hence, m E A, and A is LSAI.

The same arguement works when taking m E Z( (A**) and using the factorization

h a = g o ip a w ith g E A*. |

C o ro lla ry 2 ([17]) Let G be a locally compact, non-compact group. Then L \ (G) is

SAL

P ro o f By (Lemma 2.1, [19]) we know that L \ (G) has the Mazur property of level

n(G), where k (G) is the compact covering number of G. By [20] (Theorem 2.1)

we have that L\ (G) has the left A**-factorization property of level k (G) and minor

changes to the proof (replacing right translations w ith left and an obvious alteration

to the construction of the net) we get that L i (G) has the right A**-factorization

property of level k (G ). By the above theorem we have that L i (G) is both LSAI and

RSAI and thus SAL |

2.3 Nuclear operators

W e firs t s t a r t w ith som e m e a su re th eo ry .

D e fin it io n 2.3.1 Let (X ,s r f , f i) be a measure space. A subset A of X is called g,-

n u l l i f there is an element B £ srf such that A C B and g (B) = 0. We say that a

subset Af C X is loca lly g -n u l l i f fo r each A € srf with g (A) < oo the set A n Af is

g-null.

10


D e fin it io n 2.3.2 Let X be a space, sF be a a-algebra of subsets of X , and /a be a

measure defined on sA. A property of points in X is said to hold pb-almost every

where (abbreviated p-a.e.) i f the set of points fo r which the property does not hold

is p-null.

The next theorem contains the definition of a left Haar measure as well as the

existence of this measure for a locally compact group.

T heorem 2.3.1 ([3]) Let G be a locally compact group and sA a a-algebra of subsets

of G. Then there exists a left translation invariant regular Borel measure p on srf

which is called a left H aar measure (i.e. p (gA ) = p (A ) VA e sA, Vg e G where

gA = {ga | a e A }).

D e fin it io n 2.3.3 Let G be a locally compact group, let p be a left Haar measure on

G and let 1 < p < oo. Then

Lp (G) = { f : G ^ C f is Borel measurable and / \ f \pdp < oo[ If \ pdp J g

and the norm || ||p on LP(G) is given by ||/||£ = f G \ f \pdp. For p = oo we define

|| ||oo by letting ||/||oo = infM>o {{p € G \ \ f (g) \ > M } is locally p-null}, then

Loo (G) = { / : G —> C | / is Borel measurable and ||/||oo < oo} .

Note: The functions / in the above definition are actually equivalence classes of

functions [/], where

[/] = {h : G —> C | h is Borel measurable and / = h p — a.e.}.

Also, w ith these definitions, we get the following well known results:

L x (G)* = L x (G)

and for 1 < p, q < oo w ith ^ ^ = 1 (this is also referred to as p and q being

conjugated):

LP (G)* = L q (G ) .

11


D e fin it io n 2.3.4 Let X and Y be Banach spaces and let Y C X * , then we denote

the p re -a n n ih i la to r as Y± and the a n n ih i la to r as Y 1 , where

Y± = { x e X \ ( y , x ) = 0 V y e Y }

and

y 1 = {x e X**| (x,y) = 0 \/y e Y } .

D e fin it io n 2.3.5 Let X and Y be Banach spaces. Then B (X, Y) denotes the Banach

space of bounded linear operators from X to Y. I f X = Y then we write

B (X, Y) — B (X , X ) = : B ( X ) .

The nuclear operators on Lp (G). denoted by Af (Lp (G)), have a number of equiv

alent definitions. The one that we w ill take is given in the following.

D e fin it io n 2.3.6 Af (Lp (G)) — Lp (G)®7L q (G) where q E (1, oo) such that ^ ^ = 1

and <g>7 is the projective tensor product.

The projective tensor product norm is defined as follows: For x E Lp (G) 0 L q (G),

So, the space of nuclear operators is then the completion of Lp (G) <S> L q (G) w ith

respect to this norm. There is a subtle point one should be aware of when looking at

Lp (G) <g>7 L q (G) which is that an arbitrary element of Lv (G) Cg>7 L q (G) has the form

Fortunately, because we are using the projective tensor product, by Proposition 2.8

in [24] this element actually has the form

N ote : The way in which the nuclear operators evaluate an element in Lp (G) is

by the following: let x ® y be an elementary tensor in A f (Lp (G)) and take an element

(in norm topology).i=0

k

i=0

12


f £ Lp (G ). Then x ® y ( / ) = (y, f ) x, where (y, / } is the usual pairing of elements

in L q (G ) = Lp (G)* and Lp (G ). Since Lp (G) ®7 L q (G) is the completion of the set

of all linear combinations of elementary tensors, this evaluation extends linearly to

all elements of Lp (G) ®7 L q (G) = A f (Lp (G)). Also, as Banach spaces, it is shown

in ([24]) that A f(Lp(G))* = B (L P(G)). The dual pairing between Af (LP(G)) and

B (L P(G)) is given by the following. For T £ B (L P(G)) and an elementary tensor

p — £ <S> y in Af (Lp (G)) we have

(T ,P} = {T ,£® r]} = (T£, 77)

where the last dual pairing is the usual of Lp (G) and L q (G). As the dual brackets

are bilinear this extends linearly and by the continuity of T, it also extends to the

completion.

R em ark: When p = 2, the space A f (L 2 (G)) is also referred to as the trace class

operators T (L2 (G)) and is commonly written as T (H ).

The spaces LU C (G), of left uniformly continuous functions on G , RUG (G), of

right uniformly continuous functions on G and Co (G ), of continuous functions on G

vanishing at in fin ity w ill be used throughout this thesis and are defined as subspaces

of Cb (G ), the bounded continuous functions on G as follows.

D e fin it io n 2.3.7

LU C (G ) = { / £ Cb (G ) | the map G — > (Cb (G ) , || ||oo), x 1—»• lxf is continuous}

where lx : L 00(G) —> L 0 0 (G) defined pointwise by lxf ( y ) = f (x y ) .


RUC (G ) = { / £ Cb (G ) | the map G — >• (Cb (G ), || ||oo), x 1—> r xf is continuous}

where r x : Loo(G) —> L 00(G) defined pointwise by r xf ( y ) — f(y x ) .


Cq (G) — { / £ Cb (G) | Ve > 0 3K c G compact such that \/x ^ K : \ f (x) \ < e} .

13


Chapter 3

The topological centres of the

convolution algebra of nuclear

operators

3.1 The convolution product

In this section G denotes an arbitrary locally compact group. We start by defining a

product * on the nuclear operators as follows. For p, t e A f (Lp (G)) and

T e B (Lp (G)) = A f (Lp (G))*, we set

(p * t ,T ) := (r, T 0 p)

where the module operation © is defined to be

T © p := M (LtTLt_ltP'j

with M^LiTL _i ^ being the m ultiplication operator of the function 1 1—> (L tT L t- i ,p )

and L t is used instead of lt when T is an operator rather than a function. For example,

i f / 6 Lp (G) then

(m ( w v „ ) / ) t o = {LXTLX- I, p) f ( x) .

The next proposition explores a b it of the nature of the function (L tT L t- i ,p ) .

14


Proposition 3.1.1 Let p ^ N ( L p (G)) a n d T e B (Lp (G)). Then t f ' p G RUC(G)

where 7f ' p(t) := (L tT L t- i ,p ) .

Proof Define r f ' p : G —> C by ^J ’p (t ) := (L tT L t- i , p). Then

l l ^ l l o o = sup|7p ( t ) | teG

= sup I (L tT L t- i ,p ) IteG

< sup \\LtT L t- i IIHpIIteG

< sup ||Lt||||T ||||Lt-i||||p||teG

So we have that /'/[ 'p G Loo (G). To show that in fact 7f ' p is actually in RUC (G ) we

let e > 0 be given. Recall that J\f (Lp (G)) = Lp (G) L q (G) where ^ ^ = 1 and

for p G A f {L p(G)) then p = lim fc Y l n = i ^ ® V n for some £n G LP(G) and pn G L q(G).

Let pk — Y lkn = i ® Vn then \\p — pk\\jy — > 0 as A: — > 0. So, it suffices to check that

(L tT L t- i ,p ) G RUC (G) for an elementary tensor since

\(LtT L t~i , p) — (L tT L t- i , pk)\ < ||T||||p — Pk\\tf-

Now, because of the strong continuity of left translation on LP(G) [8] (Proposition

2.41), we know that there is a neighbourhood U of e such that for each x G U we have

IIL x£ - Clip < 2||T|||M|a and WL ^ ~ vWq < 2 ||T|||[g||p where we assume that ||T||, ||£||p,

and ||?7||9 ^ 0. Since G is a topological group, we know that there exists a symmetric

neighbourhood V C U. Thus for each x G V C U (and thus x -1 G V C 17) we have,

II T,p T,p ||Wxh - l l lo o

= sup \rxr f ’p (t) - 7;r ’p (t)te G

= sup |7J ’P (tx) - r f ' P (t) |teG

= sup | ( L txTL^txy i , p ) — (L tT L t- i , p) \ teG ' /

= sup | ( L txT L ,tx)- i , £ ® r i ) - {L tT L t- i ,£ 0 p) teG ' I

15


< \\LX- ^ - e l l P | | r | | | | 7 7 | | 9 + \\Lx-ir] - r y l U l T l l l i e l l p

< £.

