1 Introduction
The notion of a stable structure is widely studied, both in the general context of model theory and in functional analysis. In particular, the concept of a stable Banach space was introduced and studied by Krivine and Maurey [Reference Krivine and Maurey18]. This notion, which we will refer to as additive stability, plays an important role in the study of the geometry of Banach spaces (see [Reference Guerre and Lapresté10, Reference Iovino14, Reference Kalton15, Reference Raynaud21]). We recall the definition and introduce the analogous natural concept for Banach algebras of multiplicative stability.
Definition 1.1. Let
${\cal {A}}$
be a Banach algebra. We say that
${\cal {A}}$
is additively or multiplicatively stable if the following holds: for any bounded sequences
$(a_i)$
and
$(b_j)$
in
${\cal {A}}$
and any free ultrafilters
${\mathcal {U}}$
and
${\mathcal {V}}$
on
$\mathbb {N}$
, we have respectively
$$ \begin{align*}\lim_{i, {\mathcal{U}}} \lim_{j, {\mathcal{V}}} \| a_i + b_j \| = \lim_{j, {\mathcal{V}}} \lim_{i, {\mathcal{U}}} \| a_i + b_j \| \quad\mbox{or}\quad \lim_{i, {\mathcal{U}}} \lim_{j, {\mathcal{V}}} \| a_i b_j \| = \lim_{j, {\mathcal{V}}} \lim_{i, {\mathcal{U}}} \| a_i b_j \| .\end{align*} $$
Otherwise,
${\cal {A}}$
is called additively or multiplicatively unstable, respectively.
In these definitions, the use of ultrafilters can be avoided by considering ordinary double limits and asking that they coincide whenever they exist. For Banach algebras, stability in the model theoretic sense was studied by Farah et al. in [Reference Farah, Hart and Sherman7, Reference Farah, Hart and Sherman8]. We will refer to this notion simply as ‘stability’. Since stability implies in particular a condition similar to the above on all quantifier-free formulae (see, for example, [Reference Iovino, Caicedo and Montenegro13, Theorem 9.1]), it implies both additive and multiplicative stability. As noted in [Reference Farah, Hart and Sherman8, Section 6.2], for any
$p \in [ 1, \infty )$
, the Banach algebra
$\ell _p$
with pointwise product is stable. However, as we will see below,
$c_0$
(and hence
$\ell _\infty $
) is both additively and multiplicatively unstable.
Remark 1.2. (i) Considering arbitrary families (and free ultrafilters on their index sets) rather than sequences does not change the above definition (see [Reference Raynaud21, I.A(i)] or [Reference Dales and Lau5, Proposition 3.1]).
(ii) Boundedness of both sequences is crucial in the definition. Indeed, consider, as above, the stable Banach algebra
$\ell _p$
with pointwise product, where
$p \in [ 1, \infty )$
. Put
$a_i := e_i$
, the standard basis of
$\ell _p$
, and
$b_j := \sum _{k=1}^{j} e_k$
. Then,
$\lim _i \lim _j \| a_i b_j \|_p = \lim _i \| e_i \|_p = 1$
, whereas
$\lim _j \lim _i \| a_i b_j \|_p = 0$
.
