Chapter 4 Symmetry in a groupoid setting
4.1 Groupoids and groupoid actions
Definition 4.1 A groupoid is a small category in which every morphism is an isomorphism, i.e. is invertible.
The reference for the following more detailed form of the definition is (Kumjian 1998).
Definition 4.2 A groupoid \(\Xi\) is a set endowed with two maps \[\label{eq:groupoidOperations} \cdot:{\Xi}^{(2)}\rightarrow\Xi, \,\,\,\,\,\,\,\, ^* : \Xi\rightarrow \Xi\] called multiplication and inverse respectively, where \(\Xi^{(2)}\) is a subset of \(\Xi\times\Xi\) called the set of composable elements. These operations satisfy the following properties:
\((\gamma^*)^*=\gamma\) for every \(\gamma\in\Xi\),
if \((\gamma,\alpha),(\alpha,\beta)\in\Xi^{(2)}\) then \((\gamma\alpha,\beta),(\gamma,\alpha\beta)\in\Xi^{(2)}\) and \((\gamma\alpha)\beta=\gamma(\alpha\beta)\),
if \((\gamma^*,\gamma)\in\Xi^{(2)}\) and if \((\gamma,\alpha)\in\Xi^{(2)}\) then \(\gamma^*(\gamma\alpha)=\alpha\),
if \((\gamma,\gamma^*)\in\Xi^{(2)}\) and if \((\beta,\gamma)\in\Xi^{(2)}\) then \((\beta\gamma)\gamma^*=\beta\).
If \(\gamma\in\Xi\), \(\textsf{s}(\gamma)=\gamma^*\gamma\) and
\(\textsf{r}(\gamma)=\gamma\gamma^*\) are the domain and range of \(\gamma\)
respectively. The pair \((\gamma, \alpha)\) is composable if and only if
\(\textsf{s}(\gamma)=\textsf{r}(\alpha)\). We define \(\Xi^{(0)}:=\textsf{s}(\Xi)=\textsf{r}(\Xi)\) as the
unit space of \(\Xi\).
Usually a groupoid is denoted as
\[\Xi \stackrel{\textsf{s}\text{,}\textsf{r}}{\overrightarrow{\rightarrow}} \Xi^{(0)}.\]
Other useful subset of the groupoid \(\Xi\) is the isotropy group of a
given \(v\in\Xi^{(0)}\) which it is defined by
\[\Xi_v^v:=\left\{\gamma\in\Xi|\,\, \textsf{s}(\gamma)=\textsf{r}(\gamma)=v\right\}.\]
The isotropy group of \(v\) if often seen as \(\Xi_v^v=\Xi_v\cap\Xi^v\)
where \(\Xi_v:=\textsf{s}^{-1}(\{v\})\) and \(\Xi^v:=\textsf{r}^{-1}(\{v\})\) and these are
called the \(\textsf{s}\)-fibre and \(\textsf{r}\)-fibre respectively. One also uses the
isotropic subgroupoid of \(\Xi\) written as
\(\text{Iso}(\Xi):=\cup_{v\in\Xi^{(0)}}\Xi_v^v\).
4.2 Fell bundles
Definition 4.3 Let \(X\) be a Hausdorff topological space. A bundle over \(X\) is a pair \(\big<\mathscr{C},\pi\big>\), where \(\mathscr{C}\) is a Hausdorff topological space and \(\pi:\mathscr{C}\to X\) is a continuous open surjection.
The concept of Fell bundles over groups introduced here can be found in (Exel 2017 Def. 16.1).
