Stability of 2nd conjugate Banach algebras
Stability of 2nd conjugate Banach algebras
Ana L. Barrenechea 0 1
Carlos C. Peña 0 1
Mathematics Subject Classification 0 1
0 A. L. Barrenechea
1 Primary 46H99
In general the stability of normed algebras is a non hereditary property. We shall prove that second conjugate Banach algebras may be non stable even if the underlying Banach algebra is stable. We shall characterize stability of second conjugate Banach algebras. Finally, we shall study kinds of stability induced on an algebra with an stable second conjugate algebra. Throughout this article, A will be a complex Banach algebra. Then A is called left (resp., right) stable if the left (resp., right) regular representation L A (resp., antirepresentation R A) of A into B( A) is isometric. A is stable if it is left and right stable. Recently, stable Banach algebras became relevant because of their important role in the theory of almost multiplier maps [1]. It is readily seen that the class of left (resp., right) stable algebras contains the class of Banach algebras endowed with a bounded right (resp., left) approximate identity with bound 1. Likewise, any stable Banach algebra is faithful. However, there are stable Banach algebras without bounded approximate units and there are faithful non stable Banach algebras. Further, subalgebras of stable Banach algebras are not necessarily stable [2]. Let χ A be the isometric immersion χ A : A → A∗∗ of A into its second conjugate space A∗∗. It is known that A∗∗ has two Banach algebra structures with products and ♦ that generalize the underlying product of A [3, 4]. This fact gave raise to the important issue of algebraic (or Arens) regularity, related to deciding

1 Introduction
conditions of coincidence of these products. Given a∗∗, b∗∗ ∈ A∗∗ we shall write
a∗∗ b∗∗ = w∗ − lii∈mI lji∈mJ χA(ai b j ),
a∗∗♦b∗∗ = w∗ − lji∈mJ lii∈mI χA(ai b j ),
where {ai }i∈I and {b j } j∈J are bounded nets in A so that
a∗∗ = w∗ − lim χA(ai ) and b∗∗ = w∗ − lji∈mJ χA(b j )
∈I
(cf. [7], 1.4., 46–64). Since its introduction in 1951, this setting constituted an important way for the
development of Banach algebra theory. We remark the following two facts: (i) Any Banach algebra A becomes a
subalgebra of its second dual Banach algebra ( A∗∗, ) (or ( A, )). (ii) Stability is not an hereditary property.
Thus, it is of interest to seek on relationships of stability of a Banach algebra and stability of the corresponding
second dual Banach algebras (cf. [2], 4., Open Problem 2).
Our aim is to investigate the relationship between the stability of A∗∗ and the stability of A. To this end,
in Lemma 2.1 we shall relate the left regular representations of A and A∗∗. In our main result Th. 2.3 we shall
see that the underlying algebra A inherits the same kind of stability of its second conjugate BanachArens
algebras. In Ex. 2.4 we shall exhibit a stable Banach abelian algebra whose second conjugate algebra is not
stable. This example gives a negative answer to Problem (2) posed in [2]. As stability is a metric condition,
the characterization in Th. 2.5 of stable second conjugate Banach algebras relies on metric facts explored in
Lemma 2.2.
2 Algebraic stability of the 2nd conjugate algebra
Lemma 2.1 There is a contractive linear operator : B( A)∗∗ → B( A∗∗) so that L = ◦ (L A)∗∗,
R♦ = ◦ (R A)∗∗, where L and R♦ denote the left and right regular representation and antirepresentation
of A∗∗ endowed with the Arens products and ♦, respectively.
Proof Let u : A∗ × A∗∗ → B( A)∗ so that u(a∗, a∗∗)(T ) = T ∗(a∗), a∗∗ if T ∈ B( A), a∗ ∈ A∗ and a∗∗ ∈
A∗∗. It is plain that u is a well defined normone bilinear form and if n ∈ B( A)∗∗ we set a∗, (n)(a∗∗) =
u(a∗, a∗∗), n . It is readily seen that becomes a well defined contractive linear operator between B( A)∗∗
and B( A∗∗).
Let a ∈ A, a∗ ∈ A∗ and a∗∗, b∗∗ ∈ A∗∗. Then,
a, (L A)∗[u(a∗, b∗∗)] =
L A(a), u(a∗, b∗∗)
i.e., (L A)∗[u(a∗, b∗∗)] = b∗∗a∗. Therefore,
a∗, ( ◦ (L A)∗∗)(a∗∗)(b∗∗) = a∗, ((L A)∗∗(a∗∗))(b∗∗)
= (L A)∗(a∗), b∗∗
= a∗a, b∗∗
= a, b∗∗a∗ ,
= u(a∗, b∗∗), (L A)∗∗(a∗∗)
= (L A)∗[u(a∗, b∗∗)], a∗∗
= b∗∗a∗, a∗∗
= a∗, a∗∗ b∗∗
= a∗, L (a∗∗)(b∗∗) .
The other identity follows analogously.