I

Since (L tT L t- i , p ) G RUC (G) then we have that T 0 p G M RUc (g) C M Roo(g)-

P ro p o s itio n 3.1.2 (J\f (Lp (G ), *)) is a Banach algebra.

We first show that for all p , r G N (Lp (G)), p * r e N (Lp (G)) and also the relation

\\p*t\\m < \\p\W \ \ t ||aa holds. Now, it is clear that p * r e B (Lp (G))*. So, let (Ta)aeI

be a ic*-convergent net in Ba ll(23 (Lp (G))) converging to 0. I t suffices to show that

(p * r, Ta) converges to 0 in view of Theorem 3.16 in [7]. We have that (L tTaL t- i , p)

converges to 0 pointwisely and by using the surjection n : J\f(Lp(G)) -» L-RC) we

have

where A is a left Haar measure on G. As n (r) G T i(G ), then 7r(r)dA G M {G ). By

Theorem 3.1 in [16] (more precisely, part 2. of the proof), this integral goes to 0 in

P ro o f The fact that N (Lp (G)) is complete follows from the definition

N {Lp (G)) = Lp (G) ®7 L g (G) where 1 + 1 = 1.

{p * Ta) = {t , Ta © p)

= {LtTaL t- i ,p ) )

= [ {L tTaL t- i , p)tt{t )(t )d \ ( t )

16


a. Thus, p * r G M (Lp (G)). To show that || p * t||j\/- < ||p||Af IMIjV we recall that the

canonical embedding of a Banach space into its bidual is isometric. So,

\ \p*T\W = \\p * T\\B(Lp{G)y

— sup { | ( p * t , T )| ; ||T|| < 1}

= sup{| ( r ,T © p ) | ; ||T|| < 1}

= SUP { | ( T) } | ; HT H < l }< s u p { || r | |^ | |M ^ (T V i|^ || ; ||T|| < l }

< s u p { | | r | | ^ \ (L tT L t- i ,p ) \ ; ||T|| < 1}

< s u p { | | r |U \\LtT L t- 4 | H k ; | |T | |< 1 }

< s u p { | | r | k \\Lt \\ ||T|| I IV . I I |H U ; ||T|| < 1}

< IM W \\p\W-

Next we show that * is distributive and associative. Let p,T,(j) e J \ f (Lp (Gj) and

T e B (L P (G)).

Distributive: We have

((p + r ) * 0, T) = ((/), T © (p + r ) )

and

T Q ( p + t ) = M ^ LtTLt_1}P+T}

= ■^(LtTLt_1,p) + (LtTLt_i,r)

= M ( L tT L t - 1,p) + M ( L tT L t - 1,T)

= T Q p + T Q r .

17


So

{ { p + t ) * ( j> ,T ) = (<f>,T © (p + t ))

= ((f), T © p + T © r )

= (<f>,T Q p ) + (</>, T Q t )

= ( p *4> ,T ) + (t * <f>, T )

— (p * <f) + T * 4>,T) .

Also,

(0 * (p + t ) , T) = {(p + t ) , T 0 0) = (p, T © 0) + (r, T © 0) = (0 * p + <f> * r, T ) .

Associative: By Proposition 3.1.1 we have that h ( t) := (L tT L t- i ,p ) E M /G (G )

and /sh (f) = h (st) = (^LstTL^sty \ , p j . Also, for every f E Lp (G) and x E G,

L sM hL s- i f (x) = M hla- i f (sx ) = h (sx ) f (x) = lsh (x) / (s) = M hhf (x )

which gives L sM hL s- i = M ish. So,

LfTLt- i © p = M^LaLtTL _iL'_upj

= M ( L (ts)T L ( t s r l ,p)

~ ^ h ( L 3T L s^ , p )

= L tM^LsTL'_ip ^Lt-1

= Lt(T Q p)Lt- 1 .

18


Thus,

( ( p * r ) * 0 , T ) = (0, T © (p * r ) }

= ^ M ( L t T L t - l t p * T ) )

= ( ^ ^ * r , L (TLr i ) }

= ^ ( T,LtTLt_i 0p))

= ( & M (T ,L tT Q p L t - 1 )}= (0 , ( T © p ) © r )

= ( r * 0, T1 © p)

= (p * ( r *</>), T ) .

Scalar multiplication: Let a 6 C. We have

( a ( p * r ) ,T ) = a ( p * r , T)

= a ( r ,T Q p)

= (a r, T © p)

= (p * ( a r ) , T)

and

((a p ) * r , T ) = { T , T Q ( a p ) )

- { T ^ M ( L tT L t - 1,ap)

= ( T ’ M a { L tT L t - Up)

= (r , a ( T Q p ))

= a (r, T © p)

= a {p * t , T ) .

I

19


3.2 The first topological centre of J\f(Lp(G))

For this section we w ill restrict ourselves to second countable, locally compact non

compact groups. We want to show that the convolution algebra of nuclear operators

is left strongly Arens irregular, i.e. Z} {Af {Lp {G))**) = A f (LP(G)). This follows

closely [15].

To show this we cite Theorem 3.2 [17] but we first need to define a map that is

used in this theorem. We define the mapping

F : RUC {G)* -> B { B { L p {G)))

m i—► T (m)

where for T € B {Lp (G)) and p € A f (Lp (G)),

(T (m )T ,p ) = m ( {L tT L t- i ,p ) ) = {m, (LtT L t- i ,p ) )

where, as usual, (L tT L t- i ,p ) stands for the function G 3 t i—> (L tT L t- i ,p ) which is

in RUC(G).

Theorem 3.2.1 Let (Tn) be an arbitrary bounded sequence of operators in B (Lp (G)).

Then there exists a sequence {f>n) C Ball {RUG {GY) and a single operator

T E B {Lp {G)) such that the factorization Tn — F (ipn) T holds fo r all n.

W ith Theorem 2.2.1 i t is enough to show that A f{Lp {G))* has the left Af {LP{G))**-

factorization property of level d0 and A f{Lp (G)) has the Mazur property of level d0.

The last condition is true since by Lemma 7.5 of [12], Af {Lp (G)) = Lp {G) ©7 L q {G)

is even separable because G is second countable.

As can be seen from the definition of the factorization property we need to have

our sequence {f>n) in Ball {Af {Lp (G))**). I t is easy to make this change and we show

this w ith the next theorem.

Theorem 3.2.2 Let (Tn) be an arbitrary bounded sequence of operators in A f {Lp (G))*.

Then there exists a sequence {mn) C Ball{A f {LP{G))**) and a single operator

T G Af (Lp {G))* such that the factorization Tn = mn © T holds fo r all n.

20


P ro o f Given a sequence (Tn)neN C Af (Lp (G))* — B (Lp (G )), by Theorem 3.2.1 we

have a sequence (ipn) C B all (RUC (G)*) and an operator T G Af (Lp (G))* such that

Tn = T (fjn) T holds for all n. For each n, the elements tpn G RUC (G )* can be used

define ^ G M ruc(g) by (ijfn, JVR) := fd;n. h). Let mn be norm preserving Hahn-

Banach extensions of f)'n to B (Lp (G))*. Then for all p G Af (Lp (G)) we have

Thus Tn = mn Q T . |

Theorem 3.2.3 Let G be a second countable, locally compact, non-compact group.

Then A f(Lp(G) is left strongly Arens irregular.

Proof Theorem 3.2.2 shows that A f (Lp (G))* has the left A f (Lp (G))** —factorization

property of level d0 and so we have the conditions of Theorem 2.2.1, hence Af (Lp (G))

is left strongly Arens irregular. |

3.3 The second topological centre of J\f{Lp(G))

In the previous section, it was shown that A f (LP(G)) is left strongly Arens irregu

lar. Our goal for this section is to show that A f (Lp (G)) is not right strongly Arens

irregular. This is done in [15] but we shall provide the details here.

We w ill use the following two lemmas to prove this result.

Lemma 3.3.1 LetG be a locally compact group. Then (M Loo (G,)1 C Z p M ( L p ( G ) n .

to define functionals if'n on M fRUC(G) canonically by the following. For h G RUC (G)

(Tn,p) = { T { f n)T ,p )

= ( f ’n, (L tT L t-\, p))

{mn,T Q p }

(mn ® T , p ) .

21


P ro o f Let p £ Af (Lp (G)) , T £ A f (Lp (G))* and n £ A f (Lp (G))** w ith ( tq)q6J a net

in Af (Lp (G)) such that w* — lim a r a = n. Then,

(T o n ,p ) = (n, p o T )

= w* - lim a (rQ, p o T )

= w* - lim a (poT ,Ta)

= w* - lim a (T, r a * p)

= w* - lim a (T © r Q, p ) .

Hence we have that T o n — w* — lim a T 0 r a £ M Lrxp?)•

Now let m £ (M Log(G))X C A/" (Lp (G))**. We want to show that

m £ Z (2 (Af (Lp (G))**) and the way in which this is done is to prove that for every

n £ A f (Lp (G))** we have that n 0 m = O = n o m .