(iii) Given a Banach algebra
${\cal {A}}$
, its unit ball
$B_{\cal {A}}$
, with the norm topology, is a topological semigroup for the algebra product. A fundamental object associated with any topological semigroup S is the algebra
${\mathrm {WAP}}(S)$
of weakly almost periodic functions on S, that is, all
$f \in C_b(S)$
such that the set of left (equivalently, right) translates of f by elements of S is relatively weakly compact in
$C_b(S)$
; the latter denotes the algebra of all continuous bounded complex-valued functions on S. As is well known,
${\mathrm {WAP}}(S)$
, endowed with the supremum norm, is a commutative
$C^*$
-algebra, and functions in
${\mathrm {WAP}}(S)$
are characterised by Grothendieck’s double limit criterion (see, for example, [Reference Berglund, Junghenn and Milnes3, Theorem 2.3]). Thus, a Banach algebra
${\cal {A}}$
is multiplicatively stable if and only if its norm belongs to
${\mathrm {WAP}}(B_{\cal {A}})$
. An earlier version of the present paper, in particular, from the viewpoint of this observation, was the starting point of the PhD thesis [Reference Alabkary1] by N. Alabkary, written under the supervision of the second author. In [Reference Alabkary1], Banach algebras are studied whose norm belongs to various function spaces over the unit ball of the algebra. This leads to a dynamical hierarchy of Banach algebras. The main results of [Reference Alabkary1] are contained in [Reference Alabkary2] and a joint paper by N. Alakbary and M. Neufang in preparation. As we shall show, many classes of Banach algebras, such as infinite-dimensional
$C^*$
-algebras and various convolution algebras, are multiplicatively unstable. This is particularly interesting from the viewpoint of the dynamical hierarchy of Banach algebras. Indeed, the latter allows us to refine the degree of multiplicative instability through the notion of wildness (see [Reference Alabkary1]). We call a Banach algebra
$\mathcal {A}$
wild if its norm is a nontame function on the unit ball of
$\mathcal {A}$
(here, tameness is understood in the sense of topological dynamics; see [Reference Glasner, Megrelishvili, Ferenczi, Kulaga-Przymus and Lemańczyk9]). Any wild Banach algebra is obviously multiplicatively unstable, and, for instance, any infinite-dimensional
$C^*$
-algebra is wild (see [Reference Alabkary1]), but this is not known for the Fourier algebra.
(iv) In the spirit of item (iii), associated to any Banach algebra
${\cal {A}}$
is the space
${\mathrm {WAP}}({\cal {A}})$
of weakly almost periodic functionals on
${\cal {A}}$
, that is, all
$\varphi \in {\cal {A}}^*$
such that the operator
${\cal {A}} \ni a \mapsto \varphi a \in {\cal {A}}^*$
is weakly compact; here,
$\langle \varphi a , b \rangle := \langle \varphi , ab \rangle $
for all
$a,b \in {\cal {A}}$
. As is well known (see [Reference Pym20]),
$\varphi \in {\mathrm {WAP}}({\cal {A}})$
if and only if
$\varphi $
satisfies our condition of multiplicative stability, which in Definition 1.1 is imposed on the sublinear functional
$\| \cdot \|$
. Moreover, the equality
${\mathrm {WAP}}({\cal {A}}) = {\cal {A}}^*$
holds if and only if
${\cal {A}}$
is Arens regular, that is, the two Arens products on the bidual
${\cal {A}}^{**}$
coincide (see [Reference Hennefeld11, Theorem 2.1]). In a future paper, we will study stability of Banach algebras more generally. In the present note, we focus on exhibiting examples from various classes of Banach algebras.
2 Multiplicative stability for
$C^*$
-algebras, Fourier algebras and convolution algebras
It is proved in [Reference Farah, Hart and Sherman7, Lemma 5.3] that all infinite-dimensional
$C^*$
-algebras are unstable (for the witness function,
$\varphi (a,b) = \| ab-b \|$
, that is, this function witnesses different iterated limits for certain bounded sequences
$(a_i)$
and
$(b_j)$
). The following argument shows that for any infinite-dimensional
$C^*$
-algebra, instability is witnessed already at the level of the addition operation; in other words, the underlying Banach space is unstable in the sense of Krivine–Maurey.
Proposition 2.1. Every infinite-dimensional
$C^*$
-algebra is additively unstable.
Proof. Suppose there is an infinite-dimensional additively stable
$C^*$
-algebra. Take an infinite-dimensional separable
$C^*$
-subalgebra. As the latter is separable and additively stable, it is weakly sequentially complete by [Reference Guerre and Lapresté10, Théorème 1], whence finite-dimensional by [Reference Sakai22, Proposition 2], which gives a contradiction.
We now consider multiplicative stability.
Proposition 2.2. Every infinite-dimensional
$C^*$
-algebra is multiplicatively unstable.
Proof. Let
${\cal {A}}$
be an infinite-dimensional
$C^*$
-algebra. According to [Reference Ogasawara19] (see also [Reference Blackadar4, Theorem II.3.2.12]),
${\cal {A}}$
contains an infinite-dimensional abelian
$C^*$
-subalgebra
${\cal {A}}_1$
. Also by [Reference Blackadar4, Theorem II.3.2.12],
${\cal {A}}_1$
in turn contains a nonunital abelian
$C^*$
-subalgebra
${\cal {A}}_2$
. Now,
${\cal {A}}_2$
is
$C^*$
-isomorphic to
$C_0(X)$
for a locally compact noncompact Hausdorff space X. We shall find two bounded sequences
$(f_i)$
and
$(g_j)$
in
$C_0(X)$
such that
$$ \begin{align*}\lim_i \lim_j \| f_i g_j \|_\infty = 1 \quad\text{and}\quad \lim_j \lim_i \| f_i g_j \|_\infty = 0,\end{align*} $$
which will conclude the proof.
