Completely Compact Herz–Schur Multipliers of Dynamical Systems

Abstract

We prove that if G is a discrete group and $(A,G,\alpha )$ is a C*-dynamical system such that the reduced crossed product $A\rtimes _{r,\alpha } G$ possesses property (SOAP) then every completely compact Herz–Schur $(A,G,\alpha )$-multiplier can be approximated in the completely bounded norm by Herz–Schur $(A,G,\alpha )$-multipliers of finite rank. As a consequence, if G has the approximation property (AP) then the completely compact Herz–Schur multipliers of A(G) coincide with the closure of A(G) in the completely bounded multiplier norm. We study the class of invariant completely compact Herz–Schur multipliers of $A\rtimes _{r,\alpha } G$ and provide a description of this class in the case of the irrational rotation algebra.

1 Introduction

Fourier multipliers are transformations on function spaces associated with abelian topological groups that act term-wise on the Fourier series of their elements. In the case of the group $\mathbb {Z}$, such mappings have a natural significance in classical Analysis, where the approximation of a given function by trigonometric polynomials has been of paramount importance. When such truncations are realised as Fourier multipliers, these mappings have finite rank and are hence compact. In subsequent developments, compactness of Fourier multipliers has been studied from other perspectives as well, for example, in relation with the compactness of pseudo-differential operators (see e.g. [8]).

For non-commutative locally compact groups G, where Fourier transform is not readily available, the natural setting for the study of Fourier multipliers is provided by the Fourier algebra A(G) of the group G—a commutative Banach algebra consisting of all coefficients of the left regular representation of G whose Gelfand spectrum can be canonically identified with G. This study was initiated in [6], with illustrious subsequent history and some far-reaching applications to approximation techniques in operator algebra theory, where finite rank multipliers have played a cornerstone role (see [5]). In [22], Lau showed that the Fourier algebra A(G) has a non-zero weakly compact left multiplier if and only if G is discrete and that, for discrete amenable groups, A(G) coincides with the algebra of its weakly compact multipliers. We refer the reader to [7, 11, 15] for further related results.

In the case of non-abelian groups, the new property of complete boundedness—brought about by non-commutativity—becomes a natural requirement, and the associated multipliers of A(G) are known as Herz–Schur multipliers [6]. Herz–Schur multipliers are related to Schur multipliers—transformations on the algebra of all bounded operators on $L^2(G)$ that extend point-wise multiplication of integral kernels by a given fixed function—via operator transference, pioneered in the area by Bożejko and Fendler [3]. We refer the reader to [34, 35] for a survey of these results and ideas.

A natural quantised version of the notion of compactness in the non-commutative setting—complete compactness—was first introduced by Saar in his Diplomarbeit [32] under G. Wittstock’s supervision and further developed in unpublished paper [37] by Webster. More recently it was also studied in the context of operator multipliers in [16] and completely almost periodic functionals on completely contractive Banach algebras in [31]. The notion has been important for the study of various operator space analogues of the Grothendieck approximation property and, in particular, for questions revolving around finite rank approximation. The main aim of this paper is to initiate the study of complete compactness for Herz–Schur multipliers. We work in the higher generality of dynamical systems. Herz–Schur multipliers of crossed products were introduced, and their relation with the surrounding class of operator-valued Schur multipliers investigated, in [25] (see also [1] for discrete dynamical systems). The completely compact operator-valued Schur multipliers were characterised in [18]. Here we provide a characterisation of completely compact Herz–Schur multipliers of the reduced crossed product of (unital) C*-algebras A by the action of a discrete group G, under a mild approximation property.

The paper is organised as follows. In Sect. 2, we recall some results about operator-valued Schur multipliers, Herz–Schur multipliers of C*-dynamical systems $(A,G,\alpha )$ and their interrelation [26] that will be needed in the sequel. In Sect. 3, we associate with every completely bounded map $\Phi$ with domain $A\rtimes _{\alpha ,r} G$ a Herz–Schur multiplier $F_{\Phi }$, and identify some properties of the map $\Phi \rightarrow F_{\Phi }$. This is the main technical tool, used in several subsequent results. We focus on actions of discrete groups, since the dynamical systems of non-discrete groups have no non-trivial compact Herz–Schur multipliers (see Proposition 3.2). Theorem 3.7 provides a characterisation of completely compact Herz–Schur $(A,G,\alpha )$-multipliers in the case where $A\rtimes _{\alpha ,r}G$ possesses the strong operator approximation property (SOAP) [9]. As a consequence, one can choose the maps, approximating the identity, to be in this case Herz–Schur multipliers of finite rank. As an immediate corollary, we obtain that for groups with the approximation property (AP) [17], a Herz–Schur multiplier $u : G\rightarrow {\mathbb {C}}$ defines a completely compact map $S_u:C_r^*(G)\rightarrow C_r^*(G)$ if and only if u belongs to the closure $A_{\mathrm{cb}}(G)$ of A(G) in the space $M^{\mathrm{cb}}A(G)$ of Herz-Schur multipliers. Using a result of Bożejko [2], we prove that the classes of compact and completely compact Herz–Schur multipliers $u : G\rightarrow {\mathbb {C}}$ are in general different.

The rest of the paper is devoted to special classes of completely compact $(A,G,\alpha )$-multipliers. Namely, in Sect. 4 we examine the subclass of invariant multipliers of a C*-dynamical system $(A,G,\alpha )$, obtained by lifting completely bounded maps on the C*-algebra A that satisfy a natural covariance property, and exhibit a canonical way of constructing completely compact multipliers. We provide a description of the latter class in the case of the irrational rotation algebra. Finally, in Sect. 5, we consider completely compact multipliers of the dynamical system $(c_0(G),G,\alpha )$, where $\alpha$ is induced by left translations. Such multipliers induce, via the Stone-von Neumann Theorem, mappings on the space $\mathcal {K}$ of all compact operators on $\ell ^2(G)$ that are characterised in terms of the Haagerup tensor product $\mathcal {K}\otimes \mathcal {K}$. In the opposite direction, we show that any compact Schur multiplier on $\mathcal {K}$ gives rise to a natural completely compact $(c_0(G),G,\alpha )$-multiplier.

We finish this introduction with a general comment about a separability assumption. Many of our results rely on the development of the theory of Herz–Schur multipliers of C*-dynamical systems $(A,G,\alpha )$, undertaken in [25]. In [25], the C*-algebra A of the dynamical system is assumed to be separable. However, if the group G of the dynamical system is discrete, as pointed out before [26, Theorem 2.1], an inspection of the proofs from [25] reveals that the separability assumption on A can be lifted. In the present paper, all C*-dynamical systems $(A,G,\alpha )$ will be assumed to be over discrete groups and arbitrary C*-algebras A.

2 Schur and Herz–Schur Multipliers

We denote by $\mathcal {B}(H)$ the algebra of all bounded linear operators acing on a Hilbert space H, and by $I_H$ (or I when H is clear from the context) the identity operator on H. For an operator space $\mathcal {X}\subseteq \mathcal {B}(H)$, we let $M_n(\mathcal {X})$ be the space of all n by n matrices with entries in $\mathcal {X}$, and identify it with a subspace of $\mathcal {B}(H^n)$ (where $H^n$ is the direct sum of n copies of H). If $\mathcal {X}$ and $\mathcal {Y}$ are operator spaces, acting on Hilbert spaces H and K, respectively, and $\varphi : \mathcal {X}\rightarrow \mathcal {Y}$ is a linear map, we let as usual $\varphi ^{(n)} : M_n(\mathcal {X})\rightarrow M_n(\mathcal {Y})$ be the map given by $\varphi ^{(n)}\left( (x_{i,j})_{i,j}\right) = \left( \varphi (x_{i,j})\right) _{i,j}$. The map $\varphi$ is called completely bounded if $\left\| \varphi \right\| _{\mathrm{cb}} := \sup _{n\in \mathbb {N}}\left\| \varphi ^{(n)}\right\| < \infty$. We write $\mathrm{CB}(\mathcal {X},\mathcal {Y})$ for the space of all completely bounded maps from $\mathcal {X}$ into $\mathcal {Y}$. We let $\mathcal {X}\otimes _{\min }\mathcal {Y}$ be the minimal tensor product of $\mathcal {X}$ and $\mathcal {Y}$, that is, the closure of the algebraic tensor product $\mathcal {X}\otimes \mathcal {Y}$ when considered as a subspace of $\mathcal {B}(H\otimes K)$. We will use throughout the paper basic results from operator space theory, and we refer the reader to the monographs [10, 29] for the necessary background.

For a locally compact group G, we let $\lambda ^0$ be its left regular representation on $L^2(G)$; thus,

$$\begin{aligned} \lambda ^0_t g(s) = g(t^{-1}s), \ \ \ g\in L^2(G), s,t\in G. \end{aligned}$$

We use the same symbol, $\lambda ^0$, to denote the left regular representation of $L^1(G)$ on $L^2(G)$. Let

$$\begin{aligned} C_r^*(G) = \overline{\{\lambda ^0(f) : f\in L^1(G)\}} \subseteq \mathcal {B}(L^2(G)) \end{aligned}$$

be the reduced C*-algebra of G, $\mathrm{VN}(G) := \overline{C_r^*(G)}^{w^*}$ the von Neumann algebra of G (here $w^*$ denotes the weak* topology of $\mathcal {B}(L^2(G))$), and A(G) be the Fourier algebra of G, that is, the collection of the functions on G of the form $s\rightarrow (\lambda ^0_s\xi ,\eta )$, where $\xi ,\eta \in L^2(G)$. The algebra A(G) will be equipped with the operator space structure arising from its identification with the predual of $\mathrm{VN}(G)$; its norm will be denoted by $\Vert \cdot \Vert _{A}$, and by $\Vert \cdot \Vert _{A(G)}$ in cases where we need to emphasise the group. We also write B(G) (resp. $B_r(G)$) for the Fourier-Stieltjes (resp. the reduced Fourier-Stieltjes) algebra of G. The space B(G) (resp. $B_r(G)$) is generated by continuous positive-definite functions on G (resp. by positive-definite functions weakly associated to the left regular representation $\lambda ^0$ of G), and one has the inclusions $A(G)\subseteq B_r(G)\subseteq B(G)$. By [12, Proposition 2.1], B(G) and $B_r(G)$ can be identified with the dual space of the full C*-algebra $C^*(G)$ and reduced C*-algebra $C_r^*(G)$ of G, respectively. Moreover, when B(G) is equipped with the norm arising from the identification $B(G) = C^*(G)^*$, it becomes a Banach algebra with respect to the pointwise multiplication, and A(G) and $B_r(G)$ are closed ideals of B(G). The norms on A(G) and $B_r(G)$ inherited from B(G) coincide with the norms arising from the identifications $A(G)^* = \mathrm{VN}(G)$ and $B_r(G)=C_r^*(G)^*$. We refer the reader to the monograph [21] for necessary further background from Abstract Harmonic Analysis.

A function $u : G\rightarrow \mathbb {C}$ is called a multiplier of A(G) if $uv\in A(G)$ for every $v\in A(G)$. We denote by MA(G) the algebra of all multipliers of A(G). An element $u\in MA(G)$ is called a Herz–Schur multiplier of A(G) [6] if the map $v\rightarrow uv$ on A(G) is completely bounded (here, and in the sequel, we equip A(G) and B(G) with the operator space structures, arising from the identifications $A(G)^* = \mathrm{VN}(G)$ and $B(G) = C^*(G)^*$). We let $M^{\mathrm{cb}}A(G)$ be the algebra of all Herz–Schur multipliers of A(G). We note that $u\in M^{\mathrm{cb}}A(G)$ if and only if the map $S_u: C_r^*(G)\rightarrow C_r^*(G)$, $\lambda ^0(f)\mapsto \lambda ^0(uf)$, $f\in L^1(G)$, is completely bounded, which is proved using similar arguments to the ones in [6, Proposition 1.2].

