1 Introduction
Let G be a locally compact group. The Fourier–Stieltjes
$B(G)$
and the Fourier algebras
$A(G)$
of G were introduced by Eymard in his celebrating paper [Reference Eymard11]. Recall that
$B(G)$
is the linear combination of all continuous positive-definite functions on G, and as a Banach space,
$B(G)$
is naturally isometric to the predual of
$W^*(G)$
, the von Neumann algebras generated by the universal representations
$\omega _G$
of G. Moreover, it is a commutative Banach
$*$
-algebra with respect to pointwise multiplication and complex conjugation. The Fourier algebra
$A(G)$
is the closed ideal of
$B(G)$
generated by the functions with compact supports. As a Banach space,
$A(G)$
is isometric to the predual of the group von Neumann algebra
$\mathrm {VN}(G)$
, the von Neumann algebra generated by the left regular representations
$\lambda _G$
of G. It is well known that
$A(G)$
is regular and semisimple, and the Fourier and the Fourier–Stieltjes algebras are both subalgebras of
$C_b(G)$
, the algebra of continuous bounded functions on G.
Takesaki and Tatsuuma in [Reference Takesaki and Tatsuuma24] showed that there is a one-to-one correspondence between compact subgroups of G and nonzero right invariant closed self-adjoint subalgebras of
$A(G)$
. As a refinement, Bekka, Lau, and Schlichting in [Reference Bekka, Lau and Schlichting2] studied nonzero, closed, invariant
$*$
-subalgebras of
$A(G)$
. They showed that these spaces are the Fourier algebras
$A(G/K)$
of the quotient group
$G/K$
for some compact normal subgroup K of G. On the other hand, Forrest [Reference Forrest12] introduced the Fourier algebra
$A(G/K)$
of the left coset space
$G/K$
, where K is a compact (not necessary to be normal) subgroup of the locally compact group G. This algebra can simultaneously be viewed as an algebra of functions on
$G/K$
and as the subalgebra of
$A(G)$
consisting of functions in
$A(G)$
which are constants on left cosets of K. Note that
$A(G/K)$
is regular and semisimple, the spectrum
$\sigma (A(G/K))$
is
$G/K$
, and it is a norm-closed left translation invariant
$*$
-subalgebra of
$A(G)$
.
A long-standing question in harmonic analysis is to determine all homomorphisms of Fourier or Fourier–Stieltjes algebras of any locally compact groups. For any pair of locally compact abelian groups G and H, Cohen [Reference Cohen6] characterized all bounded homomorphisms from the group algebra
$L^1(G)$
to the measure algebra
$M(H)$
. In doing so, he made use of a profound discovery of his characterization of idempotent measures on the groups. Cohen’s results were generalized by Host in [Reference Host14], where he discovered the general form of idempotents in the Fourier–Stieltjes algebras, and characterized bounded homomorphisms from
$A(G)$
to
$B(H)$
when the group G has an abelian subgroup of finite index. Further generalizations were made in [Reference Ilie15, Reference Ilie and Spronk16] for any locally compact amenable group G, where completely bounded homomorphisms from
$A(G)$
into
$B(H)$
were characterized by continuous piecewise affine maps (see also [Reference Daws7]). Most general results were given by Le Pham in [Reference Le Pham20], and he determined all contractive homomorphisms from
$A(G)$
into
$B(H)$
for any locally compact groups G and H.
To describe idempotent elements in the Fourier–Stieltjes and the Fourier algebras, we first recall some terminologies. Let G be a group, and let K be a subgroup of G; we see that
$Ks= ss^{-1}Ks$
for any
$s\in G$
, which means that we need not distinguish between left and right cosets of the group G. The coset ring of G, denoted
$\Omega (G)$
, is the smallest ring of subsets of G containing all cosets of subgroups of G. We denote
$\Omega _{\text {o}}(G)$
the ring of subsets generated by open cosets of G, and similarly,
$\Omega _{\text {o}}^{\text {c}}(G)$
the ring of subsets generated by compact open cosets of G. By [Reference Host14], idempotents in the Fourier–Stieltjes algebra
$B(G)$
are the indicator functions
$1_F$
of an element F of
$\Omega _{\text {o}}(G)$
. Let
$I_B(G)$
be the set of all idempotent elements in
$B(G)$
. We denote the closure of the span of
$I_B(G)$
by
$B_I(G)$
. From [Reference Ilie and Spronk17, Proposition 1.1], we have that
$$ \begin{align*} A(G) \cap I_B(G)= \{1_Y: Y\in \Omega_{\text{o}}^{\text{c}}(G)\}, \end{align*} $$
which gives rise to idempotents in
$A(G)$
, denoted by
$I(G)$
. Let
$A_I(G)$
be the subalgebra of
$A(G)$
generated by
$I(G)$
. Note that Ilie and Spronk [Reference Ilie and Spronk16] showed that
$1_F$
is an idempotent in
$B(G)$
with
$\|1_F\|_{B(G)}= 1$
if and only if F is an open coset in G; however, there are idempotents with small norms [Reference Mudge and Le Pham23] or with large norms [Reference Anoussis, Eleftherakis and Katavolos1]. Moreover, the existence of idempotents of arbitrarily large norm implies the existence of homomorphisms of arbitrarily large norm (see [Reference Anoussis, Eleftherakis and Katavolos1] for details). Thus, idempotent elements play an essential role in the study of homomorphisms on Fourier algebras. It is of its own interest to study the norms of idempotent elements in Fourier–Stieltjes and Fourier algebras, but for our purpose, we will focus solely on operators that preserve idempotents.
In the rich literature of linear preservers, there are many works that study linear maps T on spaces X which preserve some subsets S of X, i.e.,
$T(S) \subset S$
. Dieudonné in [Reference Dieudonné8] studied semilinear maps on
$M_n(\mathbb {K})$
, the algebra of
$n \times n$
matrices over a field
$\mathbb {K}$
, which preserve the set of all singular matrices. After that, many mathematicians considered linear maps on
$M_n(\mathbb {K})$
that preserve subsets of matrices with different properties (e.g., [Reference Botta, Pierce and Watkins3, Reference Dixon9, Reference Jacob18, Reference Marcus and Moyls22] to name a few). In [Reference Brešar and Šemrl4], it is shown that every complex linear map T on
$M_n(\mathbb {C})$
which preserves the set of all idempotents is either an inner automorphism or an inner antiautomorphism. In addition, in [Reference Brešar and Šemrl5], linear maps on
$M_n(\mathbb {C})$
which send potent matrices (that is, matrices A satisfy
$A^r= A$
for some integer
$r\geq 2$
) to potent matrices were characterized. Since then, the studies of idempotent preserving maps have attracted considerable interest (see, e.g., [Reference Dolinar10, Reference Guterman, Li and Šemrl13]). Recently, in [Reference Li, Tsai, Wang and Wong21], the authors proved that every additive map from the rational span of Hermitian idempotents in a von Neumann algebra into the rational span of Hermitian idempotents in a C*-algebra can be extended to a Jordan
$*$
-homomorphism.