Construct inductively a sequence of nonempty compacta
$(K_i)$
such that
$K_i \cap K_j = \varnothing $
for all
$i \not = j$
, and functions
$(f_i) \subseteq C_c(X)$
of norm 1 with
$\text {supp} f_i \subseteq K_i$
for all i. One can clearly choose
$K_1$
and
$f_1$
as desired. Now, assume
$K_i$
and
$f_i$
have been constructed for all
$i=1, \ldots , n-1$
. As
$\bigcup _{i=1}^{n-1} K_i$
is compact and X is not, there are a nonempty compact set
$K_n$
such that
$K_n \cap \bigcup _{i=1}^{n-1} K_i = \varnothing $
, and
$f_n \in C_c(X)$
of norm 1 with
$\text {supp} f_n \subseteq K_n$
. So we have
$\text {supp} f_n \cap \bigcup _{i=1}^{n-1} \text {supp} f_i = \varnothing $
.
Put
$g_j := \sum _{k=1}^j f_k$
for all
$j \in \mathbb {N}$
. Obviously, each
$g_j$
has norm 1. We have
$\lim _i \lim _j \| f_i g_j \|_\infty = \lim _i \| f_i^2 \|_\infty = 1$
and
$\lim _j \lim _i \| f_i g_j \|_\infty = 0$
, as desired.
We now turn to algebras which are far away from being
$C^*$
-algebras, namely Fourier and Fourier–Stieltjes algebras over locally compact groups (unlike
$C^*$
-algebras, these are always weakly sequentially complete, since they are preduals of von Neumann algebras). We recall that, given a locally compact group
${\cal {G}}$
, the Fourier algebra
$A({\cal {G}})$
consists of the coefficients of the left regular representation
$\lambda $
of
${\cal {G}}$
, that is, all functions of the form
${\cal {G}} \ni x \mapsto \langle \lambda (x) \xi , \eta \rangle $
, where
$\xi , \eta \in L_2({\cal {G}})$
; here,
$(\lambda (x) \xi ) (y) = \xi (x^{-1} y)$
for all
$x,y \in {\cal {G}}$
. With the norm
$$ \begin{align*}\| f \| = \inf \{ \| \xi \|_2 \| \eta \|_2 : f = \langle \lambda (\cdot) \xi , \eta \rangle \}\end{align*} $$
and pointwise product,
$A({\cal {G}})$
becomes a commutative Banach algebra; note that
$A({\cal {G}})$
is contained (in fact, uniformly dense) in
$C_0({\cal {G}})$
. Moreover,
$A({\cal {G}}) = {\mathit {VN}}({\cal {G}})_*$
, that is,
$A({\cal {G}})$
is the predual of the group von Neumann algebra
${\mathit {VN}}({\cal {G}})$
; the latter forms a central object in the theory of operator algebras.
The Fourier–Stieltjes algebra
$B({\cal {G}})$
over a locally compact group
${\cal {G}}$
consists of all functions of the form
${\cal {G}} \ni x \mapsto \langle \pi (x) \xi , \eta \rangle $
, where
$\pi $
is a unitary representation of
${\cal {G}}$
on some Hilbert space H and
$\xi , \eta \in H$
. With the norm
$$ \begin{align*}\| f \| = \sup \bigg\{ \bigg\lvert \kern-2pt\int_{\cal{G}}\kern-2pt f(x) h(x) \,dx \bigg\rvert : h \in L_1({\cal{G}}), \|h\|_{C^*} \leq 1 \bigg\}\end{align*} $$
and pointwise product,
$B({\cal {G}})$
becomes a commutative unital Banach algebra;
$B({\cal {G}})$
contains
$A({\cal {G}})$
as a closed ideal and is contained in
${\mathrm {WAP}}({\cal {G}})$
, the algebra of weakly almost periodic functions on
${\cal {G}}$
. We have
$B({\cal {G}}) = C^*({\cal {G}})^*$
, that is,
$B({\cal {G}})$
is the dual of the full group
$C^*$
-algebra
$C^*({\cal {G}})$
. Finally,
$B({\cal {G}})$
can also be described as consisting of all finite linear combinations of continuous positive definite functions on
${\cal {G}}$
.