Definition 4.4 A Fell Bundle (also known as a C\(^*\)-algebraic bundle) over a group \({\textsf{G}}\) is a bundle \(\mathscr{C}=\big<\mathscr C,\pi\big>\) over \({\textsf{G}}\), where we define \(\mathfrak{C}_g:=\pi^{-1}(\{g\})\) as the fibers. In addition it is equipped with continuous multiplication and involution operations \[\cdot:\mathscr{C}\times\mathscr{C}\rightarrow\mathscr{C}, \,\,\,\,\,\,\,\, ^* : \mathscr{C}\rightarrow \mathscr{C},\] satisfying the following properties for all \(g,h\in{\textsf{G}}\) and all \(b,c\in\mathscr{C}\)
\(\mathfrak{C}_g\cdot \mathfrak{C}_h \subset \mathfrak{C}_{gh}\),
the product \(\cdot\) is bilinear on \(\mathfrak{C}_g\times \mathfrak{C}_h\) to \(\mathfrak{C}_{gh}\),
the product \(\cdot\) on \(\mathscr{C}\) is associative,
\(\Vert b \cdot c \Vert_{\pi(bc)} \leq \Vert b \Vert_{\pi(b)} \Vert c \Vert_{\pi(c)}\),
\((\mathfrak{C}_g)^*\subset \mathfrak{C}_{g^{-1}}\),
For each \(g\in{\textsf{G}}\), \(^*\) restricted to \(\mathfrak{C}_g\) is conjugate-linear from \(\mathfrak{C}_g\) to \(\mathfrak{C}_{g^{-1}}\),
\((bc)^*=c^*b^*\),
\(b^{**}=b\),
\(\Vert b^* \Vert_{\pi(b^*)}=\Vert b\Vert_{\pi(b)}\),
\(\Vert b^*b \Vert_{e}=\Vert b\Vert^2_{\pi(b)}\),
\(b^*b\geq 0\) in \(\mathfrak{C}_\textsf{e}\).
We will write the Fell bundle in the form \(\mathscr{C}:=(\mathfrak{C}_g)_{g\in{\textsf{G}}}\). It is important to note that \(\mathfrak{C}_\textsf{e}\) is a C\(^*\)-algebra with the restricted operations, it is often refered as the unit fiber algebra (Exel 2017, 124).
Consider the Fell Bundle \(\mathscr{C}=\big<\mathscr C,\pi\big>\) over the locally compact Hausdorff group \({\textsf{G}}\). On the space \[\textsf{C}_c(\textsf{G},\mathscr{C}):=\left\{ \varphi:\textsf{G}\to\mathscr{C}\,\, \text{ continuous };\,\, \varphi(g)\in \mathfrak{C}_g\,\,\forall g\in\textsf{G} \,\, \textrm{ s.t. supp}(\varphi) \textrm{ is compact} \right\},\] one defines the convolution product \[\varphi\star\psi(g):= \int_\textsf{G} \varphi(gh^{-1}) \psi(h)dh.\] Then we define \(L^1(\textsf{G},\mathscr{C})\) as a Banach \(^*\)-algebra by completing \(\textsf{C}_c(\textsf{G},\mathscr{C})\) with respect to the norm \[\Vert \varphi \Vert_{L^1(\textsf{G},\mathscr{C})}:=\sum_{g\in\textsf{G}}\Vert\varphi(g)\Vert_{\mathfrak{C}_g}\,\, \textrm{ for }\,\, \varphi\in \textsf{C}_c(\textsf{G},\mathscr{C}).\]
4.3 Fell bundles over groupoids
We introduce the concept of Fell bundles over groupoids which can be found in (Kumjian 1998; Muhly and Williams 2008; Yamagami 1990).