In this article, we study bounded linear operators from
$A(G)$
into
$B(H)$
which send idempotents to idempotents. We show that such an operator will give rise to an algebraic homomorphism on
$A_I(G)$
. The algebra
$A_I(G)$
will be our main object of study, namely, we will characterize linear mappings defined on the Fourier algebra
$A(G)$
or on
$A_I(G)$
which preserve
$I(G)$
. Moreover, we show that when the groups are totally disconnected, idempotent preserving operators will recover algebraic homomorphisms on the Fourier algebra.
2 Main results
Let G be a locally compact group, and let K be a closed subgroup of G. We will denote by
$G/K$
the homogeneous space of left cosets of K. Let
$$ \begin{align*} B(G: K):= \{u\in B(G): u(xk)= u(x) \, \text{for all } x\in G, k\in K\}, \end{align*} $$
that is, functions in
$B(G)$
which are constant on cosets of K, and
$$ \begin{align*} A(G: K):= \{u\in B(G: K): q(\mathrm{supp}(u)) \text{ is compact in } G/K\}^{-_{B(G)}}, \end{align*} $$
where
$\mathrm {supp}(u)$
is the support of u in G and q is the canonical quotient map from G to
$G/K$
. If, furthermore, K is a normal subgroup, by [Reference Forrest12, Proposition 3.2], we have that
$B(G: K)$
and
$A(G: K)$
are isometrically isomorphic to the Fourier–Stieltjes and the Fourier algebras
$B(G/K)$
and
$A(G/K)$
, respectively. Note that
$A(G:K)\cap A(G)\neq \{0\}$
if and only if K is compact (see [Reference Forrest12, Proposition 3.1] or [Reference Takesaki and Tatsuuma24, Theorem 9] and [Reference Bekka, Lau and Schlichting2, Theorem 2.1] for more details).
Let e be the identity of the group G, and we denote the connected component of e by
$G_e$
, which is a closed normal subgroup of G; thus,
$G/G_e$
is a totally disconnected locally compact group. The following result about the algebra
$A_I(G)$
generated by idempotents of
$A(G)$
in relation with
$A(G: G_e)$
was given in [Reference Ilie and Spronk17]; for the completion, we give a short proof in the article.
Proposition 2.1 [Reference Ilie and Spronk17, Proposition 1.1(ii)] If the connected component
$G_e$
is compact, then
$A_I(G)= A(G:G_e)$
, that is,
$A_I(G)$
consists of all functions in
$A(G)$
that are constant on cosets of
$G_e$
. On the other hand, if
$G_e$
is not compact, then
$A_I(G)= \{0\}$
.
Proof Let
$q_G: G \to G/G_e$
be the quotient map onto
$G/G_e$
. Since
$G_e$
is compact, via
$u \mapsto u\circ q_G$
, we have that
$A(G/G_e)$
is isometrically isomorphic to
$A(G:G_e)$
, which is a closed subalgebra of
$A(G)$
. Thus,
$A_I(G)= \overline {\operatorname {\mathrm {span}}} \{1_Y: Y\in \Omega _{\text {o}}^{\text {c}}(G)\} \subseteq A(G:G_e)$
. Conversely, since
$G/G_e$
is totally disconnected, we have that the span of the idempotents of
$A(G/G_e)$
is dense [Reference Forrest12, Theorem 5.3]. Moreover,
$A(G/G_e)$
is isomorphic to
$A(G:G_e)$
; thus,
$A(G:G_e)$
is generated by idempotents of
$A(G/G_e)$
, so
$A(G:G_e) \subseteq A_I(G)$
.
If the Fourier algebra which contains nontrivial idempotents, that is, the connected component
$G_e$
, is compact, then by Proposition 2.1, there is an isometric isomorphism from
$A_I(G)$
onto
$A(G/G_e)$
. More precisely, it induces an isometric isomorphism
$\varphi _G: A_I(G) \to A(G/G_e)$
as
$$ \begin{align} \varphi_G(f)(q_G(a))=f(a) \end{align} $$
for any
$f \in A_I(G)$
and
$a \in G$
, where
$q_G: G \to G/G_e$
is the quotient map onto
$G/G_e$
.
2.1 Idempotent preserving maps with
$T(I(G)) \subset I_B(H)$
Let G and H be two locally compact groups. We consider a bounded complex linear map
$T: A(G) \to B(H)$
which satisfies
$$ \begin{align} T(I(G)) \subset I_B(H). \end{align} $$
For any
$f \in \operatorname {\mathrm {span}} \{1_Y: Y\in \Omega _{\text {o}}^{\text {c}}(G)\}$
, there exist
$\alpha _i \in \mathbb {C}$
and
$Y_i \in \Omega _{\text {o}}^{\text {c}}(G)$
such that
$f= \Sigma _{k= 1}^{n} \alpha _k 1_{Y_k}$
. Thus, we have
$Tf=\Sigma ^{n}_{k=1} \alpha _k T1_{Y_k} \in \operatorname {\mathrm {span}} \{1_Y: Y\in \Omega _{\text {o}}(H)\} \subset B(H)$
. Let us recall that
$A_I(G)= \overline {\operatorname {\mathrm {span}}} \{1_Y: Y\in \Omega _{\text {o}}^{\text {c}}(G)\}$
and
$B_I(H)= \overline {\operatorname {\mathrm {span}}} \{1_Y: Y\in \Omega _{\text {o}}(H)\}$
. Since T is a bounded map, we obtain
$T(A_I(G)) \subseteq B_I(H) \subset B(H)$
.
Our aim is to obtain a representation of such a map T on
$A_I(G)$
. If
$I(G)=\{0\}$
, then
$A_I(G)=\{0\}$
. Since T is complex linear, we have
$T=0$
on
$A_I(G)$
. Thus, without loss of generality, we can assume that the Fourier algebra
$A(G)$
have nonzero idempotent elements. Hence, the connected component
$G_e$
is always a compact normal subgroup of G. On the other hand, we define the following map, which will be used in the sequel.
Definition 2.1 Let G be a locally compact group. Using the axiom of choice, let S be a set of representatives of the cosets of
$\mathop {G/G_e}$
, that is,
$G=\bigsqcup _{a\in S}aG_e$
. Then we define a map
$[\ \cdot \ ]_{\mathop {G/G_e}}$
from
$\mathop {G/G_e}$
onto S by
$$ \begin{align*} [aG_e]_{\mathop{G/G_e}}=a \end{align*} $$
for any
$a \in S$
.
We first have the following observations concerning the operator satisfying (2.2).
Lemma 2.2 The map T preserves the disjointness of idempotents. That is,
$Tf \cdot Tg= 0$
for any
$f,g \in I(G)$
with
$f\cdot g= 0$
.
Proof Let
$f, g \in I(G)$
such that
$f\cdot g= 0$
. Then we have
$(f+g)^2=f+g$
. Thus,
$f+g \in I(G)$
. By the assumption,
$Tf$
,
$Tg$
, and
$T(f+g) \in I_B(H)$
. Since we have
$(T(f+g))^2=Tf+Tg$
, we get
$Tf \cdot Tg=0$
.