Lemma 2.2 Let X , Y be complex normed spaces and let T ∈ B(X, Y ). Then T ∗ is isometric if and only if
T ([X ]1)− = [Y ]1, i.e., if and only if the closure of the image of the closed unit ball of X under T is the closed
unit ball of Y .
Proof If T ∗ is isometric it has norm one. So, T = 1 and T ([X ]1)− ⊆ [Y ]1. If this inclusión is strict let
y0 ∈ [Y ]1 − T ([X ]1)−. By the HahnBanach theorem, there exist a nonzero linear form y1∗ on Y and constants
r ∈ R and δ > 0 so that
( y, y1∗ ) < r < r + δ <
( y0, y1∗ )
(2.1)
for all y ∈ T ([X ]1)−. It is easy to see that y1∗ is bounded and with minor changes we can assume that
y1∗ = 1. It is also clear that r > 0. Now, for x ∈ [X ]1 such that T (x ), y1∗ = 0 let ς ∈ C so that  ς = 1
and T (x ), y1∗ = ς  T (x ), y1∗  . Then
 x , T ∗(y1∗)  =
T (ς¯ x ), y1∗ < r.
By (2.1) we can conclude that T ∗(y1∗) < y1∗ , which contradicts the fact that T ∗ is an isometry.
On the other hand, let us suppose that the condition holds. Hence it is plain that T = 1 and so
T ∗ = 1. Further, given ε > 0 and y2∗ ∈ Y ∗ let y ∈ [Y ]1 so that y2∗ −ε < y, y2∗  . By hypothesis
and the continuity of y2∗ we can choose x1 ∈ [X ]1 so that
y∗
2 − ε <  T (x1), y2∗  =  x1, T ∗(y2∗) ≤
T ∗(y2∗) ≤
y∗
2
and the claim follows.
Theorem 2.3 The underlying algebra A inherits the same kind of stability of its second dual conjugate algebra.
Proof By Lemma 2.1 the map becomes isometric on the range of (L A)∗∗ (or (R A)∗∗) if ( A∗∗, ) (or
( A∗∗, ♦)) is left (or right) stable. In these cases it is clear that A becomes left (or right) stable.
If a∗∗ ∈ A∗∗ we see that
where L is the representation of ( A∗∗, ) into B( A∗) so that for a∗∗ ∈ A∗∗ and a∗ ∈ A∗ acts as L(a∗∗)(a∗) =
a∗∗a∗. Consequently, ( A∗∗, ) is right stable if and only if L is isometric. Moreover, let : B( A∗∗) → B( A∗)
be the norm one projection = (χA)∗ ◦ ∗ ◦ χA∗ introduced in [6], p. 545. Following the notation of Lemma
2.1, for a ∈ A, a∗ ∈ A∗ and a∗∗ ∈ A∗∗ we have
a, [ ◦
◦ (R A)∗∗](a∗∗)(a∗) = a, [ ((R A)∗∗(a∗∗))] (a∗)
= [ ((R A)∗∗(a∗∗))](χA(a)), χA∗ (a∗)
= a∗, [ ((R A)∗∗(a∗∗))](χA(a))
= u(a∗, χA(a)), (R A)∗∗(a∗∗)
= (R A)∗[u(a∗, χA(a))], a∗∗
= a∗a, a∗∗
= a, L(a∗∗)(a∗) ,
i.e., L = ◦ ◦ (R A)∗∗. The maps , and (R A)∗∗ are contractive. So, if L is isometric then (R A)∗∗ must
be isometric. But χB(A) ◦ R A = (R A)∗∗ ◦ χA and as χB(A) and χA are isometric it is plain that R A becomes
isometric.
Left stability of A induced by left stability of ( A∗∗, ♦) follows analogously.
Example 2.4 Second conjugate algebras of stable Banach algebras could be not stable. For instance, let us
consider the abelian Banach algebra A = L1[
0, 1
] with respect to the usual convolution product. It is stable as it
is provided by the Fejér kernel of a bounded approximate identity (cf. [7], 1.8.15). However, let E ∈ L1[
0, 1
]∗∗
so that g, E = g(0) if g ∈ C[
0, 1
]. Let f ∈ L∞[
0, 1
] and g ∈ C[
0, 1
]. Then f becomes absolutely integrable,
( f ∗ g)(0) = 0 and f ∗ g ∈ C[
0, 1
] because C[
0, 1
] is an ideal of A. So,
i.e., E f = 0A∗ by the density of continuous functions within A. Now, given F ∈ A∗∗ we obtain
g, E f = f ∗ g, E = ( f ∗ g)(0) = 0,
f, R (E)(F) = f, F E = E f, F = 0.
We infer that R (E) = 0B(A∗∗) and as evidently E = 0A∗∗ then ( A∗∗, ) is not right stable.