So w ith T £ A f (Lp (G))* ,n £ A f (Lp (G))** w ith (ra)aeI a net in A f (Lp (G)) such

that w* — liniQ, r a = n we see that

T o n = w* - l im T © r a £ M Loo(G),a

and thus

(n o m ,T ) — (m, T o n) = 0.

So this gives that n o m = 0.

On the other hand w ith T £ A f (Lp (G))* and p £ Af (Lp (G)) we have,

(n ® m ,T ) = (n, m © T)

and

(m ® T ,p ) — (ra, T 0 p) = 0.

Therefore, ra © T — 0 which gives that n Q ra = (J. |

Lem m a 3.3.2 Let G be a locally compact, non-compact group. Then

0 ^ (O ) )± \ V ( L P( G ) ) ^ 0 .

22


P ro o f First recall that Af (Lp (G))* — B (L p (G)) and hence A f (Lp (G))* is unital.

By the Hahn-Banach Separation Theorem there exists an n G (M c0(G))L such that

(n, 1) = 1 since G is non-compact. Let P M p (G) denote the u>*-closure of the sub

algebra of B (Lp (G)) generated by the right regular representation of G on Lp (G).

Take R G P M p (G) \ M Loo(G) and

m G (M Loo(g) )x such that (m, R) — 1 (again using the Hahn-Banach Separation

Theorem). Set E = m © n G Af (Lp (G))**. We w ill show that

£ € (M w g ) ) ± \J V (L p (G)).

Our first step is to prove that E ^ 0. So let T G Af (Lp (G))*, then

(E ,T ) = { m O n , T )

= (m, n O T )

= {n O T , ra)

= (n, T © ra)

= (n ,M ^ LtTLt_um^ .

Now, since R G P M p (G) we have that = L ti?, Vi G G and thus from the

above calculation we get

(E, R) = (n , M (LtRLt_i my = (n, M M ) = (n , M i) - (n, 1) = 1.

So E ^ 0. To see that E G (M ioo(G)) 'L we let h G Loo (G) and again using the

calculation above and the fact that ra € (m Loo(g)) 1 we get

(E, M h) = (n , M (LtMhLt- 1,m)') = (n ,M ^ Mi^ my = 0.

Finally, to show that E £ A f (LP(G)). We shall do this by showing that for any

compact operator K we have that (E, K ) — 0. So, let K G K, (Lp (G)). Then from

[16] (Lemma 2.5) we have that (L tK L t- i ,m ) G Co(G). Thus as n G (Cc^G))1 we

h av e

(E , K ) = (n , M^LtKL^ my = 0.

I

T heo rem 3.3.1 Let G be a locally compact, non-compact group. Then Af (LP(G)) is

not right strongly Arens irregular.

23


Proof By the previous two lemmas we get that

0 # \ V(L,,(G)) c Z 2 (J V (M G )D \ V (L„(G )).

Hence, V (L P(G)) C Z 2 (V (LP(G))). |

Combining this section w ith the last section we arrive at the conclusion that for

any second countable locally compact non-compact group G, A f (Lp (G)) is LSAI bot

not RSAI. We would like to understand Z \ (Af (Lp (G))**) better. When we restrict

ourselves to discrete groups we have discovered a construction in which we can find

the second topological centre. To do this we call upon a construction developed by

Dales and Lau in [5]. This is given in the following proposition.

Proposition 3.3.1 Let (A, *) be a Banach algebra and E be a Banach A-bimodule.

Then 21 = A E is a Banach algebra where the product on 21 is as follows:

(a, 0) □ (6, r ) = (a *b ,a • r T (f ■ b) Va, b <E A V</>, r e E.

Proof We start by showing that 21 is an algebra. Let (a, <f>), (b, r ) , (c, if) € A © E.

Associative:

[(a,0)D (6,r)]D(c,'0) = (a * 6, a ■ t + f ■ b) □ (c, ip)

( ( a * b ) * c , ( a * b ) ■ i f + (a • t + ■ b) ■ c)

(a * (b * c) ,a • (b ■ if) + (a ■ r ) ■ c + (</> • b) • c)

(a * (b * c) ,a ■ (b ■ if) + a • ( r ■ c) + <p ■ (b * c))

(a * (b * c) ,a • (b ■ i f + t • c) + cf • (b * c))

(a, <p) □ (b * c, b ■ i f + t ■ c)

(a, <f)n[(b, r ) n ( c , i f ) ] .

24


Distributive:

(a, p) □ [(£>, r ) + (c, tp)] = (a, p) □ (6 + c, r + 0 )

= (a * (6 + c) ,a • ( r + -0) + P • (& + c))

= (a*fe + a * c , a - r + a- 0 + 0 - & + 0 -c)

= ( a * b, a ■ t + p ■ b) + ( a * c, a ■ p + p • c)

= (a, P) □ (6, r ) + (a, 0) □ (c, 0 )

and

[(a ,0 ) + ( & , t ) ] D ( c , 0 ) = (a + 6,0 + r ) D ( c ,0 )

= ((a + b) * c,(a + b) - p + (p + r ) ■ c)

— (a * c + b * c, a - p + b- p + p - c + r - c )

= (a * c, a • p + p ■ c) + (b * c,b • ip + r ■ c)

= (a, P) □ (c, 0 ) + (b, t ) □ (c, ip) ;

and finally for a E C

(a (a, 0)) □ (6, r ) = (aa, ap) □ (6, r )

= ((aa) * 6, (aa) • r 4- (exp) ■ b)

= (a (a * b) ,a (a • t ) + a (p • b))

— a (a * b, a ■ t + cp ■ b)

= a ((a, (p) □ (b, r ) )

= a (a * b, a ■ r + p • b)

= (a (a * b) , a (a ■ t ) + a (p ■ b))

= (a * (ab) , a • (a r) + 0 • (afe))

= (a, 0) □ (aft, a r )

= (a, 0 ) □ (a ( b , r ) ) . |

25


Our goal here is to determine the second topological centre of A f (£p (G))**, where

G is discrete. In the previous section we showed that the first topological centre

of A f (£p (G))** is A f (£p (G)). In this section we showed that Af(£p (G)) is not right

strongly Arens irregular, and indeed we w ill show that the second topological centre

is

G (G) © (Af/oo(G)) 'L while Af ( lp (G)) itself is G (G) © (M eoo{G)) r

Proposition 3.3.2 A f (£p (G)) = G (G ) © [M ioo o))± as Banach spaces.

P ro o f We first begin by noting that since G is discrete we can represent an element in

J\f (£p (G)) as a m atrix where the coordinates of the m atrix are determined using the

group elements. W ith this representation, one can view G (G) functions as operators

in which the m atrix representation is a diagonal m atrix in a canonical way. This

shows that G (G) C Af ( lp (G)). Now, let / G A f (£p (G)) and consider its canonical

embedding into A f ( ip (G))** = B { ip {G))*. Let / i = f\ex (G) € (G)* (using the

natural isometric isomorphism between (G) and M ^ G)). Since / G A f ( ip (G)),

f is w* continuous and thus / i is w* continuous which implies that f i G G (G). Set

fa = f - h - We claim that f 2 G (M €oo(g)) j_. Indeed, let T G M loo{G) then T = M h

for some h G ^oo(G) and so we have

(T, / 2> = (M h, f 2) = ( f2\£oo{G), h) = ( ( / - h ) 1 ^ (0 , h)

= (f\ioo(G),h) - </i, h) = ( f i ,h ) - ( / i , h) = 0

which gives A f { ip (G)) = G (G) + (M£oo(g)) j_. As G (G) n (M ioo{G)) ± = {0 } we have

that this sum is direct. |

We now show that ( M ^ g ) ) j_ an ideal in A f (l v (G)). Once we have this property

then we can define the action of G (G) on (AA^fo')) ± to be the convolution product

on Af (£p (G)) making (AAoo(G))x a Banach G (G)-bimodule.

Lem m a 3.3.3 A f {£p (G)) * (M loo{a)) L = 0.

26


P ro o f Let p e M {£v (G)), r G {Mloo[G)) L and S e B (£p (G)). Then

(p * t ,S ) = (r, 5 © p)

= (T’M(p, L f S L t —i ) )= 0.

I

Lemma 3.3.4 (M m g ) ) ± * A/"(fp (G)) C (M tgo{G)) L .

Proof Let p G J\f (£P(G)) , t G (M^oo(G))± ,T g M ^ g ) - We want to show that

( r * p, T ) = 0. We have T = M/, for some h G Go (G). Recall that L tM hL t-1 =

Thus

(r * P)T} = {p ,T Q t ) = 0

since

T © r = = M^Mi^ = 0.

I

Remark: The last two lemmas hold for any arbitrary locally compact group but for

our purpose here, we need only the discrete case.

So, combining the last two lemmas we see that [ M ^ o ) ) ± is an ideal in N (£p (G)).

We continue by linking the product on M (tp (G)) to the product □ on

<i (G) ® (M M O )) ± .

Proposition 3.3.3 (£, (G) ® (M ,to(G)) 1 , □ ) = (A/" (£p (G )) , *), where the product □

is defined by

(a, x) D (b,y) = (a * b, a * y + x * b) (a, b e G (G ), x, y G (M /oo(g)) ±) .