Definition 2.2 We define
$\Phi : \mathop {A(G/G_e)} \to B(H)$
by
$$ \begin{align*} \Phi (f)= T \circ \varphi_G^{-1}(f) \end{align*} $$
for any
$f \in \mathop {A(G/G_e)}$
, where
$\varphi _G$
is given in (2.1).
Then
$\Phi $
is a bounded complex linear operator from
$\mathop {A(G/G_e)}$
into
$B(H)$
. In order to achieve our main result, we consider the dual map
$\Phi ^{*}: W^*(H) \to \mathrm {VN}(G/G_e)$
and have the following lemmas.
Lemma 2.3 Let
$\lambda \in \mathrm {VN}(\mathop {G/G_e})$
and
$a \in \mathop {G/G_e}$
. Suppose that
$a \in \mathrm {supp} \lambda $
. Then, for every neighborhood V of a in
$\mathop {G/G_e}$
, there exists
$h \in I(\mathop {G/G_e})$
such that
$\mathrm {supp} h \subset V$
and
$\langle \lambda , h \rangle \neq 0$
.
Proof Since
$\mathop {G/G_e}$
is totally disconnected, every neighborhood of the identity contains an open compact subgroup. As
$a^{-1}V$
is a neighborhood of the identity, there exists an open compact subgroup
$G_{a}$
in
$\mathop {G/G_e}$
such that
$G_{a} \subset a^{-1}V$
. Thus,
$aG_{a} \subset V$
. Since
$aG_{a}$
is a compact open coset in
$\mathop {G/G_e}$
, we have that
$1_{aG_{a}} \in \mathop {A(G/G_e)}$
is an idempotent with norm
$1$
. Since
$a \in \mathrm {supp} \lambda $
, there is
$g \in \mathop {A(G/G_e)}$
such that
$\mathrm {supp} g \subset aG_{a}$
and
$\langle \lambda , g \rangle \neq 0$
. Put
$\delta = |\langle \lambda , g \rangle |$
. As
$\varphi _G^{-1}(g) \in A_{I}(G)$
, there are
$\alpha _i \in \mathbb C$
and
$f_i \in I(G)$
such that
$\|\varphi _G^{-1}(g)-\sum ^{n}_{i=1}\alpha _if_i\| < \delta / \|\lambda \|$
for some
$n\in \mathbb N$
. Since
$\varphi _G$
is an isometric isomorphism, we have
$\|g-\sum ^{n}_{i=1}\alpha _i\varphi _G(f_i)\| < \delta / \|\lambda \| $
and
$\varphi _G(f_i) \in I(\mathop {G/G_e})$
. Then we obtain
$$ \begin{align*} 1_{aG_{a}}\left(g-\sum^{n}_{i=1}\alpha_i\varphi_G(f_i)\right)=g-\sum^{n}_{i=1}\alpha_i1_{aG_{a}}\varphi_G(f_i), \end{align*} $$
and thus
$$ \begin{align*} \left\|g-\sum^{n}_{i=1}\alpha_i1_{a G_{a}}\varphi_G(f_i)\right\| \le \left\|g-\sum^{n}_{i=1}\alpha_i\varphi_G(f_i)\right\| <\frac{\delta}{\|\lambda\|}. \end{align*} $$
Suppose that, for every
$1 \le i \le n$
, we have
$\langle \lambda , 1_{aG_{a}}\varphi _G(f_i) \rangle = 0$
. Then
$$ \begin{align*} \begin{aligned} |\langle \lambda, g \rangle| &= \left|\langle \lambda, g \rangle - \sum^{n}_{i=1}\alpha_i \langle \lambda, 1_{aG_{a}}\varphi_G(f_i) \rangle \right| \\ &= \left|\left\langle \lambda, \left(g-\sum^{n}_{i=1}\alpha_i 1_{aG_{a}}\varphi_G(f_i) \right)\right\rangle \right| \\ & \le \|\lambda\| \left\|g-\sum^{n}_{i=1}\alpha_i 1_{aG_{a}}\varphi_G(f_i) \right\| \\ &< \|\lambda\| \frac{\delta}{\|\lambda\|}=\delta. \end{aligned} \end{align*} $$
This implies that
$|\langle \lambda , g \rangle |< \delta $
, which is a contradiction. Therefore, there is an
$i_0 \in \{1, \ldots , n \}$
such that
$$ \begin{align*} \langle \lambda, 1_{aG_{a}}\varphi_G(f_{i_0}) \rangle \neq 0. \end{align*} $$
We also have
$\mathrm {supp}(1_{aG_{a}}\varphi _G(f_{i_0})) \subset V$
and
$1_{aG_{a}}\varphi _G(f_{i_0}) \in I(\mathop {G/G_e})$
, and the proof is thus completed.
Proposition 2.4 For any
$a \in H$
, there exist unique
$b \in G/G_e$
and
$\alpha \in \mathbb C$
such that
$\Phi ^{*}(\omega _{H}(a))=\alpha \lambda _{\mathop {G/G_e}}(b)$
.
Proof Suppose that there are
$b_1, b_2 \in \mathop {G/G_e}$
such that
$b_1, b_2$
were both in
$\mathrm {supp}(\Phi ^{*}(\omega _{H}(a)))$
. Since
$G_e$
is a closed subgroup of G, the quotient group
$\mathop {G/G_e}$
is Hausdorff. Thus, there are neighborhoods
$V_{b_1}$
and
$V_{b_2}$
of
$b_1$
and
$b_2$
, respectively, in
$\mathop {G/G_e}$
such that
$V_{b_1} \cap V_{b_2}= \emptyset $
. By Lemma 2.3, there are
$h_i \in I(\mathop {G/G_e})$
, for
$i=1,2$
, such that
$\mathrm {supp} h_i \subset V_{b_i}$
and
$\langle \Phi ^{*}(\omega _{H}(a)), h_i \rangle \neq 0$
. As
$V_{b_1} \cap V_{b_2}= \emptyset $
, we get
$h_1h_2=0$
. Since
$\varphi _G$
is an isomorphism, we have
$\varphi _G^{-1}(h_i) \in I(G)$
, for
$i= 1, 2$
, and
$\varphi _G^{-1}(h_1)\cdot \varphi _G^{-1}(h_2)=\varphi _G^{-1}(h_1 h_2)=0$
. By Lemma 2.2, we have
$T(\varphi _G^{-1}(h_1))\cdot T(\varphi _G^{-1}(h_2)) = 0$
. On the other hand, we obtain
$$ \begin{align*} 0 \neq \langle \Phi^{*}(\omega_{H}(a)), h_1 \rangle=\Phi(h_1)(a)= T \circ \varphi_G^{-1}(h_1)(a)= T(\varphi_G^{-1}(h_1))(a) \end{align*} $$
and
$$ \begin{align*} 0 \neq \langle \Phi^{*}(\omega_{H}(a)), h_2 \rangle=\Phi(h_2)(a)=T \circ \varphi_G^{-1}(h_2)(a)= T(\varphi_G^{-1}(h_2))(a). \end{align*} $$
Therefore,
$$ \begin{align*}T(\varphi_G^{-1}(h_1))\cdot T(\varphi_G^{-1}(h_2))\neq 0;\end{align*} $$
this is a contradiction. Since
$\mathrm {supp} (\Phi ^{*}(\omega _{H}(a))) \neq \emptyset $
, there is a unique
$b \in \mathop {G/G_e}$
such that
$\mathrm {supp} (\Phi ^{*}(\omega _{H}(a)))=\{b\}$
. Consequently, by [Reference Kaniuth and Lau19, Corollary 2.5.9], there is an
$\alpha \in \mathbb C$
such that
$\Phi ^{*}(\omega _{H}(a))=\alpha \lambda _{\mathop {G/G_e}}(b)$
.