We henceforth fix a Hilbert space H and a non-degenerate C*-algebra $A\subseteq \mathcal {B}(H)$. Let G be a discrete group and $\alpha : G\rightarrow \mathrm{Aut}(A)$ be a point-norm continuous homomorphism; thus, $(A,G,\alpha )$ is a C*-dynamical system. We write $\{\delta _s : s\in G\}$ for the canonical orthonormal basis of $\ell ^2(G)$. We let $\ell ^1(G,A)$ be the convolution *-algebra of all summable functions $f : G\rightarrow A$, set $\mathcal {H} := \ell ^2(G)\otimes H$ and identify it with the Hilbert space $\ell ^2(G,H)$ of all square summable H-valued functions on G. Let $\lambda : G\rightarrow \mathcal {B}(\mathcal {H})$, $t\rightarrow \lambda _t$, be the unitary representation of G given by

$$\begin{aligned} \lambda _t \xi (s) = \xi (t^{-1}s), \ \ \ s,t\in G, \xi \in \mathcal {H}; \end{aligned}$$

note that $\lambda _t = \lambda _t^0 \otimes I$. Let $\pi : A\rightarrow \mathcal {B}(\mathcal {H})$ be the *-representation given by

$$\begin{aligned} \pi (a)\xi (s) = \alpha _{s^{-1}}(a)(\xi (s)), \ \ \ a\in A, \xi \in \mathcal {H}, s\in G. \end{aligned}$$

We note the covariance relation

$$\begin{aligned} \pi (\alpha _t(a)) = \lambda _t \pi (a) \lambda _t^*, \ \ \ a\in A, t\in G. \end{aligned}$$

(1)

The pair $(\pi ,\lambda )$ gives rise to a *-representation $\tilde{\pi } : \ell ^1(G,A) \rightarrow \mathcal {B}(\mathcal {H})$, given by

$$\begin{aligned} \tilde{\pi }(f) = \sum _{s\in G} \pi (f(s))\lambda _s, \ \ \ f\in \ell ^1(G,A). \end{aligned}$$

(2)

(Note that the series on the right hand side of (2) converges in norm for every $f\in \ell ^1(G,A)$). The reduced crossed product $A\rtimes _{\alpha ,r} G$ is defined by letting

$$\begin{aligned} A\rtimes _{\alpha ,r} G = \overline{\tilde{\pi }(\ell ^1(G,A))}, \end{aligned}$$

where the closure is taken in the operator norm of $\mathcal {B}(\mathcal {H})$. Note that, after identifying A with $\pi (A)$, we may consider A as a C*-subalgebra of $A\rtimes _{\alpha ,r} G$. It is well-known that if $\rho : A\rightarrow \mathcal {B}(K)$ is a faithful non-degenerate *-representation and $\alpha '$ is the canonical action of G on $\rho (A)$, arising from $\alpha$, then $A\rtimes _{\alpha ,r} G\cong \rho (A)\rtimes _{\alpha ',r} G$ canonically (see e.g. [30, Theorem 7.7.5]).

Identifying $\mathcal {H}$ with $\oplus _{s\in G} H$, we associate to every operator $x\in \mathcal {B}(\mathcal {H})$ a matrix $(x_{p,q})_{p,q}$, where $x_{p,q}\in \mathcal {B}(H)$, $p,q\in G$; thus,

$$\begin{aligned} \langle x_{p,q} \xi ,\eta \rangle = \langle x (\delta _q \otimes \xi ), \delta _p \otimes \eta \rangle , \ \ \ \xi ,\eta \in H, p,q\in G. \end{aligned}$$

In particular, if $a\in A$ and $t\in G$ then

$$\begin{aligned} \left( \pi (a)\lambda _t\right) _{p,q} = {\left\{ \begin{array}{ll} \alpha _{p^{-1}}(a) &{} \text {if } pq^{-1} = t\\ 0 &{} \text {if } pq^{-1} \ne t. \end{array}\right. } \end{aligned}$$

(3)

Equation (3) implies that

$$\begin{aligned} \left( \pi (a)\lambda _t\right) _{e,qp^{-1}} = \delta _{t, pq^{-1}} a \end{aligned}$$

(here $\delta _{s,r}$ is the Kronecker symbol). By linearity and continuity,

$$\begin{aligned} x_{p,q} = \alpha _{p^{-1}} (x_{e, qp^{-1}}), \ \ \ x\in A\rtimes _{\alpha ,r} G. \end{aligned}$$

(4)

Let $\mathcal {E} : \mathcal {B}(\ell ^2(G))\otimes _{\min } A \rightarrow A$ be the (unital completely positive) map given by $\mathcal {E}(x) = x_{e,e}$; note that $\mathcal {E}(\pi (a)) = a$, $a\in A$. Equation (3) implies that $x_{e,r} = \mathcal {E}(x\lambda _r)$, $x\in A\rtimes _{\alpha ,r} G$, $r\in G$. Now (4) can be rewritten as

$$\begin{aligned} x_{p,q} = \alpha _{p^{-1}}(\mathcal {E}(x\lambda _{qp^{-1}})), \ \ \ x\in A\rtimes _{\alpha ,r} G, \ p,q\in G. \end{aligned}$$

(5)

To every operator $x\in A\rtimes _{\alpha ,r} G$, one can associate the family $(a_s)_{s\in G}\subseteq A$, where $a_t = \mathcal {E}(x\lambda _{t^{-1}})$; we call $\sum _{t\in G} \pi (a_t) \lambda _t$ the Fourier series of x and write $x\sim \sum _{t\in G} \pi (a_t) \lambda _t$ (no convergence is assumed). Equation (5) shows that

$$\begin{aligned} x\sim \sum _{t\in G}\pi (a_t)\lambda _t \ \Longrightarrow \ x_{p,q} = \alpha _{p^{-1}}(a_{pq^{-1}}). \end{aligned}$$

(6)

Thus, if $x\in A\rtimes _{\alpha ,r} G$ then the diagonal of its matrix coincides with the family $(\alpha _{r^{-1}}(\mathcal {E}(x)))_{r\in G}$.

We note that

$$\begin{aligned} \alpha _{t}(\mathcal {E}(x))=\mathcal {E}(\lambda _t x\lambda _t^*), \ \ \ t\in G, x\in A\rtimes _{\alpha ,r} G. \end{aligned}$$

(7)

The latter equality follows from (1) in the case where $x = \pi (a) \lambda _s$ and follows by linearity and continuity for a general $x\in A\rtimes _{\alpha ,r} G$.

If $F : G\rightarrow \mathrm{CB}(A)$ is a bounded map and $f\in \ell ^1(G,A)$, let $F\cdot f\in \ell ^1(G,A)$ be the function given by

$$\begin{aligned} (F\cdot f)(t) = F(t)(f(t)), \ \ \ t\in G. \end{aligned}$$

Recall [25, Definition 3.1] that F is called a Herz–Schur $(A,G,\alpha )$-multiplier if the map $S_F$, given by

$$\begin{aligned} S_F(\tilde{\pi }(f)) = \tilde{\pi }(F\cdot f), \ \ \ f\in \ell ^1(G,A), \end{aligned}$$

(8)

is completely bounded. If F is a Herz–Schur $(A,G,\alpha )$-multiplier, then $S_F$ has a (unique) extension to a completely bounded map on $A\rtimes _{\alpha ,r} G$, which will be denoted in the same way. We write $\mathfrak {S}(A,G,\alpha )$ for the space of all Herz–Schur $(A,G,\alpha )$-multipliers, and set $\Vert F\Vert _{\mathrm{m}} = \Vert S_F\Vert _{\mathrm{cb}}$, $F\in \mathfrak {S}(A,G,\alpha )$. Note that, if $u\in M^{\mathrm{cb}}A(G)$ and $\tilde{u} : G\rightarrow \mathrm{CB}(A)$ is the function given by $\tilde{u}(t) = u(t)\mathrm{id}_A$, then $\tilde{u}\in \mathfrak {S}(A,G,\alpha )$ and its norm in $\mathfrak {S}(A,G,\alpha )$ coincides with its norm in $M^{\mathrm{cb}}A(G)$ [25, Remark 3.2 (ii)]. Without risk of confusion, for brevity we will denote by $S_u$ the map $S_{\tilde{u}}$.

Let $\varphi : G\times G\rightarrow \mathrm{CB}(A,\mathcal {B}(H))$ be a bounded function and $S_{\varphi }$ be the map from the space of all $G\times G$ A-valued matrices into the space of all $G\times G$ $\mathcal {B}(H)$-valued matrices, given by

$$\begin{aligned} S_{\varphi }\left( (x_{p,q})_{p,q}\right) = \left( \varphi (q,p)(x_{p,q})\right) _{p,q}. \end{aligned}$$

The map $\varphi$ is called a Schur A-multiplier if $S_{\varphi }$ is a completely bounded map from $\mathcal {B}(\ell ^2(G)) \otimes _{\min } A$ into $\mathcal {B}(\mathcal {H})$. By the paragraph after [26, Theorem 2.1], this definition is equivalent, for a discrete group G, to the definition of Schur A-multipliers in [25, Definition 2.1]. For a Schur A-multiplier $\varphi$, we write $\Vert \varphi \Vert _{\mathrm{m}} = \Vert S_{\varphi }\Vert _{\mathrm{cb}}$. Schur $\mathbb {C}$-multipliers are referred to as Schur multipliers. For a bounded function $F : G\rightarrow \mathrm{CB}(A)$, let ${\mathcal {N}}(F): G \times G \rightarrow \mathrm{CB}(A)$ be the function given by

$$\begin{aligned} \mathcal {N}(F)(s,t) (a) = \alpha _{t^{-1}}(F(ts^{-1})(\alpha _t(a))), \;\;\;\; s,t \in G, \ a \in A. \end{aligned}$$

In the case where $A = \mathbb {C}$, we write N(u) in the place of $\mathcal {N}(u)$. The following statement [25, Theorem 3.18] is a crossed product version of a classical result of Bożejko and Fendler [3] in the case $A = \mathbb {C}$.

Theorem 2.1

The map $\mathcal {N}$ is an isometric injection from $\mathfrak {S}(A,\alpha ,G)$ into the algebra of Schur A-multipliers.

Note that the image $\mathcal {N}(\mathfrak {S}(A,\alpha ,G))$ coincides with the so-called invariant Schur-A-multipliers, denoted by $\mathfrak {S}_\mathrm{inv}(G,G,A)$ in [25, p. 413].

We remark that in [25] Herz–Schur $(A,G,\alpha )$-multipliers and Schur A-multipliers were defined for general locally compact group G and separable C*-algebra A. But in the case of discrete G the separability condition on A can be removed without changing the statements from [25] (see the comment before [26, Theorem 2.1]).

The following observation will be frequently used: If G is discrete then $S_{\mathcal {N}(F)}|_{A\rtimes _{\alpha ,r}G}=S_F$ for any $F\in \mathfrak {S}(A,\alpha ,G)$. To see this, it suffices to show that

$$\begin{aligned} S_{\mathcal {N}(F)}(\pi (a)\lambda _s)=\pi (F(s)(a))\lambda _s, \ \ \ a\in A, s\in G. \end{aligned}$$

Write $\pi (a)\lambda _s=(x_{p,q})_{p,q}$ for $x_{p,q}=\delta _{s,pq^{-1}}\alpha _{p^{-1}}(a)$ (see (3)) and

$$\begin{aligned} (S_{\mathcal {N}(F)}(\pi (a)\lambda _s)_{p,q}= & {} \delta _{s,pq^{-1}}\mathcal {N}(F)(q,p)(\alpha _{p^{-1}}(a))\\= & {} \delta _{s,pq^{-1}}\alpha _{p^{-1}}(F(pq^{-1})(a))\\= & {} \delta _{s,pq^{-1}}\alpha _{p^{-1}}(F(s)(a)) = (\pi (F(s)(a))\lambda _s)_{p,q}. \end{aligned}$$

3 Characterisation of Complete Compactness

Let $\mathcal {X}$ and $\mathcal {Z}$ be operator spaces. A completely bounded map $\Psi : \mathcal {X} \rightarrow \mathcal {Z}$ is called completely compact if for very $\epsilon > 0$ there exists a finite dimensional subspace $\mathcal {Y}\subseteq \mathcal {Z}$ such that

$$\begin{aligned} \mathrm{dist}(\Psi ^{(m)}(x), M_m(\mathcal {Y})) < \epsilon , \text{ for } \text{ all } x\in M_m(\mathcal {X}), \Vert x\Vert \le 1, \text{ and } \text{ all } m\in \mathbb {N}. \end{aligned}$$

(9)

We denote by $\mathrm{CC}(\mathcal {X}, \mathcal {Z})$ the space of all completely compact linear maps from $\mathcal {X}$ to $\mathcal {Z}$, and write $\mathrm{CC}(\mathcal {X})$ for $\mathrm{CC}(\mathcal {X},\mathcal {X})$. By $\mathrm{F}(\mathcal {X})$ we denote the subspace of all (linear) maps of finite rank on $\mathcal {X}$. Clearly, $\mathrm{F}(\mathcal {X})\subseteq \mathrm{CC}(\mathcal {X})$ and any completely compact map on $\mathcal {X}$ is compact.