P ro o f By Proposition 3.3.2, M (£p (G)) = G (G )© (M too(G) )± , so for 0, A G -A/”(G (G))

there are 0 i, Ai G G (G) and 02, A2 G (AGoo(g))_|_ such that 0 = 0 i+ 0 2 and A = A i+A 2.

Let 8 be the isomorphism between J\f (£p (G)) and G (G) © ± , i.e.,

$ (0 ii 02) = 0 i + 02 = 0- We now check that 0 * A = # ((0 i, 02) □ (Ai, A2)).

27


Case 1 (A e 4 (G)): Here we have 0 = 0 i + 02 and A = Ai. We obtain

0 >Is A — (01 “I” 02) * ^1 — 01 * ^1 ~f“ 02 * Ai

and

$ ((01) 02) d (Al, 0)) = 9 (01 * Ai, 01 * 0 + 02 * Al)

= 9 (0 i * Ai, 02 * Ai)

= 01 * Ai + 02 * Ai

= 0 * A.

Case 2 (A € (A /^ g ) ) ± ): Here we have 0 = 01 + 02 and A = A2 and in this case by

Lemma 3.3.3 we have that 0 * A = 0. On the other hand using again Lemma 3.3.3,

0 ((01) 02) Cl (0, A2)) = 9 (0 i * 0, 0 i * X2 + 02 * 0)

= 9 (0 ,0 i * A2)

= 0 (0, 0)

= 0

= 0 * A.

Since □ is distributive and (Ai, A2) = (Ai, 0) + (0, A2) it is now clear that

0 * A = 9 ((0 !, 02) □ (Al, A2)) V0, A e Ar { iv (G )) .

I

Now that we have an alternate description of J\f (£P(G)), we need to determine

how the product □ extends to the bidual level. We point out here that we extend □

to the bidual level given the specific actions of A on E above; note that in [10], M.

Eshaghi Gordji and M. F ila li show how one extends a module action to the bidual

level in fu ll generality.

Note that if 21 = A © E then 21” = A** © E**.

Proposition 3.3.4 Let ($ 1, $ 2) , (Ai, A2) G h (G)** 0 {Mioo{G)y*. Then

($ 1, $ 2) (Ai, A2) = ($1 o Ai, $1 <c> A2 + $2 Ai ) .

28


Proof Fix (A<0), A f ) € h (G )© (M ,„ (C|) 1 and let 6 h (G )"® (M <oo(g)) "

Take C G (G')®(:'V//.. :y; i ) such that a / - line, ( © ' . = (® i, ®2)

for some index set I . Then

($ 1,® 2 )o (A i0),A5>))

= (w* - lim , ( j ) f ) ) o (a [°:1, A 0))

= w* - lim ( (<â), <â)) o ( A(!0), A(20)) )

= w* - lim ( (V S“ }, ^ a)) □ (a<0), A 0)) )

= w* — lim * A ^ , cf)^ * A ^ + 02° * A i°^

= w* — lim o A^°\ <fi^ o A ^ + 4>^ o A]0

= (w* — lim (f)^ o A i°\ w* — lim <p^ o A ^ + w* — lim <p^ o Aj0V a a a /

= ^ o A f ’ ^ j o A f + $ 2 o AS0)) .

Now take (A i, A2) G i \ (G)** © (M eoo(G))± and ( x ^ \ A ^ ) C t \ (G) © ( M ^ g ) ) ^\ / aEl

such that w* — lima Ââ\ A ^ ^ = (A i, A2). Then

($ 1, <f>2) o (A i, A2)

= ($ 1, $ 2) o (w* - lim Ââ), A a)^

= w* - hm ( ( $ i , $ 2) o (Aâ), A a)) )

= w* — lim <f>i o A a\ $1 o A ^ + $20 A ^ ^

= (w* — lim $1 o Ai * \w * — l im $1 o A ^ + w* — lim <f>2 o A ^ )\ a a a J

= ($1 o A i, $1 o A2 + $ 2 o A i ) .

I

Using the previous proposition we are now ready to calculate the second topolog

ical centre of J\f(£p(G))**.

29


T heorem 3.3.2 Z f (M {£p (G))**) = £x (G) © (M m g ) ) \

P ro o f We start by showing that \F o A = 0 V\F G £x (G )**, A G (Môo(G))^*. Indeed,

by Lemma 3.3.3 we have that £x (G) * (M £ao(G))± - 0. F ix A0 € (A ôo(G))_l and let

\F G t \ (G)** and (ipa)aei C G(G) such that \F = w* — lim a ^ a for some index set I .

Then we have

T o A0 — [w * — lim i£a J o A0 = w* — lim (ipa o \ 0) = w* — lim (ij)a * Ao) = 0\ a / a a

Now take A G (M eoo(G))™ and (AQ)aeJ C (M £oo(g)) x such that A = w* — lim a Aa for

some index set J. Then

'F o A = \Fofw ;* — lim A ^ = w* — lim (\F o Aa) = 0.V a J a

So, let (4>, A) e Zf (A (G)“ ® (M (oo(G)) “ ) and take a net ( , ua) in

£x (G)** © which converges w* to 0. Then by Proposition 3.3.4 we have

that for each a,

(fia, ua) o (<F, A) = (//a « $ , ^ o A + ^ o $ ) = (pQ o <P, ua o $ ) .

Now, since ($, A) G Z f (£x (G)** 0 (M ^ g ) )^ * ) , both pa o <3> and va o <F converge w*

to 0. Thus $ G Z f (G(G)**) = £x(G) and so

Zf ( £ i ( G ) " © ( M w o , p * ) c h(G) ® ( M » „ ( g ) ) " .

By Lemma 3.3.1 we have (Môo(G))± C Zf(A f(£p(G)) which implies that

< , ( G ) ® {Mu{a)f c z f ( « ; • ® ( m m o , ) ” ) .

I t remains to show that = (M iOB(G))± - By Proposition 2.6.6 in [14] weI i i

have that ( ( M ^ ( G)) ±±) = (M m g ) ) ± and since (M too{G)) x = (M m g ) )7 it

suffices to show that (Aôo(G))±± = M ^ q)- Now, as

(M £oo{g)) ±± = {T g B { ip{G)) | (T , P> = 0 VP e (M eUG)) ± }

i t is clear that M ^ G) C (Môo(G))±±. For the opposite inclusion let (ft G ( M ^ g ) ) .

Then </>|^ = 0 which implies that 0 G ((M ôo(G))1) ± . Again, by Proposition

2.6.6 in [14], ((A ôo(g))_l ) X = M e G)w C Af(£p{G))*. As M ^ (G) is w*-closed, we

get the desired inclusion. |

30


Chapter 4

Strong Arens irregularity is not

hereditary

In the last section we showed that Af (£p (G )) is LSAI but not RSAI and it is clear

that is a subalgebra that is Arens regular. So we have already provided

an example of a Banach algebra w ith a subalgebra that has a different topological

centre. In this section we w ill construct a Banach algebra that is SAI which has a

subalgebra that is non-trivially Arens regular (non-trivial in the sense that it is not

SAI) where we use the example of Af (lf) (G)) as motivation.

4.1 Strongly Arens irregular Banach algebras with

an Arens regular subalgebra

Since any finite dimensional Banach algebra is reflexive it is tr iv ia lly true that every

strongly Arens irregular Banach algebra has a subalgebra that is Arens regular. Our

goal here is to construct an algebra that is SAI that admits a subalgebra that is AR

and not SAI.

Recall that if A is a Banach algebra and E is a Banach A-bimodule then A © E

is a Banach algebra where the product is

(a, x ) □ ( b, y) = ( ab, a-y + x - b) V (a, x ) , (b, y) e A ® E.

31


Note that for G discrete we have that A f (Ip (G)) = l \ (G) © (M^oo(g))1 and by

Proposition 3.3.2 we have that (Af (£p (G )) , *) = (G (G) © {Meao(G) )1 , D ). The al

gebra AT (£P(G)) is close to our goal since it is left strongly Arens irregular w ith a

subalgebra (M^oo(G))± that is Arens regular. Our construction is modeled after the

structure of G (G) © ± - The next few propositions w ill give us the tools

needed to construct the desired algebra.

Proposition 4.1.1 Let A be a commutative Banach algebra and let E be a Banach

algebra that is also a Banach A-bimodule. I f ■ denotes an action o f A on E from the

left then

x > a = a • x Va € A ,x £ E

defines an action of A on E from the right.

Proof Let f , g 6 A and r G E. Then

( fd ) > 7- = T • (f g )

= T ■ (af)

= (t • a ) - f

= f - r ( r - a)

= f - r (a -r r ) •

The other properties are triv ia l. |

Proposition 4.1.2 Let A © E be as in the above proposition. Let

($ i, A i ) , ($ 2, A2) G A** © E**. Then

($ 1, A i) © ($ 21A2) = ($1 © $2> A2 o 4>i + A i © $ 2)

and

($ 1, A i) o ($ 2, A2) = ($1 o $2) A2 © $1 + A i o $ 2) .

Here 0 and o are used to denote the extensions of the module actions in a sim ilar

way as the product on A.