Definition 4.5 A Fell Bundle (also known as a C\(^*\)-algebraic bundle) over a discrete groupoid \(\Xi\) is a bundle \(\mathscr{C}=\big<\mathscr C,\pi\big>\) over \(\Xi\), where we define \(\mathfrak{C}_{\gamma}:=\pi^{-1}(\{\gamma\})\) as the fibers and we set \[\mathscr{C}^{(2)}:=\left\{(c_1,c_2)\in\mathscr{C}\times\mathscr{C};\,\,(\pi(c_1),\pi(c_2))\in\Xi^{(2)}\right\}.\] In addition it is equipped with continuous multiplication and involution operations \[\cdot:\mathscr{C}^{(2)}\rightarrow\mathscr{C}, \,\,\,\,\,\,\,\, ^* : \mathscr{C}\rightarrow \mathscr{C},\] satisfying the following properties
\(\mathfrak{C}_{\gamma_1}\cdot \mathfrak{C}_{\gamma_2} \subset \mathfrak{C}_{\gamma_1\gamma_2}\), if \((\gamma_1,\gamma_2)\in\Xi^{(2)}\),
the product \(\cdot\) is bilinear on \(\mathfrak{C}_{\gamma_1}\times \mathfrak{C}_{\gamma_2}\) for every \((\gamma_1,\gamma_2)\in\Xi^{(2)}\),
the product \(\cdot\) on \(\mathscr{C}\) is associative, this means \((c_1 \cdot c_2) \cdot c_3 = c_1 \cdot( \cdot c_2 \cdot c_3 )\) in \(\mathfrak{C_{\gamma_1\gamma_2\gamma_3}}\) if \((c_1,c_2)\in \mathscr{C}^{(2)}\) and \((c_2,c_3)\in \mathscr{C}^{(2)}\), where \(c_i\in\mathscr{C}_{\gamma_i}\) for \(i=1,2,3\),
\(\Vert c_1 \cdot c_2 \Vert_{\mathfrak{C}_{\gamma_1\gamma_2}} \leq \Vert c_1 \Vert_{\mathfrak{C}_{\gamma_1}} \Vert c_2 \Vert_{\mathfrak{C}_{\gamma_2}}\) if \((c_1,c_2)\in\mathscr{C}^{(2)}\),
\((\mathfrak{C}_\gamma)^*\subset \mathfrak{C}_{\gamma^*}\) for every \(\gamma\in\Xi\), which could be written as \(\pi(c^*)=\pi(c)^*\) for every \(c\in\mathscr{C}\),
For each \(\gamma\in\Xi\), \(^*\) restricted to \(\mathfrak{C}_{\gamma}\) is conjugate-linear from \(\mathfrak{C}_{\gamma}\) to \(\mathfrak{C}_{\gamma^*}\),
\((c_1c_2)^*=c_2^*c_1^*\) for every \((c_1,c_2)\in\mathscr{C}^{(2)}\),
\(c^{**}=c\) for every \(c\in\mathscr{C}\),
\(\Vert c^* \Vert_{\mathfrak{C}_{\gamma}^*}=\Vert c\Vert_{\mathfrak{C}_{\gamma}}\) for every \(c\in\mathfrak{C}_\gamma\),
\(\Vert c^*\cdot c \Vert_{\mathfrak{C}_{\gamma^*\gamma}}=\Vert c\Vert_{\mathfrak{C}_\gamma}^2\) for every \(c\in\mathfrak{C}_\gamma\),
\(c^*c\geq 0\) in \(\mathfrak{C}_{\gamma^*\gamma}\) for every \(c\in\mathfrak{C}_\gamma\).
We will continue to write the Fell bundle in the form \(\mathscr{C}:=(\mathfrak{C}_g)_{g\in\Xi}\).
Remark 4.1 Note that if \(\gamma\in\Xi^{(0)}\) then \(\mathfrak{C}_\gamma:=\pi^{-1}(\{\gamma\})\) is a C\(^*\)-algebra (with norm, multiplication and involution induced from the bundle).
4.4 Rigidly symmetric and hypersymmetric groupoids
Let \(\Xi\) be a discrete groupoid, with unit spce \(\Xi^{(0)}\!=:\!U\) , domain and range maps \(\textsf{s}\) and \(\textsf{r}\) respectively. The set of composable pairs is \(\Xi^{(2)}\!:=\{(x,y)\!\mid\!\textrm{ r}(y)=\textrm{ d}(x)\}\) .