In the case where
${\cal {G}}$
is abelian, with dual group
$\hat {{\cal {G}}}$
, the Fourier transform identifies the group algebra
$L_1({\cal {G}})$
with
$A(\hat {{\cal {G}}})$
, and the Fourier–Stieltjes transform identifies the measure algebra
$M({\cal {G}})$
with
$B(\hat {{\cal {G}}})$
, turning convolution into the pointwise product (more precisely, the transforms yield isometric Banach algebra isomorphisms). Fourier and Fourier–Stieltjes algebras are fundamental objects of study in abstract harmonic analysis (see, for example, the recent monograph [Reference Kaniuth and Lau17] devoted to these algebras).
Theorem 2.3. Let
${\cal {G}}$
be a locally compact group containing a noncompact amenable open subgroup. Then, the Fourier algebra
$A({\cal {G}})$
is multiplicatively unstable.
Proof. Let
${\cal {H}}_0$
be a noncompact amenable open subgroup of
${\cal {G}}$
. Take a noncompact
$\sigma $
-compact open subgroup
${\cal {H}}$
of
${\cal {H}}_0$
. So,
${\cal {H}}$
is a noncompact
$\sigma $
-compact amenable open subgroup of
${\cal {G}}$
. By [Reference Herz12, Proposition 5],
$A({\cal {H}})$
is a closed subalgebra of
$A({\cal {G}})$
, so it is enough to show that
$A({\cal {H}})$
is multiplicatively unstable. To this end, in
${\cal {H}}$
, take an increasing sequence
$(K_i)$
of compacta with nonempty interior
$K_i^\circ $
such that
$\bigcup K_i^\circ = {\cal {H}}$
. For all
$i \in \mathbb {N}$
, there is
$x_i \in {\cal {H}} \setminus K_i$
(as
${\cal {H}}$
is noncompact). Now, choose open neighbourhoods
$V_i$
of
$x_i$
such that
$V_i \cap K_i = \varnothing $
.
Denote by
$A_c({\cal {H}})$
the set of functions in
$A({\cal {H}})$
with compact support. Take compact symmetric neighbourhoods
$W_i$
of e such that
$x_i W_i^2 \subseteq V_i$
. Consider
$$ \begin{align*}f_i := \frac{1}{m(W_i)} \chi_{x_i W_i} * \check{\chi_{W_i}} ,\end{align*} $$
where m denotes left Haar measure of
${\cal {H}}$
. Then,
$f_i \in A_c({\cal {H}})$
with
$f_i(x_i) = 1$
and
$\mathrm {supp} (f_i) \subseteq x_i W_i^2 \subseteq V_i$
(see, for example, [Reference Kaniuth16, Lemma 2.9.5]). Also,
$\| f_i \| = 1$
since
$$ \begin{align*}1= | f_i (x_i) | \leq \| f_i \|_{\infty} \leq \| f_i \| \leq \frac{1}{m(W_i)} \| \chi_{x_i W_i} \|_2 \| \chi_{W_i} \|_2 = 1 .\end{align*} $$
Furthermore, there is a sequence
$(g_j) \subseteq A_c({\cal {H}})$
which is a bounded approximate identity for
$A({\cal {H}})$
. This can be seen by inspection of [Reference Kaniuth and Lau17, proof of Theorem 2.7.2], using the
$\sigma $
-compactness of
${\cal {H}}$
to obtain a sequence rather than a net.