32


P ro o f F ix (0 b Ax) e A@E and let ($ 2, A2) e A**®E**. Take (<j){2a\ A,a)) C A ® E

such that w* — lim a ^02q), A2a^ = ($ 2, A2) for some index set / . Then

(015 ^ l) © (*^2) A2)

— (01) ^ l) ©

= w* — lim a

lim a

lim a

— w* — lim Q

= w* — lim „

w* — lim n

= w

= w

( 0M

(0x, Ax) ^ A ^ ) )

01 * 02^) 01 ' ^2^ + - 1 ' 02^)

0X * 0 2a)) A2a) * 01 + Ax * 0 2a)^

^01 © 0 2a\ A2a o 01 + Al © 0 2 *^

= (w* — lim a 0x © 0 w* — lim a A ^ o 0x + w* — lim a Ax © 02a^

= (01 © $2, A2 <> 0x + Ax © <h2) .

Now take (S,, AO € h (G )" ® (MMO)) “ and 0 © A f ’ ) ^ C t , (G) ®

such that w* — lim Q (V ia\ A ^ ) = ($ 1, A i). Then

($ 1, A i) © ($ 2, A2)

= (w* - lim a (4>[a\ A[a)) ) © ($ 2, A2)

= w* - lim a ( ( 0 i “ \ A(iQ)) © ($ 2, A2))

= w *~ lim a ^0xa) © $ 2, A2 o 0x“ } + Aia) © $ 2 j

= (w* — lim a 0 i“ '1 © <h2, w* — lim a A2 o 0 ^ + w* — lim a A ^ © <h2)

= ($x © $2) A2 o $1 + A i © $ 2) •

Similar calculations show that the second equation holds, where first we fix

(02, A2) G A © jF and approximate ($ 1 , Ax) G A** © E** w ith a net in A 0 E to obtain

($ 1, A l) O (02, A2) = ($1 o 02, A2 © $1 + A l O 02) .

Then we approximate (<3?2) A2) € A** © E** w ith a net in A © E to get the desired

result.

We are now ready to give a classification of the topological centres of 21.

33


Lemma 4.1.1 ([9]) Given (21 = A © E, □ ) then the firs t topological centre is given

by all the elements of the form ($, A) £ 21** such that the following hold:

(1) $ £ Z.I ( A * * ) ;

(2) the map E** 3 N i—> <3? • N is w* — w* continuous;

(3) the map A** 3 M i—> A • M is w* — w* continuous.

P ro o f Let (<f>, A) £ Z ] (21**) and take (fia,ua) a net in 21** converging w * to zero.

This implies that (<h, A) © (/za, va) = (<E> © pLa, $ • ua + A • j i a) converges w* to zero.

I t is clear that (1) holds and by taking the net (0, va) we must obtain the same result

which implies that (2) holds. We also know by taking the net (pba,0) that A • p,a

converges w* to zero which gives (3). The converse is triv ia l. |

Theorem 4.1.1 Let A be a commutative Banach algebra that is strongly Arens i r

regular. Let E be a Banach space such that |.E**| > |E\ and such that E is also a

Banach A-bimodule. Define the product on E to be the zero product. Let 21 = A © E

and IB = A © 21 be constructed as in Proposition 3.3.1. Define the action of A on 21

to be

a • (b, y) = (b,y) ■ a = (b, y) □ (a, 0) Va £ A, (b, y) £ 21.

I f 21 is left strongly Arens irregular then IB is strongly Arens irregular that admits a

subalgebra that is Arens regular and not strongly Arens regular.

P ro o f We first need to show that the actions defined above are indeed actions. So,

let

f , g £ A, (b, y) £ 21 then

[fg] ■ (6, y) = (b, y ) □ (fg, 0)

= (b ,y )D (g f,0 )

= ( M ) D [ M ) □ ( /,( ) ) ]

= [(b ,y )n (g ,0 ) }a ( f ,0 )

= f ■ [(&, y) □ {g, o)]

= / • [g-(b ,y)}.

34


On the other hand,

{b, y) ■ [fg] = (6, y) □ {fg , 0)

= (6>2/)n [( / ,o )D (5,o)]

= [(b ,y )n ( f ,0 ) ]n (g , 0)

= [(6, y) □ ( / , 0)] . ^

= [(b,y) • f ] - g .

So 55 is indeed a Banach algebra. To show that IB is strongly Arens irregular, we

first look at Z \ (21**) and show that the map A** 9 $ h A 0 $ 6 21** being w* — w*

continuous is equivalent to A G Z f{21**). So, let (Aa,Ae) G 21** and take ( jia, ua) a

net in 21** converging w* to 0 . Then as

(Aa> A^;) © {Ha, Va) (Ay © Ha, Aa © Va -(- A e © /-la) ,

by Lemma 4.1.1, we have (A^, A#) G Z \ (21**) if and only i f A^ © Ha, A a © ^a, and

Ae Q Ha converge w* to 0. Since both A and 21 are LSAI, we have

(AA, AE) e Z f (21**) = 21 if and only if Aa © ^ and AE 0 Ha converge w* to 0.

Now let (<f>, A) G Z f (55**). By Lemma 4.1.1 we have that G Z \ (A**) = A and

the map A** 3 i-> A © 5/ is w* — w* continuous. Thus A G Zf($l**) = 21 which

gives us that 55 is LSAI.

Now let ($, A) G Z f (55**) and take a net (/ra, va) in 55** that converges w* to 0.

Then (na, va) o ($, A) converges w* to 0. Thus (h<x o $, A © Ha + va o $ ) converges

w* to 0 and so each of (Ua o f , A 0 Ha, and va o $ converges w* to 0. This implies

that $ G Z f (A**) = A and we have already shown that A © Ha converging w* to 0

implies that A G 21. Thus 55 is strongly Arens irregular.

Since E is a Banach space endowed w ith the zero product, then Zt (E **) = E**

a n d so is A re n s re g u la r. S ince |.E**| > \E\ th e n E is n o t s tro n g ly A re n s irre g u la r.

I t is not obvious that E is in fact a subalgebra of 55. If we identify E w ith 0 © A1

then the following simple calculation shows that it is. Let (0, f ) ,{0 ,g) G 0 © E then

35


(0, (0, / ) ) and (0, (0, g)) E 05 and we have

(0, (0, / ) ) □ (0, (0, g)) = (0,0 • (0, g) + (0, / ) • 0)

= (0, (0, g) □ (0,0) + (0, / ) □ (0,0))

= (0,0).

I

We give two examples of algebras A and E. The first is a specific example which

prompted the idea to form the algebra in an abstract manner.

Exam ple 1:

Let A = t \ (G ) and E = where G is an infinite commutative discrete group.

The product for each of these spaces is the convolution type product introduced earlier

in (Af ( ip (G))) = (G) © (M €oo(g)) ± . W ith this we see that A is strongly Arens

irregular and that E is an infinite dimensional Banach algebra that is Arens regular

and not SAI. We have seen already that A © E is left (but not right) strongly Arens

irregular. Applying the above theorem we see that l \ (G) Q A f (£r, (G)) is strongly

Arens irregular. |

Exam ple 2:

Let A = L i (G), G any infinite locally compact commutative group. Take the product

on A to be the usual convolution product and thus A is strongly Arens irregular. Let

E = A as Banach spaces and endow E w ith the tr iv ia l product. Denote this algebra

by L i (G)o to distinguish it from the usual convolution algebra L \ (G). Thus E is an

infinite dimensional Banach algebra that is Arens regular. Define the action of A on

E from the right to be a ■ x = a * x. To show that the map i-> A • T is continuous

from A** to E** i f and only i f A £ E, let A G E** and take a net / ia in A** that

converges w* to 0. Then A -pa = A 0 /rQ which forces A E Z \ (A**) = A = E. Thus we

have again satisfied the conditions of the above theorem and so the Banach algebra

L i (G) © [L i (G) © L \ (GQq] is strongly Arens irregular w ith an infinite dimensional

subalgebra that is Arens regular and not SAL |

36


One last remark on this idea is to point out that in both examples given, because

of the product defined on A ® £7, the algebra E must have the zero product. Any other

product on E would not yield a subalgebra of (A © E, □ ). I t is unknown whether

there exists an example where A is a Banach algebra that is SAI which admits a

subalgebra that is AR and not SAI where the product restricted to the subalgebra is

not the triv ia l one.

We would also like to point out that the conditions in Lemma 4.1.1 are all nec

essary. In other words, we may ask the question as to whether there is a relation

between the topological centre of A** and the projection onto the first component,

ira** ■ A** © E** —> A**, of the topological centre of A** © E**. The answer is that in

general the only relation is the triv ia l inclusion of tv a** {Z ]'2 (A** © E**)) C Z ] '2 (A**).

A simple example is similar to example 2, taking G to be any infinite locally compact

commutative group. I f we set A = L \ (G)0 and E = L \ (G) then define the action of

A on E to be the product in A. W ith this setup we can easily see that

Z I '2 (A**) = A** but from Lemma 4.1.1 we know that i f (<3>, A) E ZJ (A** © E**) then

the map M h <3> • M = $ 0 M is w* — w * continuous which imposes the condition that

$ E Z l (L i (G)**) = L i (G) and sim ilarly for the second topological centre. Thus we

have (Z t1)2 (A** © £ **)) = A ^ A**.