A net $(\Phi _i)_{i\in \mathbb {I}} \subseteq \mathrm{CB}(\mathcal {X})$ is said to converge to $\Phi \in \mathrm{CB}(\mathcal {X})$ in the strongly stable point norm topology [9, p. 197] if

$$\begin{aligned} \left\| (\mathrm{id}\otimes \Phi _i)(x) - (\mathrm{id}\otimes \Phi )(x)\right\| \rightarrow _{i\in \mathbb {I}} 0, \ \ \ x\in \mathcal {B}(K)\otimes _{\min } {\mathcal {X}}, \end{aligned}$$

for any Hilbert space K. An operator space ${\mathcal {X}}$ is said to possess the strong operator approximation property (SOAP) if the identity map on ${\mathcal {X}}$ can be approximated by finite rank maps in the strongly stable point norm topology. The notion was introduced by Effros and Ruan in [9] as an operator space analogue of the approximation property for Banach spaces. It admits a characterisation using complete compactness. Namely, $\mathcal {X}$ has the SOAP if for any operator space $\mathcal {Z}$ and any completely compact map $\Psi \in \mathrm{CC}(\mathcal {Z},\mathcal {X})$, $\Psi$ can be approximated by finite rank maps completely uniformly, i.e. in the $\Vert \cdot \Vert _{\mathrm{cb}}$-norm, see [37, Theorem 5.9] or [36, Proposition 4.4.6]. In particular, if $\mathcal X$ has the SOAP then $\overline{\mathrm{F}({\mathcal {X}})}^{\mathrm{cb}} = \mathrm{CC}({\mathcal {X}})$.

Lemma 3.1

Let $\mathcal {X}$ be an operator space, $\Psi \in \mathrm{CC}(\mathcal {X})$, $\Phi \in \mathrm{CB}(\mathcal {X})$ and $(\Phi _{i})_{i\in \mathbb {I}}\subseteq \mathrm{CB}(\mathcal {X})$ be a net such that $\Phi _i\rightarrow _{i\in \mathbb {I}} \Phi$ in the strongly stable point-norm topology. Then $\Vert \Phi _i \circ \Psi - \Phi \circ \Psi \Vert _{\mathrm{cb}}\rightarrow _{i\in \mathbb {I}} 0$.

Proof

By [37, Proposition 5.6] $\Phi _i\rightarrow _{i\in \mathbb {I}} \Phi$ completely uniformly on completely compact sets of ${\mathcal {X}}$ (see [37, Definitions 5.1 and 5.2] for the terminology). If $\Psi$ is completely compact then $(\{\Psi ^{(n)}(x) : x\in M_n(\mathcal {X}), \Vert x\Vert \le 1\})_{n\in \mathbb {N}}$ is a subset of a completely compact set and hence $\Vert \Phi _i\circ \Psi -\Phi \circ \Psi \Vert _\mathrm{cb}\rightarrow _{i\in \mathbb {I}} 0$. $\square$

The following observation shows that, when studying compact or completely compact multipliers, the interest lies only in discrete groups.

Proposition 3.2

Let G be a non-discrete locally compact group and $u : G\rightarrow \mathbb {C}$ be a Herz–Schur multiplier. If the map $S_u : C^*_r(G)\rightarrow C^*_r(G)$ is compact then $u = 0$.

Proof

Suppose that $S_u$ is compact. Let $B_r(G)$ be the reduced Fourier-Stieltjes algebra of G. Identifying the dual $C^*_r(G)^*$ of $C^*_r(G)$ with $B_r(G)$, we have that the adjoint map $S_u^* : B_r(G)\rightarrow B_r(G)$ is compact (see e.g. [28, Theorem 1.4.4]). Note that $S_u^*(v) = uv$, $v\in B_r(G)$. Since $A(G)\subseteq B_r(G)$, A(G) is the closed hull of its compactly supported elements and $A(G)\cap C_c(G)=B_r(G)\cap C_c(G)$ (see [21, Proposition 2.3.3]), the map $S_u^*$ leaves A(G) invariant. The restriction of $S_u^*$ to A(G) is thus compact, and by [22, Proposition 6.9], $u = 0$. $\square$

We henceforth assume that the group G is discrete. Recall that $A\subseteq \mathcal {B}(H)$ is a fixed non-degenerate C*-algebra, equipped with an action $\alpha : G\rightarrow \mathrm{Aut}(A)$. It will be convenient to set

$$\begin{aligned} \mathcal {V}_{A,G,\alpha } = \mathrm{span}\left\{ \pi (a)\lambda _s : a\in A, s\in G\right\} ; \end{aligned}$$

thus, $\mathcal {V}_{A,G,\alpha }$ is a (dense) subalgebra of $A\rtimes _{\alpha ,r} G$. We will write

$$\begin{aligned} \mathfrak {S}_{\mathrm{cc}}(A,G,\alpha ) = \left\{ F\in \mathfrak {S}(A,G,\alpha ) : S_F \text{ is } \text{ completely } \text{ compact }\right\} . \end{aligned}$$

If $\Phi : A\rtimes _{\alpha ,r} G \rightarrow \mathcal {B}(\ell ^2(G))\otimes _{\min } A$ is a bounded linear map, we let $F_\Phi : G\rightarrow \mathcal {B}(A)$ be the function given by

$$\begin{aligned} F_\Phi (s)(a) = \mathcal {E}(\Phi (\pi (a)\lambda _s)\lambda _s^*). \end{aligned}$$

(10)

Proposition 3.3

Let $\Phi : A\rtimes _{\alpha ,r} G \rightarrow \mathcal {B}(\ell ^2(G))\otimes _{\min } A$ be a completely bounded map.

• (i) The map $F_{\Phi }$ is a Herz–Schur $(A,G,\alpha )$-multiplier and $\Vert F_{\Phi }\Vert _{\mathrm{m}}\le \Vert \Phi \Vert _{\mathrm{cb}}$;

• (ii) If $F \in \mathfrak {S}(A,G,\alpha )$ then $F_{S_F} = F$;

• (iii) If $\Phi$ completely compact then $F_{\Phi }(s)$ is completely compact for every $s\in G$;

• (iv) If $\Phi$ has finite rank and $\mathrm{ran}\Phi \subseteq \mathcal {V}_{A,G,\alpha }$ then $F_{\Phi }(s)$ has finite rank for every $s\in G$ and $S_{F_\Phi }\in \mathrm{F}(A\rtimes _{\alpha ,r}G)$;

• (v) If $\varphi$ is a Schur A-multiplier and $\Psi$ is the restriction of $S_{\varphi }$ to $A\rtimes _{\alpha ,r} G$ then $F_{\Psi }(s) = \varphi (s^{-1},e)$, $s\in G$.

Proof

(i) By the Haagerup-Paulsen-Wittstock Theorem (see e.g. [29, Theorem 8.4]), there exist a Hilbert space K, operators $V,W : H \rightarrow K$ and a *-representation $\rho : A\rtimes _{\alpha ,r} G \rightarrow \mathcal {B}(K)$ such that

$$\begin{aligned} \Phi (x) = W^*\rho (x)V, \ \ \ x\in A\rtimes _{\alpha ,r} G. \end{aligned}$$

(11)

Using (7), we have

$$\begin{aligned} \mathcal {N}\left( F_{\Phi }\right) (s,t)(a)= & {} \alpha _{t^{-1}}\left( F_{\Phi }(ts^{-1})(\alpha _t(a))\right) \nonumber \\= & {} \alpha _{t^{-1}}\left( \mathcal {E}(\Phi (\pi (\alpha _t(a))\lambda _{ts^{-1}})\lambda _{ts^{-1}}^*)\right) \nonumber \\= & {} \alpha _{t^{-1}}\left( \mathcal {E}(\Phi (\lambda _t\pi (a)\lambda _{t^{-1}}\lambda _{ts^{-1}})\lambda _{ts^{-1}}^*)\right) \nonumber \\= & {} \mathcal {E}\left( \lambda _{t^{-1}} \Phi ((\lambda _t\pi (a)\lambda _{s^{-1}})\lambda _s\lambda _{t^{-1}})\lambda _t\right) \nonumber \\= & {} \mathcal {E}\left( \lambda _{t^{-1}} \Phi (\lambda _t\pi (a)\lambda _{s^{-1}})\lambda _s\right) \nonumber \\= & {} \mathcal {E}\left( \lambda _t^* W^* \rho (\lambda _t)\rho (\pi (a))\rho (\lambda _s)^*V\lambda _s\right) . \end{aligned}$$

(12)

Note that, if $E_{p,q}$ denotes the matrix unit in $\mathcal {B}(\ell ^2(G,H))$ with I at the (p, q)-entry, $p,q\in G$, and zero elsewhere, then $\mathcal {E}(x) = E_{e,e} x E_{e,e}$, $x\in A\rtimes _{r,\alpha } G$. Let $\tilde{V}(s) = \rho (\lambda _s)^*V\lambda _s E_{e,e}$ and $\tilde{W}(t) = \rho (\lambda _t)^*W\lambda _t E_{e,e}$, $s,t\in G$; thus, $\tilde{V}, \tilde{W} : G\rightarrow \mathcal {B}(H, K)$, and

$$\begin{aligned} \sup _{s\in G}\Vert \tilde{V}(s)\Vert \sup _{t\in G}\Vert \tilde{W}(t)\Vert \le \Vert V\Vert \Vert W\Vert . \end{aligned}$$

(13)

By (12),

$$\begin{aligned} \mathcal {N}(F_{\Phi })(s,t)(a) = \tilde{W}(t)^*(\rho \circ \pi )(a) \tilde{V}(s), \ \ \ a\in A, s,t\in G. \end{aligned}$$

(14)

By [25, Theorems 2.6 and 3.8], $F_{\Phi }$ is a Herz–Schur $(A,G,\alpha )$-multiplier. Equations (13), (14) and Theorem 2.1 show that

$$\begin{aligned} \Vert F_{\Phi }\Vert _{\mathrm{m}} = \Vert S_{\mathcal {N}(F_{\Phi })}\Vert _{\mathrm{cb}} \le \Vert V\Vert \Vert W\Vert . \end{aligned}$$

Taking the infimum over all possible choices of V and W in the representation (11) of $\Phi$, we obtain that $\Vert F_{\Phi }\Vert _{\mathrm{m}} \le \Vert \Phi \Vert _{\mathrm{cb}}$.

(ii) Let $F \in \mathfrak {S}(A,G,\alpha )$. The fact that $F_{S_F} = F$ follows from the definition (8) of $S_F$ and the definition (10) of $F_{\Phi }$.

(iii) follows from the definition (10) of $F_\Phi$ and the fact that $\mathcal {E}$ is completely bounded.

