jordan-arens irregularity
TRANSCRIPT
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.
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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.
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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
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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
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( 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.
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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.
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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.
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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 }.
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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}.
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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**}.
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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**))
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< |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.
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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.
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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 ) .
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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
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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 ) .
D e fin it io n 2.3.8
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 ) .
D e fin it io n 2.3.9
Cq (G) — { / £ Cb (G) | Ve > 0 3K c G compact such that \/x ^ K : \ f (x) \ < e} .
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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 ) .
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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
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< \\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 )
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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 .
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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 .
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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
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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.
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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 ) .
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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 .
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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.
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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 ) ] .
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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 ) ) . |
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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.
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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)).
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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 + $2 Ai ) .
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Proof Fix (A<0), A f ) € h (G )© (M ,„ (C|) 1 and let 6 h (G )"® (M <oo(g)) "
Take C G (G')®(:'V//.. :y; i ) such that a / - line, ( © ' . = (® i, ®2)
for some index set I . Then
($ 1,® 2 )o (A i0),A5>))
= (w* - lim , ( j ) f ) ) o (a [°:1, A 0))
= w* - lim ( (<^a), <^a)) 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 ^A^a\ A ^ ^ = (A i, A2). Then
($ 1, <f>2) o (A i, A2)
= ($ 1, $ 2) o (w* - lim ^A^a), A a)^
= w* - hm ( ( $ i , $ 2) o (A^a), 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))**.
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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 (M^oo(G))^*. Indeed,
by Lemma 3.3.3 we have that £x (G) * (M £ao(G))± - 0. F ix A0 € (A ^oo(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 (M^oo(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 (A^oo(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 (M^oo(G))±±. For the opposite inclusion let (ft G ( M ^ g ) ) .
Then </>|^ = 0 which implies that 0 G ((M ^oo(G))1) ± . Again, by Proposition
2.6.6 in [14], ((A ^oo(g))_l ) X = M e G)w C Af(£p{G))*. As M ^ (G) is w*-closed, we
get the desired inclusion. |
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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.
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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.
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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 © $1 + A l O 02) .
Then we approximate (<3?2) A2) € A** © E** w ith a net in A © E to get the desired
result.
We are now ready to give a classification of the topological centres of 21.
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Lemma 4.1.1 ([9]) Given (21 = A © E, □ ) then the firs t topological centre is given
by all the elements of the form ($, 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)}.
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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
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(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 |
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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**.
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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
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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.
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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 ) .
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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
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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
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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.
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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
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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
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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 / )
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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):
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— 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 -
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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 -
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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 .
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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). |
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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)). |
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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).
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