For any
$a \in H$
, by Proposition 2.4, there are unique
$b \in G/G_e$
and
$\alpha \in \mathbb C$
such that
$\Phi ^{*}(\omega _{H}(a))=\alpha \lambda _{\mathop {G/G_e}}(b)$
; thus, we have
$$ \begin{align*} \Phi(f)(a)=\alpha f(b), \end{align*} $$
for any
$f \in \mathop {A(G/G_e)}$
. We define
$\phi : H \to \mathbb C$
by
$\alpha =\phi (a)$
. We also define
$\psi : H \to \mathop {G/G_e}$
by
$b=\psi (a)$
. Then we get
$$ \begin{align} \Phi(f)(a)=\phi(a)f(\psi(a)), \end{align} $$
for any
$f \in \mathop {A(G/G_e)}$
and
$a \in H$
.
For any
$h \in I(G)$
, since we have
$\Phi (\varphi _G(h))=T(h)\in I_B(H)$
, we obtain that
$$ \begin{align*} \begin{aligned} (\Phi(\varphi_G(h)))^2&=T(h)T(h)=T(h)=\Phi(\varphi_G(h)).\\ \end{aligned} \end{align*} $$
On the other hand, since
$(\varphi _G(h))^2=\varphi _G(h^2)=\varphi _G(h)$
in
$A(\mathop {G/G_e})$
, we obtain that
$$ \begin{align} \phi(a)^2\varphi_G(h)(\psi(a))=\phi(a)\varphi_G(h)(\psi(a)) \end{align} $$
for any
$h \in I(G)$
and
$a \in H$
.
Lemma 2.5 For any
$a \in H$
, there is an idempotent
$1_{\psi (a)G_0}$
of
$A(\mathop {G/G_e})$
where
$\psi (a)G_0$
is an open compact neighborhood of
$\psi (a)$
.
Proof As
$\mathop {G/G_e}$
is totally disconnected, there is an open compact subgroup
$G_0$
in
$\mathop {G/G_e}$
. For any
$\psi (a) \in \mathop {G/G_e}$
,
$\psi (a)G_0$
is a compact open coset in
$\mathop {G/G_e}$
; hence,
$1_{\psi (a)G_0}$
is an idempotent of
$A(\mathop {G/G_e})$
with norm
$1$
.
Lemma 2.6 The map
$\Phi : A(\mathop {G/G_e}) \to B(H)$
is an algebraic homomorphism.
Proof Let
$a \in H$
. By Lemma 2.5, there is an idempotent
$1_{\psi (a)G_0}$
of
$A(\mathop {G/G_e})$
. Since
$\varphi _G: A_I(G) \to A(\mathop {G/G_e})$
is surjective, there is
$f \in A_I(G)$
such that
$\varphi _G(f)=1_{\psi (a)G_0}$
. Moreover, we have that
$f^2=(\varphi _G^{-1}(1_{\psi (a)G_0}))^2=\varphi _G^{-1}(1_{\psi (a)G_0})=f$
, and this implies that
$f \in I(G)$
. Thus, by (2.4), we have
$$ \begin{align*} \phi(a)^2&=\phi(a)^2 1_{\psi(a)G_0}(\psi(a))=\phi(a)^2\varphi_G(f)(\psi(a))\\&=\phi(a)\varphi_G(f)(\psi(a))=\phi(a)1_{\psi(a)G_0}(\psi(a))=\phi(a). \end{align*} $$
Since
$a \in H$
is arbitrary, we have
$$ \begin{align} \phi^2=\phi \end{align} $$
on H. Then we get
$\phi : H \to \{0,1\}$
. In addition, for any
$f, g \in A(\mathop {G/G_e})$
and
$a \in H$
, we have
$$ \begin{align*} \Phi(fg)(a)&=\phi(a)(fg)(\psi(a))=\phi(a)^2(fg)(\psi(a))\\ &=\phi(a)f(\psi(a))\phi(a)g(\psi(a))=(\Phi(f)\Phi(g))(a). \end{align*} $$
Hence,
$\Phi $
is an algebraic homomorphism from
$A(\mathop {G/G_e})$
into
$B(H)$
.
Lemma 2.7 The map
$\psi :\phi ^{-1}(1) \to \mathop {G/G_e}$
is continuous.
Proof For any
$a_0 \in \phi ^{-1}(1) \subset H$
, let U be an open neighborhood of
$\psi (a_0)$
in
$\mathop {G/G_e}$
. Then there is
$f_0 \in \mathop {A(G/G_e)}$
such that
$$ \begin{align*} f_0(\psi(a_0))=1 \quad \text{and} \quad f_0(b)=0 \, \, \text{ for } b \in (\mathop{G/G_e}) \backslash U. \end{align*} $$
Let
$(a_\lambda )_\lambda \subseteq \phi ^{-1}(1)$
be a net such that
$a_{\lambda } \to a_0$
. As
$\Phi (f_0) \in B(H)$
,
$\Phi f_0(a_{\lambda }) \to \Phi f_0(a_{0})=f_0(\psi (a_0))=1$
. There is a
$\lambda _0 $
such that if
$\lambda \ge \lambda _0$
, then
$|\Phi f_0(a_{\lambda })|>\frac {1}{2}$
. Since
$\Phi f_0(a_{\lambda })= f_0(\psi (a_{\lambda }))$
, we have
$\psi (a_{\lambda }) \in U$
provided
$\lambda \ge \lambda _0$
. Thus,
$\psi $
is continuous on
$\phi ^{-1}(1)$
.
Lemma 2.8 The set
$\phi ^{-1}(1)$
is an open subset of H.