(iv) Since $\Phi$ has finite rank, there exists a finite subset $E\subseteq G$ and a finite dimensional subspace $\mathcal {U}\subseteq A$, $s\in E$, such that

$$\begin{aligned} \mathrm{ran}\Phi \subseteq \mathrm{span}\{\pi (a)\lambda _s : s\in E, a\in \mathcal {U}\}. \end{aligned}$$

A direct verification shows that

$$\begin{aligned} \mathrm{ran}F_{\Phi }(s) \subseteq \left\{ \begin{array}{ll} \mathcal {U} &{} \text {if } s\in E\text {,} \\ \{0\} &{} \text {if } s\not \in E. \end{array} \right. \end{aligned}$$

Clearly, $\mathrm{ran}S_{F_\Phi } \subseteq \mathrm{span}\{\pi (a)\lambda _s : s\in E, a\in \mathcal {U}\}.$

(v) According to (3),

$$\begin{aligned} \left( \Psi (\pi (a)\lambda _s)\lambda _{s^{-1}}\right) _{p,q} = (S_{\varphi }(\pi (a)\lambda _s)_{p,s^{-1}q} = {\left\{ \begin{array}{ll} \varphi (s^{-1}q,p)(\alpha _{p^{-1}}(a)) &{} \text {if } p = q\\ 0 &{} \text {if } p \ne q. \end{array}\right. } \end{aligned}$$

Thus,

$$\begin{aligned} F_{\Psi }(s)(a) = \left( \Psi (\pi (a)\lambda _s)\lambda _{s^{-1}}\right) _{e,e} = \varphi (s^{-1},e)(a), \ \ \ a\in A.\qquad \qquad \qquad \qquad \qquad \square \end{aligned}$$

Proposition 3.3 (ii) and (iii) give the following immediate corollary.

Corollary 3.4

If $F\in \mathfrak {S}_{\mathrm{cc}}(A,G,\alpha )$ then F(s) is completely compact for every $s\in G$;

We will call a (possibly vector-valued) function $\varphi$ defined on $G\times G$ band finite if there exists a finite set $E\subseteq G$ such that $\varphi (s,t) = 0$ if $ts^{-1}\not \in E$. Let $V : \ell ^2(G)\otimes H\rightarrow \ell ^2(G)\otimes \ell ^2(G)\otimes H$ be the isometry given by

$$\begin{aligned} V(\delta _s\otimes \xi ) = \delta _s\otimes \delta _s\otimes \xi , \ \ \ s\in G, \xi \in H, \end{aligned}$$

and $\tau : A\rtimes _{r,\alpha }G \rightarrow C_r^*(G)\otimes _{\mathrm{min}} (A\rtimes _{r,\alpha }G)$ be the dual co-action to $\alpha$, that is, the *-homomorphism, given by

$$\begin{aligned} \tau \left( \pi (a)\lambda _t\right) = \lambda _t^0\otimes \pi (a)\lambda _t, \ \ \ a\in A, t\in G, \end{aligned}$$

(see e.g. [20]).

Lemma 3.5

The following hold:

• (i) $V^*\tau (x)V = x$, $x\in A\rtimes _{r,\alpha }G$;

• (ii) If $\Phi \in \mathrm{CB}(A\rtimes _{r,\alpha }G)$ then

  $$\begin{aligned} S_{F_{\Phi }}(x) = V^*(\mathrm{id}\otimes \Phi )(\tau (x))V, \ \ \ x\in A\rtimes _{r,\alpha }G. \end{aligned}$$

  (15)

Proof

(i) If $a\in A$, $s,r,t\in G$ and $\xi ,\eta \in H$ then

$$\begin{aligned}&\left\langle V^*\tau (\pi (a)\lambda _t)V(\delta _s\otimes \xi ),\delta _r\otimes \eta \right\rangle \\= & {} \left\langle (\lambda _t\otimes \pi (a)\lambda _t)(\delta _s\otimes \delta _s\otimes \xi ),\delta _r\otimes \delta _r\otimes \eta \right\rangle \\= & {} \left\langle \delta _{ts}\otimes \pi (a)\lambda _t(\delta _s\otimes \xi ),\delta _r\otimes \delta _r\otimes \eta \right\rangle = \left\langle \pi (a)\lambda _t(\delta _s\otimes \xi ),\delta _r\otimes \eta \right\rangle , \end{aligned}$$

by (3).

(ii) Let $a\in A$, $s,r,t\in G$ and $\xi ,\eta \in H$. Using (3) and (5), we have

$$\begin{aligned}&\langle V^*(\mathrm{id}\otimes \Phi )(\tau (\pi (a)\lambda _t))V(\delta _s\otimes \xi ),\delta _r\otimes \eta \rangle \\= & {} \langle (\mathrm{id}\otimes \Phi )(\lambda _t^0 \otimes \pi (a)\lambda _t)(\delta _s\otimes \delta _s\otimes \xi ),\delta _r\otimes \delta _r\otimes \eta \rangle \\= & {} \langle \left( \lambda _t^0\otimes \Phi (\pi (a)\lambda _t)\right) (\delta _s\otimes \delta _s\otimes \xi ),\delta _r\otimes \delta _r\otimes \eta \rangle \\= & {} \langle \delta _{ts}\otimes \Phi (\pi (a)\lambda _t)(\delta _s\otimes \xi ),\delta _r\otimes \delta _r\otimes \eta \rangle \\= & {} \left\{ \begin{array}{ll} 0 &{} \text {if } ts\ne r\text {,} \\ \langle \Phi (\pi (a)\lambda _t)(\delta _s\otimes \xi ),\delta _r\otimes \eta \rangle &{} \text {if } ts = r. \end{array} \right. \\= & {} \left\{ \begin{array}{ll} 0 &{} \text {if } ts\ne r\text {,} \\ \left\langle \alpha _{(ts)^{-1}}\left( \mathcal {E}(\Phi (\pi (a)\lambda _t)\lambda _t^*)\right) \xi ,\eta \right\rangle &{} \text {if } ts = r. \end{array} \right. \\= & {} \left\langle \pi \left( \mathcal {E}(\Phi (\pi (a)\lambda _t)\lambda _t^*)\right) (\delta _{ts}\otimes \xi ),\delta _r\otimes \eta \right\rangle \\= & {} \left\langle S_{F_{\Phi }}\left( \pi (a)\lambda _t\right) (\delta _s\otimes \xi ),\delta _r\otimes \eta \right\rangle . \end{aligned}$$

$\square$

Lemma 3.6

Let G be a discrete group and $(A,G,\alpha )$ be a C*-dynamical system such that $A\rtimes _{r,\alpha }G$ has the SOAP. Then there exists a net $(F_i)_{i\in \mathbb {I}}$ of finitely supported Herz–Schur $(A,G,\alpha )$-multipliers with $F_i(s)\in \mathrm{F}(A)$, $i\in \mathbb {I}$, $s\in G$, such that $(S_{F_i})_{i\in \mathbb {I}}$ converges to the identity map on $A\rtimes _{r,\alpha }G$ in the strongly stable point norm topology.

Proof

Write ${\mathfrak {A}} = A\rtimes _{r,\alpha }G$ and let $(\Phi _i)_{i\in \mathbb {I}} \subseteq \mathrm{F}({\mathfrak {A}})$ be a net such that

$$\begin{aligned} \Vert (\mathrm{id}\otimes \Phi _i)(x) - x\Vert \rightarrow _{i\in \mathbb {I}} 0, \ \ \ x\in \mathcal {B}(\ell ^2)\otimes _{\mathrm{min}}{\mathfrak {A}}. \end{aligned}$$

Following the proof of [24, Theorem 4.3], given $\varepsilon > 0$, for each $i\in \mathbb {I}$ there exists a finite rank map $\Phi _{i,\varepsilon }\in \mathrm{CB}({\mathfrak {A}})$ whose range lies in $\mathcal {V}_{A,G,\alpha }$, such that

$$\begin{aligned} \Vert \Phi _i-\Phi _{i,\varepsilon }\Vert _{\mathrm{cb}} < \varepsilon , \ \ \ i\in \mathbb {I}. \end{aligned}$$

Hence

$$\begin{aligned} \Vert (\mathrm{id}\otimes \Phi _{i,\varepsilon })(x) - x\Vert \rightarrow _{(i,\epsilon )} 0, \ \ \ x\in \mathcal {B}(H)\otimes _{\mathrm{min}}{\mathfrak {A}}, \end{aligned}$$

along the product directed set.

The function $F_{i,\varepsilon } := F_{\Phi _{i,\varepsilon }}$ is finitely supported. By Proposition 3.3 (i) and (iv), $F_{i,\varepsilon }$ is a Herz–Schur multiplier and $F_{i,\varepsilon }(s)\in \mathrm{F}(A)$, $s\in G$, $i\in \mathbb {I}$. If $y\in \mathcal {B}(H)\otimes _{\mathrm{min}} {\mathfrak {A}}$, using Lemma 3.5 we have

$$\begin{aligned} \Vert \mathrm{id}\otimes S_{F_{i,\varepsilon }}(y) - y\Vert= & {} \Vert (\mathrm{id}\otimes V^*)(\mathrm{id}\otimes \mathrm{id}\otimes \Phi _{i,\varepsilon }(\mathrm{id}\otimes \tau (y))(\mathrm{id}\otimes V)\\&- (\mathrm{id}\otimes V^*)(\mathrm{id}\otimes \tau (y)(\mathrm{id}\otimes V)\Vert \\\le & {} \Vert (\mathrm{id}\otimes \mathrm{id}\otimes \Phi _{i,\varepsilon })((\mathrm{id}\otimes \tau )(y)) - (\mathrm{id}\otimes \tau )(y)\Vert \rightarrow _{(i,\epsilon )} 0, \end{aligned}$$

that is, $S_{F_{i,\varepsilon }}$ converges to the identity map in the strongly stable point norm topology. $\square$

Theorem 3.7

Let F be a Herz–Schur multiplier of $(A,G,\alpha )$. Assume that $A\rtimes _{r,\alpha }G$ possesses the SOAP. The following are equivalent:

• (i) the map $S_F$ is completely compact;

• (ii) there exist a net $(F_i)_{i\in \mathbb {I}}\subseteq \mathfrak S(A,G,\alpha )$ and a finite dimensional subspace $\mathcal {U}_{i}\subseteq A$, such that $F_i$ is finitely supported, $\mathrm{ran}F_i(s)\subseteq \mathcal {U}_i$ for each $i\in \mathbb {I}$ and each $s\in G$, and $\Vert F_i - F\Vert _\mathrm{m}\rightarrow _{i\in \mathbb {I}} 0$;

• (iii) there exists a net $(\varphi _i)_{i\in \mathbb {I}}$ of band finite Schur A-multipliers and a finite dimensional subspace $\mathcal {U}_{i}\subseteq A$, $i\in \mathbb {I}$, such that $\mathrm{ran}\alpha _s \circ \varphi _i(s,e) \subseteq \mathcal {U}_{i}$ for all $i\in \mathbb {I}$ and all $s\in G$, and $\Vert S_{\mathcal {N}(F)} - S_{\varphi _i}\Vert _{\mathrm{cb}} \rightarrow _{i\in \mathbb {I}} 0$.

Proof

(i)$\Rightarrow$(ii) Let $(\tilde{F}_i)_{i\in \mathbb {I}}$ be a net as in Lemma 3.6 and $F_i = \tilde{F}_i \circ F$, $i\in \mathbb {I}$. By Lemma 3.1, $(F_i)_{i\in \mathbb {I}}$ satisfies the conditions of (ii).

(ii)$\Rightarrow$(iii) Set $\varphi _i = \mathcal {N}(F_i)$; by Theorem 2.1, $\varphi _i$ is a Schur A-multiplier, $i\in \mathbb {I}$. Since $F_i$ is finitely supported, $\varphi _i$ is band finite. Moreover,

$$\begin{aligned} \varphi _i(s,t)(a) = \alpha _{t^{-1}} (F_i(ts^{-1})(\alpha _t(a))), \ \ \ a\in A, \end{aligned}$$

and hence

$$\begin{aligned} \mathrm{ran}\alpha _s\circ \varphi _i(s,e) \subseteq \mathrm{ran}F_i(s^{-1}) \subseteq \mathcal {U}_i, \ \ \ s\in G. \end{aligned}$$

(iii)$\Rightarrow$(i) Let $\Phi _i$ be the restriction of the map $S_{\varphi _i}$ to $A\rtimes _{r,\alpha } G$, $i\in \mathbb {I}$. By Proposition 3.3 (i) and (ii), $\Vert F - F_{\Phi _i}\Vert _{\mathrm{m}} \rightarrow _{i\in \mathbb {I}} 0$. Since $\varphi _i$ is band finite, $F_i:=F_{\Phi _i}$ is supported on a finite set, say $E_i\subseteq G$. By Proposition 3.3 (v),

$$\begin{aligned} \mathrm{ran}F_i(s)\subseteq \mathrm{ran}\varphi _i(s^{-1},e) \subseteq \alpha _s(\mathcal {U}_i), \ \ \ s\in G. \end{aligned}$$

Let $\mathcal {V}_i = \mathrm{span}\left( \cup _{s\in E_i} \alpha _s(\mathcal {U}_i)\right)$; then $\mathcal {V}_i$ is finite dimensional and

$$\begin{aligned} \mathrm{ran}S_{F_i}\subseteq \mathrm{span}\{\pi (a)\lambda _s : a\in \mathcal {V}_i, s\in E_i\}. \end{aligned}$$

Thus, $S_{F_i}$ has finite rank, $i\in \mathbb {I}$. Since the set $\mathrm{CC}(A\rtimes _{r,\alpha } G)$ is closed in the completely bounded norm, $S_F\in \mathfrak {S}_{\mathrm{cc}}(A\rtimes _{r,\alpha } G)$. $\square$

Corollary 3.8

If $A\rtimes _{r,\alpha }G$ possesses the SOAP then every completely compact Herz–Schur $(A,G,\alpha )$-multiplier is a limit of $(A,G,\alpha )$-multipliers of finite rank.