Theorem 2.5 Let ζ1, ζ2 ∈ B[ A⊗ˆ A∗, A∗], η1, η2 ∈ B[ A∗ ⊗ˆ A∗∗, A∗] be the unique operators so that
ζ1(a ⊗ a∗) = a∗a, ζ2(a ⊗ a∗) = aa∗, η1(a∗ ⊗ a∗∗) = a∗∗a∗, η2(a∗ ⊗ a∗∗) = a∗a∗∗,
for all a ∈ A, a∗ ∈ A∗, a∗∗ ∈ A∗∗. Then:
(1) ( A∗∗, ) is RSBA if and only if ζ1([ A⊗ˆ A∗]1)− = [ A∗]1.
(2) ( A∗∗, ) is LSBA if and only if η1([ A∗⊗ˆ A∗∗]1)− = [ A∗]1.
(3) ( A∗∗, ♦) is RSBA if and only if η2([ A∗⊗ˆ A∗∗]1)− = [ A∗]1.
(4) ( A∗∗, ♦) is LSBA if and only if ζ2([ A⊗ˆ A∗]1)− = [ A∗]1.
Proof (1) It is worth mentioning the isometric isomorphism of Banach spaces ( A⊗ˆ A∗)∗ ≈ B( A∗). For, if
Q ∈ B( A∗) there is a unique θQ ∈ ( A⊗ˆ A∗)∗ so that a ⊗ a∗, θQ = a, Q(a∗) on tensors. On the other
hand, given θ ∈ ( A⊗ˆ A∗)∗ then Qθ ∈ B( A∗) if we set a, Qθ (a∗) = a ⊗ a∗, θ . It is easy to see that the
linear mappings Q → θQ and θ → Qθ are isometric and inverse of each other (cf. [5], Ch. VIII.2, Th. 1).
In particular, ζ1∗ ∈ B[ A∗∗, ( A⊗ˆ A∗)∗] and given a, a∗ and a∗∗ we have
i.e., Qζ1∗(a∗∗)(a∗) = a∗∗a∗ for all a∗ and
Qζ1∗(a∗∗) = ζ1∗(a∗∗) . Moreover,
a, a∗∗a∗ = a∗a, a∗∗
= ζ1(a ⊗ a∗), a∗∗
= a ⊗ a∗, ζ1∗(a∗∗)
= a, Qζ1∗(a∗∗)(a∗) ,
and the assertion follows by Lemma 2.2.
(2) Following the notation of Lemma 2.1, by the universal property of projective tensor products there is a
unique u∧ ∈ B[ A∗ ⊗ˆ A∗∗, B( A)∗] so that uˆ = u B[A∗,A∗∗;B(A)∗] and for any a∗ and a∗∗ the identity
u∧(a∗ ⊗ a∗∗) = u(a∗, a∗∗) holds. If n ∈ B( A)∗∗ we see that
Hence, (n) = (uˆ)∗(n) (A∗ ⊗ˆA∗∗)∗ . Indeed, η1∗ = (uˆ)∗ ◦ (L A)∗∗. Now, if A∗∗ is a left stable Banach
algebra by Lemma 2.1 the operators (L A)∗∗ and R[(L A)∗∗] become both isometric. Hence, if a∗∗ ∈ A∗∗
we have
a∗∗
= (L A)∗∗(a∗∗)
and η1∗ becomes isometric. Therefore, by Lemma 2.2 we have
and the necessity follows.
Reciprocally, if b∗∗ ∈ A∗∗ by Lemma 2.2 we can write
=
=
=
((L A)∗∗(a∗∗))
(uˆ )∗((L A)∗∗(a∗∗))
η1∗(a∗∗)
η1([ A∗⊗ˆ A∗∗]1)− = [ A∗]1
L (b∗∗)
=
=
=
=
[(L A)∗∗(b∗∗)]
(uˆ )∗[(L A)∗∗(b∗∗)]
η1∗(b∗∗)
b∗∗
.
(3) Here, we argue as in (2) since by Lemma 2.1 it is readily seen that R♦ = η∗.
2
(4) Similar argument as in (1).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://
creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided
you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate
if changes were made.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
1. AnsariPiri , E. ; Nouri , S. : Almost multipliers and some of their properties . J. Adv. Math . 11 ( 7 ), 5397  5402 ( 2015 ) 2 . AnsariPiri , E. ; Nouri , S. : Stable normed algebras . arXiv:1509.08142 , 1  8 ( 2015 ) 3 . Arens , R.: Operations induced in function classes . Monatsh. Math. 55 , 1  19 ( 1951 ) 4 . Arens , R.: The adjoint of a bilinear operation . Proc. Am. Math. Soc. 2 , 839  848 ( 1951 ) 5 . Diestel , J.; Uhl , J. J. , Jr.: Vector measures . Am. Math. Surveys and Monogr . 15. AMS ( 1977 ) 6 . Grosser , M. : Arens semiregularity of the algebra of compact operators . Illinois J. Math . 31 ( 4 ), 544  573 ( 1987 ) 7 . Palmer , T.W.: Banach algebras and the general theory of *algebras , vol. I. Cambridge University Press, Cambridge ( 2004 ) 8 . Palmer , T.W.: The bidual of the compact operators . Trans. Am. Math. Soc . 288 ( 2 ), 827  839 ( 1985 )