Proof Let
$a \in \phi ^{-1}(1)$
be arbitrary. By Lemma 2.5, there is an idempotent
$1_{\psi (a)G_0}$
of
$A(\mathop {G/G_e})$
where
$\psi (a)G_0$
is an open compact neighborhood of
$\psi (a)$
. Since
$\Phi (1_{\psi (a)G_0}) \in B(H) \subset C_b(H)$
, there exists an open neighborhood V of a in H such that if
$b \in V$
, then
$$ \begin{align*} |\Phi(1_{\psi(a)G_0})(a)-\Phi(1_{\psi(a)G_0})(b)|\le \frac{1}{2}. \end{align*} $$
We have
$$ \begin{align*} |1-\phi(b)1_{\psi(a)G_0}(\psi(b))|=|\Phi(1_{\psi(a)G_0})(a)-\Phi(1_{\psi(a)G_0})(b)| \le \frac{1}{2}. \end{align*} $$
Since either
$\phi (b)1_{\psi (a)G_0}(\psi (b))= 1$
or
$\phi (b)1_{\psi (a)G_0}(\psi (b))= 0$
, this implies that
$$ \begin{align*} \phi(b)1_{\psi(a)G_0}(\psi(b))=1. \end{align*} $$
Hence, we have
$\phi (b)=1$
for any
$b \in V$
. Thus,
$V \subset \phi ^{-1}(1)$
. It follows that
$\phi ^{-1}(1)$
is an open subset of H.
Theorem 2.9 Let G and H be two locally compact groups, and let
$T:A(G) \to B(H)$
be a bounded complex linear operator. Suppose T satisfies that
$T(I(G)) \subset I_B(H)$
. Then there are an open subset U of H and a continuous map
$\psi $
from U into
$\mathop {G/G_e}$
such that
$$ \begin{align} Tf(a)= \begin{cases} f([\psi(a)]_{G_e}), \quad & \text{if }a \in U, \\ 0, & \text{if } a \in H \backslash U, \end{cases} \end{align} $$
for any
$f \in A_I(G)$
and
$a \in H$
.
Proof Let
$U=\phi ^{-1}(1)$
. By Lemma 2.8, U is an open subset of H. Moreover, Lemma 2.7 shows that
$\psi : U \to \mathop {G/G_e}$
is a continuous map. Applying (2.3), for any
$f \in A_I(G)$
and
$a\in H$
, we have
$$ \begin{align*} Tf(a)=\Phi(\varphi_G(f))(a)=\phi(a)\varphi_G(f)(\psi(a)). \end{align*} $$
Thus, we get
$$ \begin{align*} \begin{aligned} Tf(a)&= \begin{cases} \varphi_G(f)(\psi(a)), \quad & \text{if }a \in U, \\ 0, & \text{if } a \in H \backslash U, \end{cases}\\ &= \begin{cases} f([\psi(a)]_{G_e}), \quad & \text{if }a \in U, \\ 0, & \text{if } a \in H \backslash U, \end{cases}\\ \end{aligned} \end{align*} $$
for any
$f \in A_I(G)$
and any
$a \in H$
.
The following example shows that the assumption in Theorem 2.9 does not imply
$T(I(G))\subset I(H)$
. This observation is in line with the well-known fact that
$f\circ \psi $
may not be in the Fourier algebra
$A(H)$
in general (see Remark 2.11).
Example 2.10 Let
$G=\{0\}$
be the trivial group. Then we define a bounded linear operator
$T:A(G) \to B(\mathbb {Z})$
by
$T(1_{G})=1_{\mathbb {Z}}$
. Then it satisfies
$T(I(G)) \subset I_B(H)$
. Note that in this case, we have
$U=\mathbb {Z}$
and the continuous map
$\psi : \mathbb {Z} \to \mathop {G/G_e}$
is
$\psi (n)=0$
for any
$n \in \mathbb {Z}$
. On the other hand, since
$1_{\mathbb {Z}} \notin A(\mathbb {Z})$
, we have
$T(I(G))\nsubseteq I(H)$
.
Remark 2.11 The converse statement of the above theorem may not hold since we do not know if
$Tf \in A(H)$
for any
$f \in A_I(G)$
, even T has a representation of the form (2.6). If we only have
$\psi : U \subseteq H \to G$
being continuous, then
$f\mapsto f\circ \psi $
maps
$A(G)$
into
$\ell ^{\infty }(H)$
in general. For abelian groups G and H, Cohen [Reference Cohen6] showed that
$f\mapsto f\circ \psi $
maps
$A(G)$
into
$B(H)$
if and only if
$\psi $
is a continuous piecewise affine map from a set in the open coset ring of H into G. This characterization was extended by Host [Reference Host14] to the case when G has an abelian subgroup of finite index and H is arbitrary, and by [Reference Le Pham20] to general groups.
Under the additional assumptions such as positivity or contractivity on T, we obtain algebraic structures for the open set U and algebraic properties on the map
$\psi $
. Let us first recall positive operators on the Fourier algebra.
A bounded linear operator
$T:A_I(G) \to B(H)$
is said to be positive if
$T(u)$
is positive-definite whenever
$u \in A_I(G)$
is a positive-definite function.
Corollary 2.12 Let G and H be two locally compact groups. Let
$T: A_I(G) \to B(H)$
be a positive bounded complex linear operator. If T satisfies that
$T(I(G)) \subset I_B(H)$
, then there exist an open subgroup U of H and a continuous group homomorphism or antihomomorphism
$\psi $
from the open subgroup U of H into
$\mathop {G/G_e}$
such that
$$ \begin{align*} Tf(a)= \begin{cases} f([\psi(a)]_{G_e}), \quad & \text{if }a \in U,\\ 0, & \text{if } a \in H \backslash U, \end{cases} \end{align*} $$
for any
$f \in A_I(G)$
and
$a \in G$
.
Proof Since the isometric isomorphism
$\varphi _G$
preserves positivity,
$u \in A_I(G)$
is a positive-definite function if and only if
$\varphi _G(u)$
is positive-definite. This implies that T is positive if and only if
$\Phi $
is positive; thus,
$\Phi :A(\mathop {G/G_e}) \to B(H)$
is a positive homomorphism by Lemma 2.6. It follows from [Reference Le Pham20, Theorem 4.3] that there exist an open subgroup U of H and a continuous group homomorphism or antihomomorphism
$\psi $
from U into
$\mathop {G/G_e}$
such that for any
$f \in A(\mathop {G/G_e})$
,
$\Phi f$
is either equal to
$f\circ \psi $
in U, or
$0$
otherwise. Thus, we have
$$ \begin{align*} Tf(a)= \begin{cases} f([\psi(a)]_{G_e}), \quad & \text{if }a \in U,\\ 0, & \text{if } a \in H \backslash U, \end{cases} \end{align*} $$
for any
$f \in A_I(G)$
and
$a \in H$
.
Corollary 2.13 Let G and H be two locally compact groups, and let
$T: A_{I}(G) \to B(H)$
be a contractive complex linear operator. If T satisfies that
$T(I(G)) \subset I_B(H)$
, then there exist an open subgroup U of H, a continuous group homomorphism or antihomomorphism
$\psi $
from U into
$\mathop {G/G_e}$
, and elements
$b \in G$
and
$c \in H$
such that
$$ \begin{align*} Tf(a)= \begin{cases} f(b[\psi(ca)]_{G_e}), \quad & \text{if }a \in c^{-1}U,\\ 0, & \text{if }a \in H \backslash c^{-1}U. \end{cases} \end{align*} $$
Proof Since
$\varphi _G$
is an isometric isomorphism,
$\Phi $
is a contractive operator provided that so is T. By Lemma 2.6,
$\Phi $
is a contractive homomorphism from
$A(\mathop {G/G_e})$
into
$B(H)$
. It follows from [Reference Le Pham20, Theorem 5.1] that there exist an open subgroup U of H, a continuous group homomorphism or antihomomorphism
$\psi $
from U into
$\mathop {G/G_e}$
, and elements
$bG_e \in \mathop {G/G_e}$
and
$c \in H$
such that for any
$f \in A(\mathop {G/G_e})$
and
$a \in H$
,
$\Phi f(a)= f(bG_e\psi (ca))$
provided
$a \in c^{-1}U$
; otherwise,
$\Phi f(a)= 0$
. Recalling the definition of
$\Phi $
, we have the characterization of T.