Remark

By [37, Theorem 5.9], any completely compact map on a $C^*$-algebra with SOAP is a limit of finite rank maps. Corollary 3.8 is a refinement of this statement.

Let $A_{\mathrm{cb}}(G)$ be the closure of A(G) within $M^{\mathrm{cb}}A(G)$ with respect to $\Vert \cdot \Vert _{\mathrm{m}}$. The algebra was first introduced and studied in [13]. It is a regular commutative Tauberian Banach algebra whose Gelfand space can be canonically identified with G (see [14, Proposition 2.2]).

It is known that $M^{\mathrm{cb}} A(G)$ is a dual Banach space [6, Proposition 1.10]. Recall that G has the approximation property (AP) [17] if there exists a net $(u_i)_{i\in \mathbb {I}}$ of finitely supported Herz–Schur multipliers such that $u_i \rightarrow _{i\in \mathbb {I}} 1$ in the weak* topology of $M^{\mathrm{cb}}A(G)$. By [17, Theorem 1.9] (see also [5, Theorem 12.4.9], G has (AP) if and only if $C_r^*(G)$ has SOAP. We note that, by [33, Theorem 3.6], $A\rtimes _{r,\alpha } G$ has the SOAP if A has the SOAP and G has the AP. The SOAP for $A\rtimes _{r,\alpha } G$ was also recently studied in [27].

Theorem 3.7 has the following immediate consequence.

Corollary 3.9

Let G be a discrete group possessing property (AP) and $u : G\rightarrow \mathbb {C}$ be a Herz–Schur multiplier. The following are equivalent:

• (i) the map $S_u : C^*_r(G) \rightarrow C^*_r(G)$ is completely compact;

• (ii) there exists a net $(u_i)_{i\in \mathbb {I}}$ of finitely supported elements of A(G) such that $\Vert S_u - S_{u_i}\Vert _{\mathrm{cb}}\rightarrow _{i\in \mathbb {I}} 0$;

• (iii) there exists a net $(\varphi _i)_{i\in \mathbb {N}}$ of band finite Schur multipliers such that $\Vert S_{N(u)} - S_{\varphi _i}\Vert \rightarrow _{i\in \mathbb {I}} 0$;

• (iv) $u\in A_{\mathrm{cb}}(G)$.

Proof

The equivalences (i)$\Leftrightarrow$(ii)$\Leftrightarrow$(iii) follow from Theorem 3.7. The equivalence (ii)$\Leftrightarrow$(iv) follows from the facts that the finitely supported functions in A(G) form a dense set in A(G) and $\Vert S_u\Vert _{\mathrm{cb}} = \Vert u\Vert _{\mathrm{m}}$. $\square$

Remark 3.10

Let G be a discrete group possessing (AP) and $u\in MA(G)$. The map $S_u$ is compact if and only if $u\in A_M(G):=\overline{A(G)}^{\Vert \cdot \Vert _{MA(G)}}$.

Proof

By [17, Theorem 1.9], there exists a net $(u_i)_{i\in \mathbb {I}}$ consisting of finitely supported functions such that $S_{u_i}(x) \rightarrow _{i\in \mathbb {I}} x$, $x\in C^*_r(G)$. If $S_u$ is compact then $S_{u_i} S_{u}\rightarrow _{i\in \mathbb {I}} S_u$ in norm; thus, $\Vert u - u u_i\Vert _{MA(G)} \rightarrow _{i\in \mathbb {I}} 0$.

Conversely, suppose that $u\in A_M(G)$. Since $\Vert \cdot \Vert _{MA(G)} \le \Vert \cdot \Vert _{A(G)}$, there exists a net $(u_i)_{i\in \mathbb {I}}$ of finitely supported functions such that $\Vert u_i - u\Vert _{MA(G)} \rightarrow _{i\in \mathbb {I}} 0$. Thus, $\Vert S_{u_i} - S_u\Vert \rightarrow _{i\in \mathbb {I}} 0$; since $S_{u_i}$ has finite rank, $i\in \mathbb {I}$, we have that $S_u$ is compact. $\square$

If the group G is amenable then $B(G) = M^{\mathrm{cb}}A(G) = MA(G)$ and the norms on these three spaces coincide, ( [6, Corollary 1.8 (ii)]). It was shown in [22, Proposition 6.10] that, in this case, the map $S_u$ is compact precisely when $u\in A(G)$. By Corollary 3.9, automatic complete compactness holds:

Corollary 3.11

Let $u:G\rightarrow {\mathbb {C}}$ be a Herz–Schur multiplier. If G is a discrete amenable group then the following are equivalent:

• (i) the map $S_u$ is completely compact;

• (ii) $u\in A(G)$.

Proposition 3.12

Let G be a discrete group containing the free group $F_\infty$ on infinitely many generators. If G has the (AP), then there exists a multiplier $u\in MA(G)$, for which the map $S_u$ is compact but not completely compact.

Proof

Write $H = F_\infty$, considering it as a subgroup of G. We note first that if $\varphi \in A(H)$ and ${\tilde{\varphi }}$ is the extension by zero of $\varphi$ to G, then $\Vert \varphi \Vert _{A(H)}=\Vert {\tilde{\varphi }}\Vert _{A(G)}$. In fact, we have $\varphi (s)=(\lambda _H(s)\xi ,\eta )$ for some $\xi ,\eta \in \ell ^2(H)$ such that $\Vert \varphi \Vert _{A(H)} = \Vert \xi \Vert _2\Vert \eta \Vert _2$. Considering $\ell ^2(H)$ as a subspace of $\ell ^2(G)$ and letting ${\tilde{\xi }}$ and ${\tilde{\eta }}$ be the extensions by zero to $\ell ^2(G)$ of $\xi$ and $\eta$, respectively, we have that $\tilde{\varphi }(s)=(\lambda _G(s){\tilde{\xi }},{\tilde{\eta }})$ and hence $\Vert \varphi \Vert _{A(G)}\le \Vert {\tilde{\xi }}\Vert _2\Vert {\tilde{\eta }}\Vert _2=\Vert \varphi \Vert _{A(H)}$. As the restriction map $r:A(G)\rightarrow A(H)$, $u\mapsto u|_H$, is contractive [12, Proposition 3.21], we have also $\Vert \varphi \Vert _{A(H)}\le \Vert {\tilde{\varphi }}\Vert _{A(G)}$, giving $\Vert \varphi \Vert _{A(H)} = \Vert {\tilde{\varphi }}\Vert _{A(G)}$.

In the proof of [2, Theorem 2], Bożejko constructed functions $\varphi _n\in A(H)$ with finite supports $E_n\subseteq H \subseteq G$ such that $\Vert \varphi _n\Vert _{MA(H)}=1$ but $\Vert \varphi _n\Vert _{M^{\mathrm{cb }}A(H)}\ge C\sqrt{n}$, for some constant $C > 0$. Given $u\in A(G)$, we now have

$$\begin{aligned} \Vert \varphi _n u\Vert _{A(G)} = \Vert \varphi _n u|_{H}\Vert _{A(H)}\le \Vert u|_{H}\Vert _{A(H)}\le \Vert u\Vert _{A(G)}; \end{aligned}$$

for the last inequality we use the contractability of r. Thus, $\Vert \varphi _n\Vert _{MA(G)} \le 1$. Let $\tilde{\varphi }_n$ be the extension by zero of $\varphi _n$ to a function on G, $n\in \mathbb {N}$, and $\psi _n = N(\tilde{\varphi }_n)$. By [3], $\psi _n$ is a Schur multiplier, $n\in \mathbb {N}$, and

$$\begin{aligned} \Vert \varphi _n\Vert _{M^{\mathrm{cb}}A(G)} = \Vert S_{\psi _n}\Vert _{\mathrm{cb}} \ge \Vert S_{\psi _n\chi _{H\otimes H}}\Vert _\mathrm{cb} = \Vert \varphi _n\Vert _{M^{\mathrm{cb}}A(H)}\ge C\sqrt{n}. \end{aligned}$$

This shows that the norms $\Vert \cdot \Vert _{MA(G)}$ and $\Vert \cdot \Vert _{M^{\mathrm{cb}}A(G)}$ are not equivalent on A(G). As the completely bounded norm dominates the multiplier norm, by applying the open mapping theorem one obtains that $A_{\mathrm{cb}}(G) \ne A_M(G)$. By Corollary 3.9 and Remark 3.10, there exists a compact multiplier which is not completely compact. $\square$

We remark that, for the free group $\mathbb {F}_2$ on two generators, the inequality $A_{\mathrm{cb}}(\mathbb {F}_2)\ne A_M(\mathbb {F}_2)$ was proved in [4, Proposition 4.2].

4 Subclasses of Completely Compact Herz–Schur Multipliers

In this section, we exhibit some canonical ways to construct completely compact Herz–Schur multipliers of dynamical systems, and describe them explicitly in an important special case. We assume throughout the section that $(A,G,\alpha )$ is a C*-dynamical system, where G is a discrete group. A linear map $T: A\rightarrow A$ will be called $\alpha$-invariant if

$$\begin{aligned} \alpha _t\circ T = T\circ \alpha _t, \ \ \ t\in G. \end{aligned}$$

(16)

Note that, if $T \in \mathrm{CB}(A)$ is $\alpha$-invariant then

$$\begin{aligned} (T\otimes \mathrm{id}_{\mathcal {B}(\ell ^2(G))})(\pi (a)) = \pi (T(a)), \ \ \ a\in A. \end{aligned}$$

It follows that the map $T\otimes \mathrm{id}_{\mathcal {B}(\ell ^2(G))}\in \mathrm{CB}(\mathcal {B}(\mathcal {H}))$ leaves $A\rtimes _{\alpha ,r}G$ invariant, and hence its restriction to $A\rtimes _{\alpha ,r}G$, which will be denoted by $\tilde{T}$, is well-defined. We have that

$$\begin{aligned} \tilde{T}(\pi (a)\lambda _t) = \pi (T(a))\lambda _t, \ \ \ a\in A, t\in G. \end{aligned}$$

(17)

Thus, letting $F_T : G\rightarrow \mathrm{CB}(A)$ be the function given by $F_T(s) = T$, $s\in G$, we have that $\tilde{T} = S_{F_T}$, and that $F_T(s)$ is $\alpha$-invariant for every $s\in G$. This leads us to introducing the following classes of Herz–Schur multipliers:

$$\begin{aligned} \mathfrak {S}^{\mathrm{inv}}(A,G,\alpha ) = \left\{ F\in \mathfrak {S}(A,G,\alpha ) : F(s) \text{ is } \alpha {-invariant for every } s\in G\right\} , \end{aligned}$$

and

$$\begin{aligned} \mathfrak {S}^{\mathrm{inv}}_{\mathrm{cc}}(A,G,\alpha ) = \mathfrak {S}^{\mathrm{inv}}(A,G,\alpha )\cap \mathfrak {S}_{\mathrm{cc}}(A,G,\alpha ). \end{aligned}$$

We remark that $\mathfrak {S}^{\mathrm{inv}}(A,G,\alpha )$ should not be confused with $\mathfrak {S}_{\mathrm{inv}}(G,G,A)$ which appears in [25] and denotes the so-called invariant Schur-A-multipliers (see the remark after Theorem 2.1); while invariance is a property of the map $\varphi : G\times G\rightarrow \mathrm{CB}(A)$, $\alpha$-invariance is a property of the maps F(s), $s\in G$.