37


Chapter 5

Jordan Topological Centres

In this section we use A f (£p (G)) as motivation to investigate what happens to the

topological centre of a Banach algebra that is highly non-commutative when we force

commutivity. This forcing of commutivity is done by looking at the Jordan algebra

of the original Banach algebra. We first start by introducing the Jordan product

and defining the notion of a Jordan topological centre. We then give some general

results for any Banach algebra. Finally we determine the Jordan centre for £\ (G )

and A f (£p (G)) for a few classes of groups by developing a strengthened version of the

factorization technique used in section 3.

5.1 General results for Banach algebras

We start by defining the Jordan product for a given algebra.

D e fin it io n 5.1.1 Let (A, *) be an algebra. We denote by * j the Jordan product and

i t is defined by. a * b + b * a

a * j b — ---------------- Va, b € A.Z

The Jordan product is a bilinear form that symmetrizes a noncommutative alge

bra. I t is easy to see that this product is not necessarily associative, for if a, b, c G A

38


then

. . / a * b + b * a \(,a * j b) * j c = I ------- J * j c

(2 !*±*!2 ) * c + c * (o*6±6*a)

2(a * b) * c + (b * a) * c + c * (a * b) + c * (b * a)

and

. ( b * c + c * ba * j ( b * j c ) = a * j

2a ^(^b*c±c*b^ + ( b*c±c*b j + a

2a * (6 * c) + a * (c * b) + (b * c) * a + (c * b) * a

4(a * b) * c + a * (c* b) + (b * c) * a + c * (b * a)

Now,

(a * j b) * j c — a * j (b * j c) = 0

-w- (b* a) * c + c * (a *b ) — a * (c*b) — (b* c) * a = 0

O- b * (a * c) + (c * a) * b — (a * c) * b — b * (c * a) — 0

<=>• b * ((a * c) — (c * a)) — ((a * c) — (c * a)) * b = 0

Taking A = J\f(£p(G)) = £\{G) © (M e^G ))^ and © c £ ^i(G0, & ^ (Me0o(G))_L

we see that this product may not be associative. I t is clear however, that if A is

commutative then the Jordan product is the same is the original product on A.

Fortunately the Arens extensions of a product do not require associativity. The next

proposition shows that the Arens extensions of the Jordan product give the expected

commutivity in the bidual level.

39


P ro p o s itio n 5.1.1 Let (A ,* ) be a Banach algebra. Then the firs t and second Arens

extensions o f the Jordan product, denoted Q j and Oj respectively, are

$ 0 A + A o $$ 0 J A = u ^ ----- — A o j $ V $ , A e r

P ro o f F ix 0O £ A and let A 6 A**. Take a net Aa in A such that A = w* — lima Aa.

Then

0o © j A = w* - lim 0o © j Aaa

= w* — lim 0o * j Xaa

= w* - lim \ (00 * A« + Aa * 0o) a Z

= w* - lim i (00 © Aa + Aa c> 0O)a Z

= - (0o © A + A o 0o) •

Now let T e A** and approximate $ by a net 0a in A. Then using the above

calculation

$ 0 j A = w* — l im 0a 0 j Aa

l= w* - l im - ( 0 a © A + A O 0 a )

a Z

= ^ ($ © A + A o $ ) .z

To show that $ © j A = A O j $ fix 0o £ A and let A € A**. Take a net AQ in A

such that A — w* — lim A a. Then

A Oj 0o = w* - lim Aa oj 0oa

= w* lim Aa 0oa

= w* - lim \ (Aa * 00 + 00 * Aa) a Z

= w* - lim i (Aa O 00 + 00 © Aa) a Z

= ^ (A o 00 + 00 © A ) .

40


Now let (j)a be a net in A such that $ = w* — limQ <pa. Then

A Oj $ = w* — lim A Oj 4>aa

= w* - lim ^ (A o <j)a + 4>a © A)a Z

= ^ (A o $ + $ © A)

= $ © j A

*

R em ark: For any algebra (A, *) we w ill denote the Jordan algebra (A , * j ) simply

by A j. The Jordan product produces a commutative (possibly non-associative) alge

bra and thus in light of the previous proposition we have that Z \ (A }*) = Z 2 (A }*).

Also from the previous proposition it is easy to see that Z \ (A*/) consists of all the

elements of the form <J> 6 A** such that the map n t - ^ & Q n + n o & is w* — w

continuous.

Definition 5.1.2 Let A be a Banach algebra. The Jo rd a n topo log ica l centre o f

A** is the set

Zt (A j*) = {m € A** | the map n i—► m Q j n is w* — w* continuous} .

The next proposition shows how the toplogical centre of the bidual of the Jordan

algebra relates to the first and second centres of the bidual of the original algebra.

Proposition 5.1.2 Let A be a Banach algebra. Then

z\ (A**) n z2t (A " ) = zt (A y ) n (z\ (A**) u (a **)).

P ro o f Clearly Z\ (A**) n Z2t (A**) c (Z\ (A ” ) U Z2t (A**)). To show that

Zl (A**) n Z 2 (A**) C Zt (Ay) let $ G Z} (A **) n Z 2 (A**) and let pa be a net in A**

converging w* to zero. Then

$ © j /A* = ^ ($©/©* + /A* o 0

41


since both terms converge to zero (10*) which gives the desired inclusion.

To show that Z% (A*/) n (Z f (A**) U Z f (A**)) C Z f (A**) n Z f (A **) assume the

contrary that there exists A £ Zt (A *f) f l (Z f (A**) U Z f (A**)) such that

A £ Z f (A**) n Z f (A**). This implies that A is either in Z f (A**) or Z f (A**) but not

both. W ithout loss of generality assume that A € Z f (A**). As A ^ Z f (A**) there

exists a net / ia converging w* to zero such that A (•) / ia: does not converge to zero

(w*). Since A 6 Zt (A *f) we have that A © / ia + y a o A converges to zero. Therefore

lim a A © na = — lim a fia o A (w*) but because A £ Z f (A**) the right side of the

equality is zero, a contradiction. |

Definition 5.1.3 Let A be a Banach algebra. We say that A is Jordan strongly

Arens irregular (JSAI) i f Z t (A*/) = A and A is Jordan Arens regular (JA R )

i f Zt (A*f) = A**.

5.2 Jordan topological centre of t\ (G)

The techniques that we w ill use to determine the Jordan centre of l \ (G) are very

similar to those that we used when calculating the first and second centre of t \ (G).

We state now the definition of the kind of factorization that w ill be needed and then

prove a theorem very much like Theorem 2.2.1.

Definition 5.2.1 Let A be a Banach algebra and let k be a cardinal number. We

say that A* has the simultaneous A** -factorization o f level k i f fo r every net

(fa )a<zi with |/| < ft there exists a net ( 'fa)aGi c Ball (A**) and a single function

f E A* such that the factorizations

f a = i ’a O f and f a = f o i)a

hold fo r every a £ I .

We would just like to point out that t \ (G)* has both the left t \ (G)** and right

t \ (G)**-factorization properties of level k (G). This does not imply a priori that

i \ (G)* has the simultaneous l \ (G)**-factorization property of level k (G) because a

42


sequence ( /a)aej C l \ (G )* factorizes in the following way, f a = ipa © / = g o ipa, and

generally the functionals / and g are not equal.

Theorem 5.2.1 Let A be a Banach algebra with the Mazur property of level k > N0.

I f A* has the simultaneous A**-factorization property of level k then A is JSAI.

P ro o f Let m £ Zt (A*/) and take a net (ha)aGl C Ball(A*) w ith |/| < k which

converges w* to 0. Since A has the Mazur property of level k , we only need to

show that {m, ha) converges to 0. I t is sufficient to show that every convergent

subnet ((m ,ha))aeI converges to 0. Let ((ra, ha/3)) be a convergent subnet. As

A* has the simultaneous A**-factorization property of level k we have that ha =

tfa Q h = h o © j h where (4’a)aei C Ball (A**) and h £ A*. Since the

net (ipQ/3) C Ball (A**), by Alaoglu’s Theorem, there exists a to*-convergent subnet

(fPa0 j • Let E = w* — lim 7 ipa^ e Ball (A**). Now, using the fact that ha converges

w* to 0 we have for all a £ A

Hence, m £ A, and A is JSAI. |

Now that we have an analogous result to Theorem 2.2.1 we just need to show that

i \ (G)* enjoys the simultaneous t \ ((T)**-factorization property of level k (G). The

lim (fpa^ © j h, a ) = lim (h7 \ ^ / 7

This implies that E Q j h — 0. So, as m £ Zp (A**);

( m Q j E, h)

(m, E Q j h)

0.

43


next theorem show that we can get this simultaneous factorization for any countable

ICC group. G being an ICC group means that any element g G G w ith g ^ e, belongs

to an infinite conjugacy class. A large class of examples of countable ICC groups are

the free groups on n generators, where 2 < n < ft0.

T heorem 5.2.2 Let G be a countable IC C Group. Then there exists a sequence

(V’n)neN c c Ball(£oo (G)*) such that fo r every sequence ( / „ ) neN C B a ll(£ ^ (G))

there is a single function f G B a ll(£ ^ (G)) such that the factorizations

fn = V’n © / = / O V’n = V’n © J /

hold, fo r all n G N.

P ro o f Let {A n} ngN be a covering of G by finite sets such that Vn < m G N we have

e G An C A m, where e is the identity element in G.