Note that when the group G is totally disconnected, we have
$A_I(G)=A(G)$
. In such case, positive or contractive complex linear idempotent preserving operators from
$A(G)$
to
$B(H)$
are algebraic homomorphisms; thus, our results recover Theorems 4.3 and 5.1 in [Reference Le Pham20].
2.2 Idempotent preserving maps with
$T(I(G)) \subset I(H)$
Let us assume that the bounded linear operator
$T:A(G) \to B(H)$
satisfies
$T(I(G)) \subset I(H)$
. Then, naturally, we obtain
$T(A_I(G)) \subseteq A_I(H)$
.
We define
$T_q: \mathop {A(G/G_e)} \to \mathop {A(H/H_e)}$
by
$$ \begin{align*} T_q(f)=\varphi_H \circ T \circ \varphi_G^{-1}(f)=\varphi_H \circ \Phi(f), \end{align*} $$
for any
$f \in \mathop {A(G/G_e)}$
, where
$\varphi _H: A_I(H) \to \mathop {A(H/H_e)}$
is an isometric isomorphism defined similarly as in (2.1). Note that
$T_q$
is an algebraic homomorphism.
Lemma 2.14 Let
$a \in \phi ^{-1}(1) \subset H$
and
$b\in H$
such that
$a^{-1}b \in H_e$
. Then
$\phi (b)=1$
and
$\psi (a)=\psi (b)$
.
Proof Suppose that
$\psi (a)\neq \psi (b)$
. By (2.3), we have
$\Phi : \mathop {A(G/G_e)} \to B(H)$
such that for any
$f \in \mathop {A(G/G_e)}$
,
$$ \begin{align*} \Phi(f)(a)=f(\psi(a)) \end{align*} $$
and
$$ \begin{align*} \Phi(f)(b)=\phi(b)f(\psi(b)). \end{align*} $$
Since
$\mathop {G/G_e}$
is Hausdorff, there are disjoint open neighborhoods
$V_a$
and
$V_b$
of
$\psi (a)$
and
$\psi (b)$
, respectively, in
$\mathop {G/G_e}$
. By Lemma 2.3, for
$\lambda _{\mathop {G/G_e}}(\psi (a)) \in VN(\mathop {G/G_e})$
, there is
$h \in I(\mathop {G/G_e})$
such that
$\mathrm {supp}\ h \subset V_a$
and
$h(\psi (a))\neq 0$
. Since
$a \in \phi ^{-1}(1)$
, we get
$$ \begin{align*} \Phi(h)(a)=h(\psi(a))\neq 0 \end{align*} $$
and
$$ \begin{align*} \Phi(h)(b)=\phi(b)h(\psi(b))=0. \end{align*} $$
By the assumption that
$T(I(G)) \subset I(H)$
and
$\varphi _G^{-1}(h)\in I(G)$
, we have
$\Phi (h)=T(\varphi _G^{-1}(h)) \in I(H)$
, an idempotent in
$A(H)$
. Hence, there is
$Y \in \Omega _{\text {o}}^{\text {c}}(H)$
such that
$1_Y=\Phi (h)$
. Since
$1_Y(a)=\Phi (h)(a)=h(\psi (a))\neq 0$
, we have
$a \in Y$
. In addition, Y is a clopen subset of H and
$H_e$
is a connected component containing e; thus,
$aH_e \subset Y$
. This implies that
$b=aa^{-1}b \in Y$
. It follows that
$$ \begin{align*} 1=1_Y(b)=\Phi(h)(b)=0. \end{align*} $$
This is a contradiction. Thus, we have
$\psi (a)=\psi (b)$
. Furthermore, suppose that
$\phi (b)=0$
. There is an
$h \in I(\mathop {G/G_e})$
such that
$h(\psi (b))\neq 0$
. Thus, there is
$Y \in \Omega _{\text {o}}^{\text {c}}(H)$
such that
$1_Y=\Phi (h)$
. By a similar argument, we have
$1_Y(a)=\Phi (h)(a)=h(\psi (a))\neq 0$
,
$a \in Y$
and
$b \in Y$
. We obtain that
$$ \begin{align*} 1=1_Y(b)=\Phi(h)(b)=\phi(b)h(\psi(b))=0. \end{align*} $$
This is a contradiction. Therefore,
$\phi (b)=1$
and
$\psi (a)= \psi (b)$
.
For any
$a,b \in H$
, the condition
$a^{-1}b \in H_e$
induces an equivalence relation on H. Lemma 2.14 shows that
$\phi : H \to \{0,1\}$
and
$\psi :H \to \mathop {G/G_e}$
are constant functions on each equivalence class. Thus, these induce maps
$\phi ':\mathop {H/H_e} \to \{0,1\}$
and
$\psi ': \phi ^{{\prime }-1}(1) \to \mathop {G/G_e}$
by
$$ \begin{align*} \phi'(aH_e)=\phi(a) \quad \text{for any }a \in H \end{align*} $$
and
$$ \begin{align*} \psi'(aH_e)=\psi(a) \quad \text{for any }aH_e \in \phi^{{\prime}-1}(1). \end{align*} $$
By Lemma 2.7, the map
$\psi :\phi ^{-1}(1) \to \mathop {G/G_e}$
is continuous. As we have
$\phi ^{{\prime }-1}(1)=q_{H}(\phi ^{-1}(1))$
, we obtain that
$\psi ': \phi ^{{\prime }-1}(1) \to \mathop {G/G_e}$
is continuous.
Theorem 2.15 Let G and H be two locally compact groups, and let
$T: A(G) \to B(H)$
be a bounded complex linear operator. Suppose that T satisfies
$T(I(G)) \subset I(H)$
. Then there exist an open subset U of H and a continuous map
$\psi '$
from an open subset
$q_{H}(U)$
of
$\mathop {H/H_e}$
into
$\mathop {G/G_e}$
such that
$$ \begin{align} Tf(a)= \begin{cases} f([\psi'(aH_e)]_{G_e}), \quad & \text{if }a \in U, \\ 0, & \text{if } a \in H \backslash U, \end{cases} \end{align} $$
for any
$f \in A_I(G)$
.