We will denote by Z(A) the centre of A, and by $Z(A)^+$ the cone of its positive elements. Recall that the action of a discrete group G on A is amenable [5, Definition 4.3.1] if there exists a net $(\xi _i)_{i\in \mathbb {I}}$ of finitely supported functions $\xi _i : G\rightarrow Z(A)^+$, such that $\sum _{i\in \mathbb {I}}\xi _i(t)^2 =1_A$ and

$$\begin{aligned} \left\| \sum _{s\in G} \left( \xi _i(s) - \alpha _t(\xi _i(t^{-1}s))\right) ^2\right\| \rightarrow _{i\in \mathbb {I}} 0 \end{aligned}$$

for every $t\in G$.

Theorem 4.1

Let $\alpha$ be an amenable action of a discrete group G on a unital C*-algebra A. Let $u\in M^{\mathrm{cb}}A(G)$, $T\in \mathrm{CB}(A)$ be $\alpha$-invariant and $F_{T,u} : G\rightarrow \mathrm{CB}(A)$ be given by $F_{T,u}(t)(a) = u(t)T(a)$, $a\in A$, $t\in G$. The following hold:

• (i) $F_{T,u}\in \mathfrak {S}^{\mathrm{inv}}(A,G,\alpha )$;

• (ii) $F_{T,u}\in \mathfrak {S}_{\mathrm{cc}}^{\mathrm{inv}}(A,G,\alpha )$ if $u\in A_{\mathrm{cb}}(G)$ and $T\in \mathrm{CC}(A)$;

• (iii) $F_{T,u}\in \overline{\mathrm{F}(A\rtimes _{\alpha ,r}G)\cap \mathfrak {S}(A,G,\alpha )}^{\mathrm{cb}}$ if $u\in A_{\mathrm{cb}}(G)$ and $T\in \overline{\mathrm{F}(A)}^{\mathrm{cb}}$.

Proof

(i) Let $\tilde{u} : G\rightarrow \mathrm{CB}(A)$ be given by $\tilde{u}(t) = u(t)\mathrm{id}_A$. The map $S_{\tilde{u}}$ is the restriction to $A\rtimes _{\alpha ,r}G$ of the map $\mathrm{id}_{\mathcal {B}(H)}\otimes S_u\in \mathrm{CB}(\mathcal {B}(H)\otimes _{\min } C_r^*(G))$ and hence $\tilde{u}\in \mathfrak {S}(A,G,\alpha )$. Since $\tilde{T} = S_{F_T}$ and $F_T\in \mathfrak {S}^{\mathrm{inv}}(A,G,\alpha )$, we conclude that

$$\begin{aligned} F_{T,u} = \tilde{T} \circ S_{\tilde{u}} \in \mathfrak {S}^{\mathrm{inv}}(A,G,\alpha ). \end{aligned}$$

(ii) Since the space of finitely supported functions in A(G) is dense in A(G) [12] and $\Vert \cdot \Vert _{\mathrm{m}}\le \Vert \cdot \Vert _{A}$, given $\varepsilon > 0$, there exists $v\in A(G)$ supported in a finite set $E\subseteq G$ such that $\Vert u-v\Vert _{\mathrm{m}} < \varepsilon /\Vert T\Vert _{\mathrm{cb}}$. Let $\mathcal {F}\subseteq A$ be a finite dimensional space such that $\text {dist}(T^{(n)}(a), M_n(\mathcal {F})) < \varepsilon /|E|$ for every $a\in M_n(A)$, $\Vert a\Vert \le 1$, and every $n\in {\mathbb {N}}$.

Set

$$\begin{aligned} \mathcal {Y} = \left\{ \sum _{s\in E}\pi (a_s)\lambda _s: a_s\in \mathcal {F}\right\} \text{ and } \mathcal {Z} = \left\{ \sum _{s\in E}\pi (a_s)\lambda _s: a_s\in A\right\} ; \end{aligned}$$

we have that $\mathcal {Y}$ is a finite-dimensional subspace of $\mathcal {Z}$. Let $x = [x_{k,l}]\in M_n(\mathcal {Z})$, $\Vert x\Vert \le 1$, and write $L_s = \text {diag}(\lambda _s,\ldots ,\lambda _s)\in M_n(\mathcal {Z})$. Then

$$\begin{aligned}{}[x_{k,l}] = \sum _{s\in E} (\pi \circ \mathcal {E})^{(n)}\left( [x_{k,l}]L_{s^{-1}}\right) L_s; \end{aligned}$$

as $\mathcal {E}$ is completely contractive, $\left\| \mathcal {E}^{(n)}\left( [x_{k,l}]L_{s^{-1}}\right) \right\| \le 1$. For each $s\in E$, choose $y_s = [y_{k,l}^s]\in M_n(\mathcal {F})$ such that

$$\begin{aligned} \left\| T^{(n)}\left( \mathcal {E}^{(n)}\left( [x_{k,l}]L_{s^{-1}}\right) \right) - y_s\right\| < \varepsilon /|E| \end{aligned}$$

(18)

and set $y = \sum _{s\in E} \pi ^{(n)}(y_s) L_s\in M_n(\mathcal {Y})$. By (18),

$$\begin{aligned} \left\| \tilde{T}^{(n)}\left( [x_{k,l}]\right) -y\right\| = \left\| \sum _{s\in E}\left[ T\left( \mathcal {E}\left( x_{k,l}\lambda _{s^{-1}}\right) \right) \lambda _s-y_{k,l}^s\lambda _s\right] \right\| \le \varepsilon . \end{aligned}$$

Thus, $\tilde{T}|_{\mathcal {Z}}$ is completely compact. Since the image of $S_v$ is in ${\mathcal {Z}}$ we obtain that $\tilde{T}\circ S_v$ is completely compact. The statement now follows from the inequalities

$$\begin{aligned} \Vert \tilde{T}\circ S_v - \tilde{T}\circ S_u\Vert _{\mathrm{cb}} \le \Vert \tilde{T}\Vert _{\mathrm{cb}} \Vert S_v - S_u\Vert _{\mathrm{cb}} = \Vert \tilde{T}\Vert _\mathrm{cb} \Vert u - v\Vert _{\mathrm{m}} \le \varepsilon \end{aligned}$$

and the fact that the space of completely compact maps is closed with respect to the completely bounded norm.

(iii) Let $(T_k)_{k\in \mathbb {N}}\subseteq \mathrm{F}(A)$ be a sequence such that $\Vert T_k - T\Vert _{\mathrm{cb}}\rightarrow _{k\rightarrow \infty } 0$. We follow the idea in the proof of [26, Corollary 4.6]. Let $(\xi _i)_{i\in I}$ be a net as in definition of amenable action and the maps $F_{i,k}(s) : A\rightarrow A$, $s\in G$, $i\in \mathbb {I}$, $k\in \mathbb {N}$, be given by

$$\begin{aligned} F_{i,k}(s)(a) = \sum _{q\in G} \xi _i(q)\alpha _q(T_k(\alpha _q^{-1}(a))\alpha _s(\xi _i(s^{-1}q)), \ \ \ a\in A. \end{aligned}$$

(19)

Note that if $E_i\subseteq G$ is a finite set with $\mathrm{supp}\xi _i\subseteq E_i$ then the sum on the right hand side of (19) is over the (finite) set $E_i\cap sE_i$, for every $k\in \mathbb {N}$. We have

$$\begin{aligned} \mathcal {N}(F_{i,k})(s,t)(a)= & {} \alpha _{t^{-1}}\left( \sum _{q\in G} \xi _i(q)\alpha _q(T_k(\alpha _q^{-1}(\alpha _t(a)))\alpha _{ts^{-1}}(\xi _i(st^{-1}q))\right) \\= & {} \sum _{q\in G} \alpha _{t^{-1}}(\xi _i(q))\alpha _{t^{-1}q} (T_k(\alpha _{q^{-1}t}(a)))\alpha _{s^{-1}}(\xi _i(st^{-1}q))\\= & {} \sum _{p\in G} \alpha _{t^{-1}}(\xi _i(tp))\alpha _{p} (T_k(\alpha _{p^{-1}}(a)))\alpha _{s^{-1}}(\xi _i(sp)). \end{aligned}$$

Since $T_k : A\rightarrow A$ is a completely bounded map, by Haagerup-Paulsen-Wittstock Theorem, there exist a Hilbert space $H_{p,k}$, a *-representation $\pi _{p,k} : A\rightarrow \mathcal {B}(H_{p,k})$ and bounded operators $V_{p,k}$, $W_{p,k} : H \rightarrow H_{p,k}$, such that

$$\begin{aligned} \alpha _p(T_k(\alpha _p^{-1}(a))) = W_{p,k}^*\pi _{p,k}(a)V_{p,k}, \ \ \ a\in A, \end{aligned}$$

and $\Vert T_k\Vert _{\mathrm{cb}} = \Vert \alpha _p\circ T_k\circ \alpha _p^{-1}\Vert _{\mathrm{cb}} = \Vert V_{p,k}\Vert \Vert W_{p,k}\Vert$. Renormalising we may assume that $\Vert V_{p,k}\Vert =\Vert W_{p,k}\Vert =\Vert T_k\Vert _{\mathrm{cb}}^{1/2}$ for all p.

Set $\rho _k := \oplus _{p\in G}\pi _{p,k}$ and let $\mathbf{V}_{i,k}(s) : H\rightarrow \oplus _{p\in G} H_{p,k}$ and $\mathbf{W}_{i,k}(t) : H\rightarrow \oplus _{p\in G} H_{p,k}$ be the column operators given by

$$\begin{aligned} \mathbf{V}_{i,k}(s) = (V_{p,k}\alpha _s^{-1}(\xi _i(sp)))_{p\in G} \ \text { and } \ \mathbf{W}_{i,k}(t) = (W_{p,k}\alpha _t^{-1}(\xi _i(tp)))_{p\in G}. \end{aligned}$$

Then

$$\begin{aligned} \mathcal {N}(F_{i,k})(s,t)(a) = \mathbf{W}_{i,k}^*(t)\rho _k(a)\mathbf{V}_{i,k}(s) \end{aligned}$$

and

$$\begin{aligned} \Vert \mathbf{W}_{i,k}(t)\Vert \Vert \mathbf{V}_{i,k}(s)\Vert= & {} \left\| \sum _{p\in G}\alpha _{t^{-1}}(\xi _i(tp))W_{p,k}^*W_{p,k}\alpha _{t^{-1}}(\xi _i(tp))\right\| ^{1/2}\\&\times \left\| \sum _{p\in G}\alpha _{s^{-1}}(\xi _i(sp))V_{p,k}^*V_{p,k}\alpha _{s^{-1}}(\xi _i(sp))\right\| ^{1/2}\\\le & {} \Vert T_k\Vert _{\mathrm{cb}} \left\| \sum _{p\in G}\alpha _{s^{-1}}(\xi _i(sp))^2\right\| \\= & {} \Vert T_k\Vert _{\mathrm{cb}} \left\| \sum _{p\in G}\xi _i(p)^2\right\| = \Vert T_k\Vert _{\mathrm{cb}}. \end{aligned}$$

Hence, by [25, Theorem 2.6], $\mathcal {N}(F_{i,k})$ is a Schur A-multiplier, for all $i\in \mathbb {I}$ and all $k\in \mathbb {N}$. By Theorem 2.1, $F_{i,k}$ is a Herz–Schur $(A,G,\alpha )$-multiplier; moreover, as the sequence $(\Vert T_k\Vert _{\mathrm{cb}})_{k\in \mathbb {N}}$ is bounded, there exists $C > 0$ such that $\Vert S_{F_{i,k}}\Vert _{\mathrm{cb}} \le C$, $i\in \mathbb {I}$, $k\in \mathbb {N}$.