Finally,
$| \| f_i g_j \| - \| f_i \| | \leq \| f_i g_j - f_i \| \to _j 0$
for any fixed i, so we obtain
$$ \begin{align*}\lim_i \lim_j \| f_i g_j \| = \lim_i \| f_i \| = 1.\end{align*} $$
Now, fix any j. Then,
$\mathrm {supp} (g_j)$
is compact, and hence contained in some
$K_{i_0}$
. For any
$i\hspace{-0.5pt} \geq\hspace{-0.5pt} i_0$
, we have
$\mathrm {supp} (f_i)\hspace{-0.5pt} \subseteq\hspace{-0.5pt} V_i$
and
$V_i\hspace{-0.5pt} \cap\hspace{-0.5pt} K_i\hspace{-0.5pt} =\hspace{-0.5pt} \varnothing $
, whence
$f_i g_j\hspace{-0.5pt} =\hspace{-0.5pt} 0$
(as
$\mathrm {supp} (g_j)\hspace{-0.5pt} \subseteq\hspace{-0.5pt} K_{i_0}\hspace{-0.5pt} \subseteq\hspace{-0.5pt} K_i$
). So we obtain
$$ \begin{align*}\lim_j \lim_i \| f_i g_j \| = 0.\end{align*} $$
Thus,
$A({\cal {H}})$
is multiplicatively unstable, as claimed.
Remark 2.4. (i) In view of Remark 1.2(i), we could have used families in the proof of Theorem 2.3, but we prefer to indicate how to work with sequences directly.
(ii) Theorem 2.3 obviously covers all noncompact amenable groups and also many nonamenable ones. In particular, it covers every discrete group with an infinite amenable subgroup, such as the free group
$\mathbb {F}_2$
, the prototypical example of a nonamenable group.
(iii) As Theorem 2.3 applies to all noncompact abelian groups
${\cal {G}}$
and since
$A({\cal {G}}) \cong L_1(\hat {{\cal {G}}})$
(isometrically isomorphic Banach algebras), with
$\hat {{\cal {G}}}$
denoting the dual group, it follows that
$L_1(\mathcal {H})$
is multiplicatively unstable for all nondiscrete abelian groups
$\mathcal {H}$
. For instance,
$L_1(\mathbb {T})$
as well as
$L_1(\mathbb {R})$
are multiplicatively unstable. This obviously implies their (model theoretic) instability. The latter property for
$L_1(\mathbb {R})$
was first established in [Reference Farah, Hart and Sherman8]. Indeed, as shown there (see the proof of Proposition 6.2 and the following remark), a commutative Banach algebra is unstable (in the model theoretic sense) if it has a norm
$1$
element g such that the range of the Gelfand transform
$\hat {g}$
contains scalars with modulus arbitrarily close to but different from
$1$
. In this result, instability is established by means of the witness function
$\varphi (x,y) = \inf _{\|z\|_1 \leq 1} \| xz-y \|$
, and the criterion applies for example to
$\ell _1(\mathbb {Z})$
and
$L_1(\mathbb {R})$
. Note that the criterion does not apply to
$L_1(\mathcal {H})$
when
$\mathcal {H}$
is a compact abelian group, such as
$\mathbb {T}$
, since the Gelfand transform
$\hat {g}$
of any
$g \in L_1(\mathcal {H})$
satisfies
$\hat {g} \in A(\hat {\mathcal {H}}) \subseteq c_0(\hat {\mathcal {H}})$
, with
$\hat {\mathcal {H}}$
discrete. We close by remarking that
$L_1(\mathbb {T})$
and
$L_1(\mathbb {R})$
provide natural examples of Banach algebras that are additively stable (as proved in [Reference Krivine and Maurey18]), but multiplicatively unstable (as we have shown above).
Finally, we turn to measure algebras. A deep theorem of Zelmanov [Reference Zelmanov23] allows us to deduce the multiplicative instability of measure algebras over any infinite compact group from Theorem 2.3.
Theorem 2.5. Let
${\cal {G}}$
be an infinite compact group. Then,
$M({\cal {G}})$
is multiplicatively unstable.
Proof. By [Reference Zelmanov23, Theorem 2],
${\cal {G}}$
contains an infinite compact abelian subgroup
${\cal {H}}$
. Now, by [Reference Dales, Lau and Strauss6, Proposition 5.3],
$M({\cal {H}})$
is a closed subalgebra of
$M({\cal {G}})$
. However,
$M({\cal {H}}) \cong B(\hat {{\cal {H}}})$
(isometrically isomorphic Banach algebras) and the Fourier–Stieltjes algebra
$B(\hat {{\cal {H}}})$
in turn contains
$A(\hat {{\cal {H}}})$
as a closed subalgebra. As
$\hat {{\cal {H}}}$
is infinite discrete abelian, Theorem 2.3 entails that
$A(\hat {{\cal {H}}})$
is multiplicatively unstable; hence, the same holds for
$M({\cal {G}})$
.