Proof Define
$U=\phi ^{-1}(1)$
. Recall that
$q_{H}: H \to \mathop {H/H_e}$
is the quotient map and U is an open subset of H by Lemma 2.8. By (2.3), for any
$f \in \mathop {A(G/G_e)}$
and
$a \in H$
, we have
$$ \begin{align} T_q(f)(aH_e)=\varphi_H\circ \Phi(f)(aH_e)=\Phi(f)(a) =\phi'(aH_e)f(\psi(a)). \end{align} $$
We shall show that
$\phi ^{{\prime }-1}(1)$
is an open subset of
$\mathop {H/H_e}$
. Let
$a \in \phi ^{{\prime }-1}(1)$
. By Lemma 2.5, there is an idempotent
$1_{\psi '(a)G_0}$
of
$A(\mathop {G/G_e})$
where
$\psi '(a)G_0$
is an open compact neighborhood of
$\psi '(a)$
. Since
$T_q(1_{\psi '(a)G_0}) \in A(\mathop {H/H_e}) \subset C_0(\mathop {H/H_e})$
, the space of all continuous functions on
$\mathop {H/H_e}$
vanishing at infinity, there exists an open neighborhood V of a in
$\mathop {H/H_e}$
such that if
$b \in q_H^{-1}(V)$
, then
$$ \begin{align*} |T_q(1_{\psi'(a)G_0})(a)- T_q(1_{\psi'(a)G_0})(bH_e)|\le \frac{1}{2}. \end{align*} $$
We have
$$ \begin{align*} |1-\phi'(bH_e)1_{\psi'(a)G_0}(\psi(b))|= |T_q(1_{\psi'(a)G_0})(a)- T_q(1_{\psi'(a)G_0})(bH_e)| \le \frac{1}{2}. \end{align*} $$
Since either
$\phi '(bH_e)1_{\psi '(a)G_0}(\psi (b))= 1$
or
$\phi '(bH_e)1_{\psi '(a)G_0}(\psi (b))= 0$
, this implies that
$$ \begin{align*} \phi'(bH_e)1_{\psi'(a)G_0}(\psi(b))=1. \end{align*} $$
Hence, we have
$\phi '(bH_e)=1$
for any
$b \in q_H^{-1}(V)$
. Thus,
$V \subset \phi ^{{\prime }-1}(1)$
. It follows that
$\phi ^{{\prime }-1}(1)$
is an open subset of
$\mathop {H/H_e}$
. Let us recall that
$\psi ':q_{H}(U) \to \mathop {G/G_e} $
is a continuous map. Applying (2.8), we have
$$ \begin{align*} \begin{aligned} T_q(f)(aH_e)&=\phi'(aH_e)f(\psi(a))\\ &= \begin{cases} f(\psi'(aH_e)), \quad & \text{if }a \in U, \\ 0, & \text{if } a \in H \backslash U, \end{cases}\\ \end{aligned} \end{align*} $$
for any
$f \in A(\mathop {G/G_e})$
and
$a \in H$
. As we have
$$ \begin{align*} T_q(\varphi_G(f))(aH_e)=\varphi_H\circ T\circ \varphi_G^{-1}(\varphi_G(f))(aH_e)=\varphi_H \circ T(f)(aH_e)=Tf(a) \end{align*} $$
for any
$f \in A_I(G)$
and
$a \in H$
, we get
$$ \begin{align*} \begin{aligned} Tf(a)&= \begin{cases} \varphi_G(f)(\psi'(aH_e)), \quad & \text{if }a \in U, \\ 0, & \text{if }a \in H \backslash U. \end{cases}\\ \end{aligned}\\[-34pt] \end{align*} $$
3 Idempotent preserving bijections on
$A_I(G)$
In this section, we assume that the bounded linear operator
$T: A(G) \to B(H)$
satisfies that
$T(I(G)) \subset I(H)$
and
$T|_{A_I(G)}$
is a bijection onto
$A_I(H)$
.
Theorem 3.1 Let G and H be two locally compact groups, and let
$T:A(G) \to B(H)$
be a bounded complex linear operator. Suppose that the operator T satisfies that
$T(I(G)) \subset I(H)$
and
$T|_{A_I(G)}:A_I(G) \to A_I(H)$
is bijective. Then there exists a homeomorphism
$\psi : \mathop {H/H_e} \to \mathop {G/G_e}$
such that
$$ \begin{align*} Tf(a)=f([\psi(aH_e)]_{G_e}) \end{align*} $$
for all
$f \in A_{I}(G)$
and
$a \in H$
.
Proof Since
$T|_{A_I(G)}$
is a bijective linear map, and
$\varphi _G$
and
$\varphi _H$
are isometric isomorphisms, by the proof of Proposition 2.1, we have
$T_q:= \varphi _H \circ T|_{A_I(G)} \circ \varphi _G^{-1}$
is an isomorphism from
$A(\mathop {G/G_e})$
onto
$A(\mathop {H/H_e})$
.
Applying Theorem 2.15, there are an open subset U of H and a continuous map
$\psi $
from an open subset
$q_{H}(U)$
of
$\mathop {H/H_e}$
into
$\mathop {G/G_e}$
such that
$$ \begin{align*} T_q(f)(a)= \begin{cases} f(\psi(a)), \quad & \text{if }a \in q_{H}(U), \\ 0, & \text{if } a \in (\mathop{H/H_e}) \backslash q_{H}(U), \end{cases} \end{align*} $$
for any
$f \in A(\mathop {G/G_e})$
. Since
$T_q:A(\mathop {G/G_e}) \to A(\mathop {H/H_e})$
is surjective and the Fourier algebra
$A(\mathop {H/H_e})$
separates the points in
$\mathop {H/H_e}$
, we have
$q_{H}(U)=\mathop {H/H_e}$
. Thus,
$U=H$
and we have
$$ \begin{align*} T_q(f)(a)=f(\psi(a)) \end{align*} $$
for every
$f \in A(\mathop {G/G_e})$
and
$a \in \mathop {H/H_e}$
. For any
$h \in I(H)$
, there exists
$h_q \in A(\mathop {H/H_e})$
with
$h_q^2= h_q$
such that
$$ \begin{align*} \varphi_H(h)= h_q. \end{align*} $$
Since
$T_q$
is bijective, there exists
$f_q \in A(\mathop {G/G_e})$
such that
$$ \begin{align*} T_q(f_q)= h_q. \end{align*} $$
Moreover, since
$T_q$
is an algebraic homomorphism, we have
$T_q(f_q^2)=(T_q(f_q))^2= h_q^{2}= h_q= T_q(f_q)$
. By the injectivity of
$T_q$
, we get
$f_q^2= f_q$
. On the other hand, as
$\varphi _G$
is an isometric isomorphism from
$A_I(G)$
onto
$A(\mathop {G/G_e})$
, there exists
$f \in I(G)$
such that
$$ \begin{align*} \varphi_G(f)= f_q. \end{align*} $$
Hence, we have
$$ \begin{align*} T(f)=(\varphi_H^{-1}\circ T_q \circ \varphi_G)(f)=\varphi_H^{-1}\circ T_q(f_q)= \varphi_H^{-1}(h_q)= h. \end{align*} $$
This implies that
$T(I(G))= I(H)$
. In particular, we have
$T^{-1}(I(H)) \subset I(G)$
. Thus, we can apply similar arguments to
$T|_{A_I(G)}^{-1}: A_I(H) \to A_I(G)$
and to
$T_q^{-1}=\varphi _G \circ T|_{A_I(G)}^{-1} \circ \varphi _H^{-1}$
on
$A(\mathop {H/H_e})$
, and we can then define a continuous map
$\tilde {\psi }: \mathop {G/G_e} \to \mathop {H/H_e}$
such that
$$ \begin{align*} T_q^{-1}(g)(b)=g(\tilde{\psi}(b)) \end{align*} $$
for any
$g \in \mathop {A(H/H_e)}$
and
$b \in \mathop {G/G_e}$
.