Now consider the set I = N x N. For h = (n, h) G I we set A := A n. We

construct a net {yh)n&i w ith the following properties:

*) AnVn C AnPyfn = 0 if h m

a) yaAfi f i yrhArh = 0 if h + m

Hi) yhAfi n Afnym = 0 if h rh

iv) yh n AnPy-fh ~ 0 if h rh

v ) yh C yrhAm = 0 if h m

v i) yhAn n Affijn - {Vn} \/h G: /

For this we well order / by < w ith the property that for each n G I the set

| m G / | m < n | i s finite. F ix h G / . We construct the desired net by induction

by assuming that we have found ]jm satisfying the above six properties Vm < h. For

each m < h set

Km. ~ A^ ArnVm C yfriA^A^ U Aj^y^A^ U Afhym U ymAfrj.

Enumerate An as A fl = {1, g1} g2, ..., gkj- As G is ICC we have that [gf\ is infinite for

each 1 < i < k (where (//,] is the conjugacy class of gf). Since Afl is finite there exist

44


infin ite ly many h G G such that hgih_1 ^ An for every 1 < i < k. Choose one of

these elements h e G such that h £ Vm < h. This is possible since K fn is a finite

union of finite sets. Set Un = h, then ya satisfies the 6 properties listed above.

We define a relation on I by

h -< rh n < m.

Clearly, this relation is transitive and directs / . Note that whenever h -< m we have

that XasXaa = XA*-

For h = (n,h) e I let = W(n,fc) := XA(n>h) and consider the functions

v h — v {n ,h) : = i u ( ,n ,h ) fh )

and

(u(n,h)Ift) •

For n = (n,h) ,rh — (m, k) G I w ith h ^ rh by (5.1) we have the following results:

supp ( v n ) f l supp (Vrn) = S U p p ( « ( „ , / , ) / / » ) ? /(„ ,/,) C S U p p ( lU{m<k)f k ) y (m ,fc)

C supp (w(„,h)) C supp (u(m,fc)) y(m,fc)

= A [n ,h )V {n ,h ) Fl ^4(m,fc)y(m,fc)

= AfiUa n

= 0

and

supp (t%) n supp (rwm) =

c

45

y(n,f t )S U p p (w(n,h)//») n y(m,fe )S U pp (u(m,fe)/fc)

2 /(n ,h )S U p p (« (» ,? » )) n ?/(>«,fc )s u p p (« (m ,fc ) )

y(n,h)A(nth) C y(m,k)A(m,k)

ynAfi n ymAfn

0


and

S U p p ( n ^ ) n S U p p (W rh) = S U p p (U (n>h) f h ) V(„,h) Id ?/(m )fc )S U p p ( l t ( m ,fe )/fc )

C S U P P («(„,ft)) J/(„,fc) n J/(m ,fe )S U pp (W(m,fc))

= ■^■{n,h)U{n,h) C V{m,k)-^-{m,k)

~ -^nVn C Vm-A-m

= 0

and for each h £ I we have

S U p p (lOfl) PI S U p p (vn) = J /(n ,h )S U p p («(„,/»)/ft) H S U p p ( « ( „ , f t ) / f c ) ?/(„,/,)

c J/(„,h)SUpp («(»,/»)) n SUpp («(„,h)) V(n,h)

= y(n,h)A(n,h) C 0 (n,h)y{n,h)

= IJh- -n n d-fi J/ft

= O/n}-

Now for each h e N we set gh (x) = f h (e) Vrr e G, and then we define / pointwise

by

f := X / [r »(; ‘k) { u ( n , h ) f h ) + l y - ' h) ( u ( n , h ) f h ) - X y { n ,h ) 9 h ^ ■

We claim that / e Ball ( qq (G)). Indeed, let x £ G. From the above relations we

have that supp (xy (n,h)9h) = {y(n,h)} and supp (w(„>h)) n supp (u(„,h)) = {s/(n,/i)}-

Case l: ( x = y(n,h) for some (n, h) e / )

46


Then

f (X) = '^2 [VV(n,h) (U(n<h) fh) + lV(n,h) i U(n’h) fH) ~ Xy(n,h)9h (X)(n,h)el

— r y^h) (u(n’h) fh) {y(n>h)) (u(n’h) fh) {y(n’h) ) ~ ‘X-V(n,h)9h {y(n,h))

= (u(n,h)fh) (e) + (U(n,h)fh) (e) - 9h (e)

= fh (e) + fh (e) - gh (e)

= 2 f h (e) - f h (e)

= f h( e) .

Thus, | / (y(n,h)) | < 1 for all (n, h) G / .

Case 2:(x ^ i/(n,/p V (n, h) G / )

Case 2a: (a; G supp (i%) for some n G I )

Then \ f (x ) | = |tU(n,h)| < \ f h (x) | < 1.

Case 2b: (x G supp (v„) for some n G I )

Then |/ (a:) | = \vM \ < | f h (x) | < 1.

Case 2c: (x £ supp (wa) U supp (ufi))

Then / (x) = 0.

Hence, / G Ball (£00 (£?))•

Let T be an u ltra filter on N which dominates the (natural) order filter. For h G N,

we define

4>h := ic* - lim Sy G <5GW C Ball (G) * ) .

We now show that the factorizations

fh = A © /

and

fh = f oiph

hold for all h G N.

We first show the following results for all (n, h ) , (m, fc), (£, c) G / w ith

(Z, c) -< (m, k):

47


— 3 (n ,h ) , (m ,k )u ( l ,c)u ( m , k ) f k

u (l,c)

= “ ».«) ( r »(~.» ( r» 5 , f e * .

(Z,c) ( X,A(rn,k)y(m,k)SJ ( X-V(n,h)A(n,h) ^!/(n

= S ^h)^m,k)U(l,c) ( ^ { i / (m,fc)} ) i}y'[n,h/ h) )

— f i ( n ,h ) , (m ,k )U ( l , c )X {e } ( j~ i i (m,k) ( ^ jfc)^ fc) )

= &{n,h),{'m-,k)'Uj { l , c ) X { e } f k -

- l h)

W (*,e) ( ^ / ( m , k ) ^ / (“ *h) ( 'U( n , h ) f h ) ) — « ( / ,c ) ( ' U' ( m , f c ) V m , f c ) ^ h) ( u ( n , h ) f h ) )

= “ (»,c) (iy(m,fc) ( Zy{ fc)U<ro'fe>) V -,h)

— n ,h ) , (m ,k )u (l,c) ( ly (m ,k ) ( ^ ( m , fc) U (TO’ fc^ fe) )

= $ (n ,h ) , (m ,k )u ( l ,c)u { m , k ) f k

— fi(n,h),(m,k) 'U ,( l , c ) f k -

48


u(l,c) ( ly(m,k)ry-n\h) (u(n,h)fh)) - «(I,c) ( u(m,k)ly(m,k)ry ^h) (u(n,h)fh) )

= «(Z,c) ( V , * ) S » ,«

= W(Z,c) (ly(m,k) (xy(m,k)A(m,k)) ('X-A(n,h)V(n,h))

= 5(n,h),(m,k)u(l,c) ( i t J ( m ,k) (^ { j/ (m,fc)} ) ( S n ^ ) ^ ) )

= ${n,h),(m,k)u (l,c)X{e} (jy(m,k)

U(l,c)ry{m,k) (x.y(n,h)9h) — u(l,c) ( M(m,fc)r ?/(m,/o)

= «(/,C) (j'yim.k) ( r y[Jl!k)U(m’k')) (xv(n,h)9h

= M(i,C) ( r 2 / ( m , fc ) ( X.A(m,k)y(m,k)' ^2/(n,h)^h) )

= 8 ( n , h ) , ( m , k ) u ( l , c ) ( ^ ’y ( m tk) X y ( m , k ) 9 k J

— ^ ( n , h ) , ( m , k ) U ( l , c ) X { e } r y ( m : k ) 9 k

— ^ ( n , h ) , ( m , k ) u ( l , c ) X { e } 9 k ■

u(l,c)ly(m,k) (x.V(n,h)9h) = U(l,c) {^irn,k)h(m,k) {xyin,h)9hj^j

= « (/,c ) ( l y (mtk) ( ^ ifc)U (™ >*o) ( x ? / ( „ , f t ) ^ ) )

= W(jiC) (ly{mk) {x.y(m,k)A(m,k}) (xy(n,h)9hj^j

= fi(n,h),(m,k)u (l,c) ( j i i(m,k)Xy(m,k)9kJ

= f i ( n , h ) , { m , k ) u ( l , c ) X { e } l y ( m t k ') 9 k

= & ( n , h ) , ( m , k ) ' U { l , c ) X { e } 9 k -

49


F ix x € G. By the above results we obtain for all k € N and (7, c) G / :

U(l,c) (X) (T’fc, I x f )

= ^ (x)

= X / “ (*.«) ^ r »C«*.*> K " A) + V»U) ~ (*)neN heJ

= J & E E h * < * ) * * . . (y,(n,h)fh) (x) "FneN fteJ

+W(/,c) (x ) r y(m,k)ly-n]h) (U(n,h)fh) ( x ) - U(i,c) (x ) r y(m<k)Xy{nih)9h (x )

= ^ ^ 0^) fk (x ) "F 3(n,h),(m,k)u(l,c) (x ) X{e}/fc (x ) —neN /ieJ

&(n,h),(m,k)'U'(l,c) (x ) X{e}i7fc (x )]

= S 5(n,h),(m,k)U(l,c) (x) [ fk (x) + X{e}/fc (x) ~ X{e}3fc (x)]neN he J

$(n,h),(m,k)'Uj(l,c) (x) fk (x)neN he J

= W(i,c) (x) f k (x) .