Next we prove that $\Vert F_{i,k}(s) - T\Vert _{\mathrm{cb}}\rightarrow 0$ for each $s\in G$. As T is $\alpha$-invariant, if $a = [a_{r,l}]\in M_n(A)$ then

$$\begin{aligned}&\Vert F_{i,k}(s)^{(n)}(a)-T^{(n)}(a)\Vert \\= & {} \left\| \sum _{p\in G}[\xi _i(p)\alpha _s(\xi _i(s^{-1}p))\alpha _p(T_k(\alpha _p^{-1}(a_{r,l})))- \alpha _p(T(\alpha _p^{-1}(a_{r,l})))]\right\| \\\le & {} \sum _{p\in G} \left\| \xi _i(p)\alpha _s(\xi _i(s^{-1}p))\right\| \left\| [\alpha _p(T_k(\alpha _p^{-1}(a_{r,l})) - \alpha _p(T(\alpha _p^{-1}(a_{r,l})))]\right\| \\+ & {} \left\| \sum _{p\in G} \xi _i(p)\alpha _s(\xi _i(s^{-1}p))-1\right\| \left\| T^{(n)}(a)\right\| \\\le & {} \left\| \sum _{p\in G}\xi _i(p)^2\right\| \left\| T_k - T\right\| _{\mathrm{cb}} \Vert a\Vert + \left\| \sum _{p\in G} \xi _i(p)\alpha _s(\xi _i(s^{-1}p))-1\right\| \Vert T\Vert _{\mathrm{cb}}\Vert a\Vert , \end{aligned}$$

where in the last line we used the Cauchy-Schwarz inequality and the fact $\xi _i(p) \in Z(A)^+$, $p\in G$. By [5, Lemma 4.3.2], $\Vert \sum _p\xi _i(p)\alpha _s(\xi _i(s^{-1}p)) - 1\Vert \rightarrow _{i\in \mathbb {I}} 0$; the facts that $\Vert T_k - T\Vert _{\mathrm{cb}}\rightarrow _{k\rightarrow \infty } 0$ and $\sum _{p\in G}\xi _i(p)^2 = 1$ now imply that

$$\begin{aligned} \Vert F_{i,k}(s)-T\Vert _{\mathrm{cb}}\rightarrow _{(i,k)\in \mathbb {I}\times \mathbb {N}} 0, \ \ \ s\in G. \end{aligned}$$

(20)

Since $T_k\in \mathrm{F}(A)$ and the maps $F_{i,k}$ are finitely supported, the map $S_{F_{i,k}}$ on $A\rtimes _{\alpha ,G} G$ has finite rank. In order to prove the statement, it hence suffices to show that

$$\begin{aligned} \Vert S_{F_{i,k}}\circ S_u - S_{F_{T,u}}\Vert _{\mathrm{cb}}\rightarrow _{(i,k)\in \mathbb {I}\times \mathbb {N}} 0. \end{aligned}$$

(21)

Let $\varepsilon > 0$. Since $u\in A_{\mathrm{cb}}(G)$, there exists $v\in A(G)$ with finite support E such that

$$\begin{aligned} \Vert S_u - S_v\Vert _{\mathrm{cb}} \le \Vert u - v\Vert _{A} \le \varepsilon . \end{aligned}$$

Set $\mathcal {Z} = \left\{ \sum _{s\in E}\pi (a_s)\lambda _s: a_s\in A\right\}$ and let $x = [x_{r,l}]\in M_n(\mathcal {Z})$. As in the proof of (ii), write $L_s$ for $\text {diag}(\lambda _s,\ldots ,\lambda _s)\in M_n(\mathcal {Z})$. Then $x = \sum _{s\in E}(\pi \circ \mathcal {E})^{(n)}(x L_{s^{-1}})L_s$ and

$$\begin{aligned} \Vert S_{F_{i,k}}^{(n)}(x) - S_T^{(n)}(x)\Vert = \Vert \sum _{s\in E}[\pi ((F_{i,k}(s) - T)(\mathcal {E}(x_{r,l}\lambda _{s^{-1}})))\lambda _s]\Vert \\ \le \sum _{s\in E}\Vert [(F_{i,k}(s) - T)(\mathcal {E}(x_{r,l}\lambda _{s^{-1}})]\Vert \le \sum _{s\in E}\Vert (F_{i,k}(s) - T)^{(n)}\Vert \Vert [\mathcal {E}(x_{r,l}\lambda _{s^{-1}})]\Vert \\ \le \sum _{s\in E}\Vert (F_{i,k}(s) - T)^{(n)}\Vert \Vert x\Vert \le |E| \Vert x\Vert (\text {max}_{s\in E} \Vert F_{i,k}(s)^{(n)} - T^{(n)}\Vert ). \end{aligned}$$

By (20),

$$\begin{aligned} \Vert (S_{F_{i,k}} - S_T)|_{\mathcal {Z}}\Vert _{\mathrm{cb}}\rightarrow _{(i,k) \in \mathbb {I}\times \mathbb {N}} 0. \end{aligned}$$

We now have

$$\begin{aligned}&\Vert S_{F_{i,k}}\circ S_u - S_{F_{T,u}}\Vert _{\mathrm{cb}}\\&\quad \le \Vert S_{F_{i,k}}\circ (S_u - S_v)\Vert _{\mathrm{cb}} + \Vert S_{F_{i,k}}\circ S_v - S_T \circ S_v\Vert _{\mathrm{cb}} + \Vert S_T \circ (S_v - S_u)\Vert _{\mathrm{cb}}\\&\quad \le \Vert (S_{F_{i,k}} - S_T)|_{\mathcal {Z}}\Vert _{\mathrm{cb}}\Vert v\Vert + \varepsilon (\Vert S_{F_{i,k}}\Vert _{\mathrm{cb}} + \Vert S_T\Vert _{\mathrm{cb}}), \end{aligned}$$

which implies (21). $\square$

Theorem 4.1 exhibits a large class of elements of $\mathfrak {S}^{\mathrm{inv}}_{\mathrm{cc}}(A,G,\alpha )$. In the next theorem, we provide a precise description of the latter class of Herz–Schur multipliers in the case of the irrational rotation algebra. Let $\theta \in {\mathbb {R}}$ be irratrional and $\alpha : \mathbb {Z}\rightarrow \mathrm{Aut}(C({\mathbb {T}}))$ be given by

$$\begin{aligned} \alpha _n(f)(z)=f(e^{2\pi in\theta }z), \ \ \ f\in C({\mathbb {T}}), z\in {\mathbb {T}}. \end{aligned}$$

Let $M({\mathbb {T}})$ denote the Banach algebra of all complex Borel measures on the unit circle $\mathbb {T}$, and note that $L^1(\mathbb {T})$ is a (closed) ideal of $M({\mathbb {T}})$. For $\mu \in M({\mathbb {T}})$, let $T_{\mu } : C({\mathbb {T}})\rightarrow C({\mathbb {T}})$ be the completely bounded map given by $T_\mu (f) = \mu *f$. Note that $T_{\mu }$ is $\alpha$-invariant; indeed,

$$\begin{aligned} \alpha _n(\mu *f)(z)= & {} (\mu *f)(e^{2\pi in\theta }z)=\int f(e^{2\pi in\theta }w^{-1}z)d\mu (w)\\= & {} \int \alpha _n(f)(w^{-1}z)d\mu (w)=\mu *\alpha _n(f)(z). \end{aligned}$$

Theorem 4.2

The functions $F_{T_f,u}$, where $f\in L^1(\mathbb {T})$ and $u\in A(\mathbb {Z})$, have a dense linear span in $\mathfrak {S}^{\mathrm{inv}}_\mathrm{cc}(A,G,\alpha )$.

Proof

Let F be a completely compact Herz–Schur $(C({\mathbb {T}}),\mathbb Z,\alpha )$-multiplier such that $F(n)\circ \alpha _m = \alpha _m\circ F(n)$ for all $m,n\in {\mathbb {Z}}$. We show that $F(n) = T_{f_n}$ for some $f_n\in L^1({\mathbb {T}})$. In fact, let $\Phi _n : M({\mathbb {T}})\rightarrow M({\mathbb {T}})$ be the dual map of F(n). As $\alpha _n(f)(z) = f(e^{2\pi in\theta }z) = (\delta _{e^{-2\pi in\theta }}*f)(z)$, where $\delta _s$ is the point mass measure at $s\in {\mathbb {T}}$, we obtain

$$\begin{aligned} \left\langle \Phi _n(\delta _{e^{2\pi im\theta }}*\mu ), f\right\rangle= & {} \left\langle \delta _{e^{2\pi im\theta }}*\mu , F(n)(f)\right\rangle = \left\langle \mu ,\alpha _{-m}(F(n)(f))\right\rangle \\= & {} \left\langle \mu ,F(n)(\alpha _{-m}(f))\right\rangle = \left\langle \delta _{e^{2\pi im\theta }}*\Phi _n(\mu ),f\right\rangle , \end{aligned}$$

giving

$$\begin{aligned} \Phi _n(\delta _{e^{2\pi im\theta }}*\mu )=\delta _{e^{2\pi im\theta }}*\Phi _n(\mu ), \ \ m,n\in \mathbb {Z}, \mu \in M({\mathbb {T}}). \end{aligned}$$

In particular, $\Phi _n(\delta _{e^{2\pi im\theta }})=\delta _{e^{2\pi im\theta }}*\Phi _n(\delta _1)$. Using the weak* continuity of $\Phi _n$, the density of $\{e^{2\pi im\theta }: m\in {\mathbb {Z}}\}$ in ${\mathbb {T}}$ and the fact that every measure $\mu$ is a weak* limit of linear combinations of point mass measures, we obtain

$$\begin{aligned} \Phi _n(\mu ) = \mu *\Phi _n(\delta _1), \ \ \ \mu \in M({\mathbb {T}}). \end{aligned}$$

It now follows that, if $\mu _n = \Phi _n(\delta _1)$ then $F(n)(f) = \mu _n*f$, $f\in C({\mathbb {T}})\simeq C_r^*({\mathbb {Z}})$. By Corollary 3.4, F(n) is completely compact for every $n\in \mathbb {Z}$; by Corollary 3.11, $\mu _n\in L^1({\mathbb {T}})$.

Fix $F\in \mathfrak {S}^{\mathrm{inv}}_{\mathrm{cc}}(A,G,\alpha )$; then F(n) is $\alpha$-invariant for every $n\in \mathbb {Z}$. We show that $S_F\in \overline{[S_u\circ S_{F(n)}: u\in A({\mathbb {Z}}), n\in \mathbb Z]}^{\Vert \cdot \Vert _{\mathrm{cb}}}$. By the amenability of ${\mathbb {Z}}$, there exists a bounded sequence $(u_i)_{i\in \mathbb {N}}\subseteq A(\mathbb Z)$ of finitely supported functions such that $S_{u_i}\rightarrow \mathrm{id}$ in the strong point norm topology (see [17, Theorem 1.12, Theorem 1.9]). Since $u_i$ is finitely supported,

$$\begin{aligned} S_{u_i}\circ S_F\in \mathrm{span}\{S_u\circ S_{F(n)}: u\in A({\mathbb {Z}}), n\in {\mathbb {Z}}\}. \end{aligned}$$

Since $S_F$ is completely compact, Lemma 3.1 implies $\Vert S_{u_i}\circ S_F - S_F\Vert _{\mathrm{cb}}\rightarrow _{i\rightarrow \infty } 0$. $\square$

5 Herz–Schur Multipliers of $\mathcal {K}$

Let G be a discrete group and $\alpha : G\rightarrow \mathrm{Aut}(c_0(G))$ be the homomorphism given by $\alpha _t(f)(s) = f(t^{-1}s)$, $s\in G$, $f\in c_0(G)$. For $a\in c_0(G)$, we write $M_a$ for the operator on $\ell ^2(G)$ given by $(M_a\xi )(s) = a(s) \xi (s)$, $\xi \in \ell ^2(G)$, $s\in G$, and let

$$\begin{aligned} \mathcal {C} = \{M_a : a\in c_0(G)\}; \end{aligned}$$

the map $\iota : c_0(G)\rightarrow \mathcal {C}$ given by $\iota (a) = M_a$ is a *-isomorphism. By abuse of notation, we write $\alpha _t$ for the corresponding automorphism of $\mathcal {C}$. Recall that, for $t\in G$, we denote by $\lambda _t^0$ the left regular unitary acting on $\ell ^2(G)$. The pair $(\iota ,\lambda ^0)$ of representations is covariant, and hence gives rise to a (faithful) representation $\iota \rtimes \lambda ^0 : \mathcal {C}\rtimes _{\alpha ,r} G \rightarrow \mathcal {B}(\ell ^2(G))$. According to the Stone-von Neumann Theorem [38, Theorem 4.24], the image of $\iota \rtimes \lambda ^0$ coincides with the C*-algebra $\mathcal {K}$ of all compact operators on $\ell ^2(G)$. Thus, the Herz–Schur $(\mathcal {C},G,\alpha )$-multipliers give rise, in a canonical way, to a certain class of completely bounded maps on $\mathcal {K}$. The aim of this section is to formalise this correspondence and examine the complete compactness of the resulting maps on $\mathcal {K}$.