For any
$g\in \mathop {A(H/H_e)}$
and
$a \in \mathop {H/H_e}$
, we have
$$ \begin{align*} g(a)=T_q(T_q^{-1}g)(a)=g(\tilde{\psi}(\psi(a))). \end{align*} $$
Since the Fourier algebra
$\mathop {A(H/H_e)}$
separates points in
$\mathop {H/H_e}$
, we get
$$ \begin{align} a=\tilde{\psi}(\psi(a)) \quad \text{for} \quad a \in \mathop{H/H_e}. \end{align} $$
Moreover, we obtain
$$ \begin{align*} f(b)=T_q^{-1}(T_q f)(b)=f(\psi(\tilde{\psi}(b))), \end{align*} $$
for any
$f \in \mathop {A(G/G_e)}$
and
$b \in \mathop {G/G_e}$
. Similarly, as
$\mathop {A(G/G_e)}$
separates points in
$\mathop {G/G_e}$
, we have
$$ \begin{align} b=\psi(\tilde{\psi}(b)) \quad \text{for} \quad b \in \mathop{G/G_e}. \end{align} $$
By (3.1) and (3.2), we have that
$\psi :\mathop {H/H_e} \to \mathop {G/G_e}$
is a bijection and
$\tilde {\psi }=\psi ^{-1}$
. Let us recall that
$\psi $
and
$\tilde {\psi }$
are continuous on
$\mathop {H/H_e}$
and
$\mathop {G/G_e}$
, respectively. As
$\tilde {\psi }=\psi ^{-1}$
, we have that
$\psi $
is a homeomorphism. In addition, we obtain
$$ \begin{align*} T_q(f)(a)=f(\psi(a)) \quad \text{for }\, f \in \mathop{A(G/G_e)}, \, a \in \mathop{H/H_e}\kern-1pt. \end{align*} $$
Since
$T=\varphi _H^{-1} \circ T_q \circ \varphi _G$
, we get
$$ \begin{align*} Tf(a)=f([\psi(aH_e)]_{G_e}) \end{align*} $$
for all
$f \in A_{I}(G)$
and
$a \in H$
.
Note that the bijectivity in Theorem 3.1 is an essential assumption for the function
$\psi : \mathop {H/H_e} \to \mathop {G/G_e}$
to be a homeomorphism.
Example 3.2 Let
$G=\{1,2\}$
be a multiplicative group equipped with the discrete topology. Let
$H=\{0\}$
be the trivial group. We define
$T: A(G) \to A(H)$
by
$Tf(0)=f(1)$
for any
$f \in A(G)$
. Then T is a bounded complex linear operator on
$A(G)$
and for any
$1_Y \in A(G)$
,
$T(1_Y)=1_H$
if
$1 \in Y$
; otherwise,
$T(1_Y)=0$
. Thus,
$T(I(G))=I(H)$
. On the other hand,
$T(1_{\{1\}})=1_H=T(1_G)$
, this implies that
$T|_{A_I(G)}:A_I(G) \to A_I(H)$
is not injective. In addition,
$\psi : \mathop {H/H_e}= H \to \mathop {G/G_e}= G$
satisfying
$$ \begin{align*} \psi(0)=1. \end{align*} $$
is not a homeomorphism.
Under additional assumptions such as positivity or contractivity on T, a characterization of linear idempotent preserving maps between two Fourier algebras follows from Corollaries 2.12 and 2.13. Note that since the continuous map
$\psi $
in the following two corollaries is either a group isomorphism or an anti-isomorphism, we naturally have
$f\circ \psi \in A_I(H)$
for any
$f \in A_I(G)$
(see [Reference Walter25]); thus, we obtain a necessary and sufficient condition for the idempotent preserving operator T on
$A_I(G)$
.
Corollary 3.3 Let G and H be two locally compact groups. A surjective complex linear contraction
$T: A_I(G) \to A_I(H)$
satisfies
$T(I(G)) \subset I(H)$
if and only if there exist a continuous group isomorphism or anti-isomorphism
$\psi : \mathop {H/H_e} \to \mathop {G/G_e}$
and an element
$b \in G$
such that
$$ \begin{align*} Tf(a)=f(b[\psi(aH_e)]_{G_e}) \end{align*} $$
for all
$f \in A_{I}(G)$
and
$a \in H$
.
Proof Since the operator T defined on
$A_I(G)$
is contractive and satisfies
$T(I(G))\subset I(H)$
, we have a characterization of T by Corollary 2.13, and now the result follows from Theorem 3.1 as T is onto
$A_I(H)$
.
Similarly, if the operator
$T: A_I(G) \to A_I(H)$
is positive and preserves the idempotents, we have a characterization of T by Corollary 2.12 and thus the following corollary follows from Theorem 3.1.
Corollary 3.4 Let G and H be two locally compact groups. A positive bounded complex linear bijection
$T: A_I(G) \to A_I(H)$
satisfies
$T(I(G)) \subset I(H)$
if and only if there exists a continuous group isomorphism or anti-isomorphism
$\psi : \mathop {H/H_e} \to \mathop {G/G_e}$
such that
$$ \begin{align*} Tf(a)=f([\psi(aH_e)]_{G_e}) \end{align*} $$
for all
$f \in A_{I}(G)$
and
$a \in H$
.
We will end our article with a special case when the groups are totally disconnected. In such case, the idempotent preserving operators between Fourier algebras recover the results of algebraic homomorphisms. More precisely, Theorem 2.15 and Corollaries 3.3 and 3.4 are followed by the following remark.
Remark 3.5 Suppose that G and H are totally disconnected locally compact groups. Let
$T: A(G) \to A(H)$
be a bounded complex linear operator satisfying
$T(I(G))\subset I(H)$
. Then there exists a continuous map
$\psi $
from an open subset U of H into G such that
$$ \begin{align*} Tf(a)= \begin{cases} f(\psi(a)), \quad & \text{if }a \in U, \\ 0, & \text{if }a \in H \backslash U, \end{cases} \end{align*} $$
for any
$f \in A(G)$
and
$a \in H$
. In addition, if T is a surjective contraction or T is a positive bijection, then it is equivalent to
$Tf = f \circ (b \psi )$
for some
$b \in G$
or
$Tf= f \circ \psi $
, respectively, where
$\psi : H\to G$
is a continuous group isomorphism or group anti-isomorphism, and
$b\psi : H\to G$
is defined by
$b\psi (\cdot ):= b \psi (\cdot )$
; in particular, T is an algebraic homomorphism.
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