Since it(j)C) — ► 1 pointwise, we obtain

fk = fa O f

for all K N .

50


Similarly,

u (i,c) (x) (il>k, r xf )

= l % U(l ,c){x){ly(m,k)f ) ( x )

= X X Ud,c) (X) him*) [ \ f h) (u(n,h)fh) + ly £ h) (u{n,h)fh) - Xy(n,h)9h] (x)neN heJ

= i ^ X X h ,c ) (x ) ly(m,k)\n\h) (U(n,h)fh) (®) +neN feeJ

+ w(Z,c) (x ) l y ( m , k ) l y ^ h) i U ( n , h ) f h ) ( x ) ~ U( / jC) ( X ) l y { m t k ) X y ( n ,h ) 9 h (x )

= lin^_ ^ [( (n,/i),(m,fc), '(/,c) (x) X { e } f k (x) “I- ^ ( n , h ) , { m , k ) ' ^ j { l , c ) (x) f k (a?)

( (n,h),(m,A;)' (i,c) (x) X{e}l/fc (x)]

- f e E E w , m,fc)«(Z,c) (x) [X{e}/fc (x) + fk (x) ~ X{e}9k (x)] neN heJ

= X X! 5 ( n , h ) , ( m , k ) U {l,c ) (x) f k (x)neN fceJ

= «(i,c) (x) fk (x)

and again as u ^ c) — > 1 pointwise, we obtain

fk = f o ipk

for all k £ N.

Finally we have,

fk = ^ (fk + fk) = ^ (i>k 0 / + / o fe) = ipk®j f

which completes the proof. |

We now come to our main result.

Theorem 5.2.3 Let G be a locally compact, noncompact abelian group and let H

be a countable IC C group. Then L x (G) and £x (H ) are both Jordan strongly Arens

irregular.

P ro o f Since G is an abelian group then L x(G) j = L X(G) and thus L x (G) is Jordan

strongly Arens irregular. By the previous theorem we have that £x (H )* enjoys the

simultaneous £x (ff)**-factorization property of level tt0- Thus by Theorem 5.2.1 we

have that Zt (£x = £x (H). |

51


5.3 Jordan topological centre of the convolution

algebra of nuclear operators

From the last section we have found that for G an abelian group or a countable ICC

group we have that G (G) is JSAI. Since Af (£p (G)) = G (G) © ± we can

now determine the Jordan centre for M (£p (G)) for these two classes of groups.

T heorem 5.3.1 Let G be a countable IC C group or a discrete abelian group. Then

A f (£p (G)) is JSAI.

P ro o f Recall that J\f (£p (G)) = G (G) © ( M ^ q ) ) L and

M ( e , ( G ) r = h (G)“ e (M(„ (g|) x Let <f = (p,r) e Z, (AT&(<?))?)• We

first show that p G G(G) . So, let (/ia)aeJ C G (G)** be a w*-convergent net

converging to 0. Then (pa, 0) is a net in G{G)** ffi converging w* to

0. As (p, r ) G Zt (Af (£p (G ))j*) then (p, r ) © j (pa, 0) converges w* to 0. Since

(p, r ) © j (pa, 0) = (p © j p.a, r Q j pa) we have that p Q j pa converges w* to 0 which

gives p E Z t (G (G )j*) = G (G). Thus (p ,r ) G G (G) © (M eoo{G)) X. By Proposition

5.1.2 we get that (p, r ) G ^ (G) © (M ioa{G)) ± = A/- (^p (G)). |

52


5.4 Conclusion and main results

As we stated in the introduction, there were three main objectives for our research.

These objectives were:

i)to determine the second topological centre of (Af (tv (G ) ) , *) which demonstrates

concretely that (Af (£p ( G j ) , *) is not right strongly Arens irregular;

ii)to construct a Banach algebra that is strongly Arens irregular that admits a

subalgebra that is Arens regular (non-triv ia lly);

iii) to investigate the notion of the Jordan topological centre for l \ (G) and

( V ( « P ( G ) ) , . ) .

We have successfully shown all three parts except for part (iii) where we have

only shown this when G is a discrete abelian group or a countable ICC group. Our

technique used to determine (iii), a simultaneous factorization result, is also of interest

on its own. This result is a strengthening of the factorization result developed by

Neufang in [18]. So far, Neufang’s factorization result is the most powerful tool

in determining the topological centres of a Banach algebra, and the simultaneous

factorization result is an analogous tool for that of Jordan-Banach algebras.

In the investigation of the Jordan topological centre of (Af (£p (G) ) , * ) i t was

natural to look at the Jordan centre of l \ (G). This has inspired some questions

that are s till unanswered. These questions are:

1) Is L i (G) Jordan strongly Arens irregular for every locally compact group G?

2) I f G is not isomorphic to a locally compact abelian group or a countable ICC

group, can L% (G)* s till have the simultaneous L \ (G)**-factorization property of level

k (G), the compact covering number of G? In particular, when G is the product of a

locally compact abelian group and a countable ICC group.

3) Does there exist a Banach algebra where the Jordan centre is strictly bigger

than the intersection of the first and the second centre? (Note: this question is

tr iv ia lly yes if we relax the condition of associativity of the original algebra since any

Lie algebra is Jordan Arens regular).

53


Bibliography

[1] O.Y. Aristov. Banach Algebras and Their Applications. Contemporary Mathe

matics 363. Sixteenth International Conference on Banach Algebras, University

of Alberta 2003. pp.15-37.

[2] B. Bollobas. Linear Analysis. Cambridge University Press 1999.

[3] D.L. Cohn. Measure Theory. Boston, Birkhauser 1980.

[4] G. Dales. Introduction to Banach algebras, operators, and harmonic analysis.

New York, Cambridge University Press 2003.

[5] G. Dales and A.T. Lau,. The Second Duals of Beurling Algebras. Memoirs of the

American Mathematical Society 177 (2005) no.836, pp.40-41.

[6] E.G. Effros and Z.Ruan. Operator Spaces. New York, Oxford University Press

Inc. 2000.

[7] M. Fabian et al. Functional Analysis and Infinite-Dimensional Geometry.

Springer-Verlag New York, Inc. 2001.

[8] G.B. Folland. A course in Abstract Harmonic Analysis. Boca Raton, CRC Press.

1995.

[9] F. Ghahramani, J.P. McClure, and M. Meng. On Asymmetry of topological Cen

ters o f the Second Duals of Banach Algebras. Proceedings of the American Math

ematical Society 126 (1998), no.6, pp.1765-1768

[10] M.E. Gordji and M. Filali. Arens regularity o f module actions. Preprint.

54


[11] E. Hewitt and K.A. Ross. Abstract Harmonic Analysis. Volume 1. Berin, Springer

1970.

[12] Z. Hu and M. Neufang. Decomposability o f von Neumann Algebras and the Mazur

Property o f Higher Level. Canadian Journal of Mathematics 58 (2006) no.4,

pp.768-795.

[13] A.T. Lau and V. Losert. On the second conjugate algebra of L\ {G) of a locally

compact group. Journal of the London Mathematical Society. Second Series 37

(1988) no.3,

pp 464-470.

[14] R.E. Megginson. An Introduction to Banach Space Theory. Springer-Verlag New

York, Inc. 1998.

[15] M. Neufang. On one-sided strong Arens irregularity. Preprint.

[16] M. Neufang. Isometric Representation of Convolution Algebras as Completely

Bounded Module Homomorphisms and a Characterization of the Measure Alge

bra. Preprint.

[17] M. Neufang. On a conjecture by Ghahramani-Lau and related problems concern

ing topological centres. Journal of Functional Analysis 224 (2005) no.l, pp.217-

229.

[18] M. Neufang. Abstrakte Harmonische Analyse und Modulhomomorphismen iiber

von Neumann-Algebren. Ph.D. Thesis. Univeritat des Sarlandes (2000).

[19] M. Neufang. A unified approach to the topological centre problem fo r certain

Banach algebras arising in abstract harmonic analysis. Archiv der Mathematik

(Basel) 82 (2004) no.2, pp.164-171.

[20] M. Neufang. Solution to a conjecture by Hofmeier-Wittstock. Journal of Func

tional Analysis 217 (2004) no.l, pp.171-180.

55


[21] T.W . Palmer. Banach Algebras and the General Theory of -Algebras. New York,

Cambridge University Press 1994.

[22] A.Y. Pirkovskii. Biprojectivity and Biflatness fo r Convolution Algebras of Nuclear

Operators. Canadian Mathematical Bulletin 47 (2003) no.3, pp.445-455.

[23] S. W illard. Introduction to general topology. Addison-Wesley Publishing Com

pany 1970.

[24] R.A. Ryan. Introduction to Tensor Products o f Banach Spaces. Springer-Verlag

London Lim ited 2002.

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