Set $\mathcal {B} = (\pi \rtimes \lambda )(\mathcal {C}\rtimes _{\alpha ,r} G)$; thus $\mathcal {B}$ is a C*-subalgebra of $\mathcal {B}(\ell ^2(G\times G))$. By [38, Theorem 4.24], the map

$$\begin{aligned} \theta : \pi (a)\lambda _t \longrightarrow M_a \lambda _t^0, \ \ \ a\in \mathcal {C}, t\in G, \end{aligned}$$

is a *-isomorphism from $\mathcal {B}$ onto $\mathcal {K}$. For an element $\Phi \in \mathrm{CB}(\mathcal {K})$, let $\widetilde{\Phi } = \theta ^{-1}\circ \Phi \circ \theta$; thus, $\widetilde{\Phi }\in \mathrm{CB}(\mathcal {B})$, and the correspondence $\Phi \longrightarrow \widetilde{\Phi }$ between $\mathrm{CB}(\mathcal {K})$ and $\mathrm{CB}(\mathcal {B})$ is bijective. For a Herz–Schur $(\mathcal {C},G,\alpha )$-multiplier F, let $\Phi _F = \theta \circ S_F \circ \theta ^{-1}$ note that $\Phi _F\in \mathrm{CB}(\mathcal {K})$ and

$$\begin{aligned} \Phi _F(M_a \lambda _t^0) = M_{F(t)(a)} \lambda _t^0, \ \ \ a\in \mathcal {C}, t\in G. \end{aligned}$$

We call the maps of the form $\Phi _F$ the Herz–Schur $\mathcal {K}$-multipliers. We let

$$\begin{aligned} \mathfrak {S}(\mathcal {K}) = \{\Phi _F : F \text{ is } \text{ a } \text{ Herz--Schur } (\mathcal {C},G,\alpha )\text{-multiplier }\}, \end{aligned}$$

and $\mathfrak {S}_{\mathrm{cc}}(\mathcal {K}) = \mathfrak {S}(\mathcal {K})\cap \mathrm{CC}(\mathcal {K})$. We note that the elements of $\mathfrak {S}(\mathcal {K})$ are precisely those completely bounded maps on $\mathcal {K}$ which leave its diagonals globally invariant, when elements of $\mathcal {K}$ are viewed as $G\times G$-matrices.

We recall that, if K is a Hilbert space and $\mathcal {A}\subseteq \mathcal {B}(K)$ is a C*-algebra, the Haagerup tensor product $\mathcal {A}\otimes _{\mathrm{h}}\mathcal {A}$ consists of (convergent) series $u = \sum _{i=1}^{\infty } a_i\otimes b_i$, where $(a_i)_{i\in \mathbb {N}}\subseteq \mathcal {A}$ and $(b_i)_{i\in \mathbb {N}}\subseteq \mathcal {A}$ are sequences for which the series $\sum _{i=1}^{\infty } a_i a_i^*$ and $\sum _{i=1}^{\infty } b_i^* b_i$ converge in norm. Each such u gives rise to a completely bounded map $\Gamma _u : \mathcal {B}(K)\rightarrow \mathcal {B}(K)$ given by

$$\begin{aligned} \Gamma _u(x) = \sum _{i=1}^\infty a_ixb_i, \ \ \ x\in \mathcal {B}(K). \end{aligned}$$

The following fact was noted in [16, Corollary 3.6].

Theorem 5.1

Any completely compact map on $\mathcal {K}$ has the form $\Gamma _u$ for some element $u \in \mathcal {K}\otimes _{\mathrm{h}}\mathcal {K}$.

Theorem 5.2

The mapping

$$\begin{aligned} \Phi \longrightarrow \Phi _{F_{\widetilde{\Phi }}} \end{aligned}$$

(22)

is a linear contractive surjection

• (i) from $\mathrm{CB}(\mathcal {K})$ onto $\mathfrak {S}(\mathcal {K})$;

• (ii) from $\mathrm{CC}(\mathcal {K})$ onto $\mathfrak {S}_{\mathrm{cc}}(\mathcal {K})$.

Proof

(i) By Proposition 3.3 (i), if $\Phi \in \mathrm{CB}(\mathcal {K})$ then $F_{\widetilde{\Phi }}$ is a Herz–Schur $(\mathcal {C},G,\alpha )$-multiplier and $\Vert F_{\widetilde{\Phi }}\Vert _{\mathrm{m}}\le \Vert \Phi \Vert _{\mathrm{cb}}$. Thus, $\Phi _{F_{\widetilde{\Phi }}}\in \mathfrak {S}(\mathcal {K})$ and $\Vert \Phi _{F_{\widetilde{\Phi }}}\Vert _{\mathrm{cb}}\le \Vert \Phi \Vert _{\mathrm{cb}}$. On the other hand, if F is a Herz–Schur $(\mathcal {C},G,\alpha )$-multiplier then, by Proposition 3.3 (ii), $F_{S_F} = F$ and hence $\Phi _F$ is the image of itself under the map (22). The linearity of the map (22) is straighforward.

(ii) Let $\Phi \in \mathrm{CC}(\mathcal {K})$. Since $\mathcal {K}$ has SOAP, Lemma 3.1 shows that $\Phi$ can be approximated in the completely bounded norm by finite rank maps $\Phi _n : \mathcal {K}\rightarrow \mathcal {K}$, $n\in \mathbb {N}$. Clearly, ${\tilde{\Phi }}_n$ has finite rank, $n\in \mathbb {N}$. As in the proof of Lemma 3.6, we can assume that the maps ${\tilde{\Phi }}_{n}$ have range in $\text {span}\{\pi (a)\lambda _s: s\in G, a\in \mathcal {C}\}$. By Proposition 3.3 (iv), $S_{F_{{\tilde{\Phi }}_n}}$ has finite rank and hence so has $\Phi _{F_{{\tilde{\Phi }}_n}}$. Moreover, by Proposition 3.3 (i), $\Vert F_{{\tilde{\Phi }}}-F_{{\tilde{\Phi }}_n}\Vert _{\mathrm{m}}\rightarrow 0$, giving $\Vert \Phi _{F_{{\tilde{\Phi }}}} - \Phi _{F_{{\tilde{\Phi }}_n}}\Vert _{\mathrm{cb}} \rightarrow _{n\rightarrow \infty } 0$. Hence $\Phi _{F_{{\tilde{\Phi }}}}$ is completely compact. The rest is similar to (i). $\square$

For $b\in \mathcal {K}$ and $s\in G$, let $b_s = \lambda _s^0b\lambda _{s^{-1}}^0$. For $u = \sum _{i=1}^\infty a_i\otimes b_i\in \mathcal {K}\otimes _{\mathrm{h}}\mathcal {K}$, write $u_s = \sum _{i=1}^\infty a_i\otimes (b_i)_s$; it is clear that $u_s$ is a well-defined element of $\mathcal {K}\otimes _{\mathrm{h}}\mathcal {K}$.

Theorem 5.3

Let $F : G\rightarrow \mathrm{CB}(\mathcal {C})$. The following are equivalent:

• (i) F is a completely compact Herz–Schur $(\mathcal {C}, G,\alpha )$-multiplier;

• (ii) there exists $u \in \mathcal {K}\otimes _{\mathrm{h}}\mathcal {K}$ such that $\Gamma _{u_s}(a) = F(s)(a)$, for all $a \in \mathcal {C}$ and $s\in G$.

Proof

(i)$\Rightarrow$(ii) The map $M_a\lambda _s^0\mapsto M_{F(s)(a))}\lambda _s^0$ extends to a completely compact map on $\mathcal {K}$. By Theorem 5.1, there exists $u = \sum _{i=1}^\infty a_i\otimes b_i\in \mathcal {K}\otimes _{\mathrm{h}}\mathcal {K}$ such that

$$\begin{aligned} M_{F(s)(a)}\lambda _s^0 = \sum _{i=1}^\infty a_iM_a \lambda _s^0 b_i = \sum _{i=1}^\infty a_iM_a\lambda _s^0 b_i (\lambda _s^0)^*\lambda _s^0, \ \ \ a\in \mathcal {C}. \end{aligned}$$

Thus, $\mathcal {C}$ is an invariant subspace for $\Gamma _{u_s}$ and $\Gamma _{u_s}(a) = F(s)(a)$, $a\in \mathcal {C}$.

(ii)$\Rightarrow$(i) By the previous paragraph, $\Gamma _u(M_a\lambda _s^0) = \Gamma _{u_s}(M_a)\lambda _s^0$. Thus, $\theta ^{-1}\circ \Gamma _u \circ \theta$ coincides with $S_F$ and so F is a completely compact Herz–Schur multiplier. $\square$

Suppose that F is a Herz Schur $(\mathcal {C},G,\alpha )$-multiplier of the form $F(s)(M_a) = M_{h_s a}$, where $h_s : G\rightarrow \mathbb {C}$ is a function, $s\in G$. We can associate with it the function $\psi : G\times G\rightarrow \mathbb {C}$, given by $\psi (s,t) = h_s(t)$. This is the identification made the next statement.

Corollary 5.4

Let $\varphi : G\times G \rightarrow \mathbb {C}$ be a compact Schur multiplier. Then the function $\psi : G\times G \rightarrow \mathbb {C}$, given by $\psi (s,t) = \varphi (t,s^{-1}t)$, is a completely compact Herz Schur $(\mathcal {C},G,\alpha )$-multiplier.

Proof

By [19], there exists an element $u = \sum _{i=1}^\infty a_i\otimes b_i \in c_0(G)\otimes _{\mathrm{h}} c_0(G)$ such that $\varphi (p,q) = \sum _{i=1}^\infty a_i(p) b_i(q)$, $p,q\in G$. By the injectivity of the Haagerup tensor product [10, Proposition 9.2.5], $u\in \mathcal {K}\otimes _{\mathrm{h}}\mathcal {K}$. Note that

$$\begin{aligned} \Gamma _{u_s}(M_a)(t) = \sum _{i=1}^\infty a_i(t) b_i(s^{-1}t)a(t), \ \ \ a\in c_0(G), t\in G; \end{aligned}$$

in other words, $\Gamma _{u_s}(M_a) = M_{h_s a}$, where $h_s(t) = \varphi (t,s^{-1}t)$, $t\in G$. The conclusion follows from Theorem 5.3. $\square$

6 Some Remarks and Open Questions

In Corollary 3.11, we showed that the amenability of a discrete group G is a sufficient condition for automatic complete compactness: every compact multiplier is in this case completely compact. For such automatic complete compactness it suffices, instead of amenability, to assume that the completely bounded multiplier norm is equivalent to the multiplier norm. By a result of Losert [23], there exist non-amenable groups such that $M^{\mathrm{cb}}A(G) = MA(G)$, for instance $SL(2,{\mathbb {R}})$. However we do not know whether there exists a discrete group with this property.

In Proposition 3.12, we exhibited an example of a multiplier $u\in MA(G)$, for which the map $S_u$ is compact but not completely compact. This multiplier however may not be completely bounded, as we only guarantee the boundedness of $S_u$ if $u\in MA(G)$. We do not know if there exists a completely bounded compact multiplier which is not completely compact.

Finally, in Corollary 3.9, we showed that if G is a discrete group possessing property (AP) then every completely compact multiplier on G is the limit, in the completely bounded norm, of finitely supported multipliers. It would be interesting to know if (AP) is in fact equivalent to the latter approximation property; we do not know if this holds true.