Definition 4.6 Let \(\Gamma:\mathfrak{B}\rightarrow \textrm{End}(\mathcal{E})\) be a representation of the Banach \(^*\)-algebra \(\mathfrak{B}\) in the Banach space \(\mathcal{E}\). The representation is called preunitary if there exists a Hilbert space \(\mathcal{H}\), a topologically irreducible \(^*\)-representation \(\Pi:\mathfrak{B}\rightarrow \mathbb{B}(\mathcal{H})\) and an injective linear bounded operator \(V:\mathcal{E}\rightarrow \mathcal{H}\) such that \[V\Gamma(\phi)=\Pi(\phi)V,\,\,\forall \phi\in\mathfrak{B}.\]
Our interest in this notion lies in the next characterization, taken from (Leptin 1976):
Theorem 4.1 The Banach \(^*\)-algebra \(\mathfrak{B}\) is symmetric if and only if all its non-trivial algebraically irreducible representations are preunitary.
An interesting object on the representation of Fell Bundles is the Hahn algebra \(\ell^{\infty,1}(\Xi\!\mid\!\mathscr C)\) adapted to this structure (Muhly and Williams 2008 Ch. 1), which in our discrete case is formed by the sections \(\Phi:\Xi\to\mathfrak C\) (thus satisfying \(\Phi(x)\in\mathfrak C_x\) for every \(x\in\Xi\)) such that the Hahn-type norm \[\begin{equation} \,\big\Vert\,\!\Phi\,\big\Vert\,_{\infty,1}\,:=\max\Big\{\sup_{u\in U}\sum_{\textrm{ r}(x)=u}\!\,\big\Vert\,\!\Phi(x)\!\,\big\Vert\,_{\mathfrak C_x}\,,\,\sup_{u\in U}\sum_{\textrm{ d}(x)=u}\!\,\big\Vert\,\!\Phi(x)\!\,\big\Vert\,_{\mathfrak C_x}\!\Big\} \tag{4.1} \end{equation}\] is finite. It is a Banach \(^*\)-algebra under the multiplication \[\begin{equation} (\Phi\star \Psi)(x):=\sum_{yz=x}\Phi(y)\bullet\Psi\big(z) \tag{4.2} \end{equation}\] and the involution \[\begin{equation} \Phi^\star(x):=\Phi\big(x^{-1}\big)^\bullet. \tag{4.3} \end{equation}\] The space \(C_\textrm{ c}(\Xi\!\mid\!\mathscr C)\) of finitely-supported sections forms a dense \(^*\)-algebra of the Hahn algebra. The complexity of the multiplication, largely responsable for the generality of the emerging algebras, comes both from the complexity of the ‘inner’ Fell multiplication \(\bullet\) and from the groupoid-type convolution inherent to the formula.
We need some special Fell bundles associated to Hilbert bundles \(\mathscr H\!:=\bigsqcup_{u\in U}\mathcal{H}_u\) over the unit space; here the fact that \(\Xi\) is discrete will be crucial. For \(u,v\in U\) we set \(\mathbb B(\mathcal{H}_u,\mathcal{H}_v)\equiv\mathbb B(u,v)\) for the Banach space of all bounded linear operators \(A:\mathcal{H}_u\to\mathcal{H}_v\) . Taking advantage of the norm, the multiplication \[\begin{aligned} \mathbb B(\mathcal{H}_w,\mathcal{H}_v)\!\times\!\mathbb B(\mathcal{H}_u,\mathcal{H}_w)&\to\mathbb B(\mathcal{H}_u,\mathcal{H}_v)\\ (A,B) &\mapsto A\circ B \end{aligned}\] and the involution \[\begin{aligned} \mathbb B(\mathcal{H}_u,\mathcal{H}_v)&\to\mathbb B(\mathcal{H}_v,\mathcal{H}_u)\\ A&\mapsto A^*, \end{aligned}\] one constructs the Fell bundle \[\mathbb B^\mathscr H\!:=\!\!\bigsqcup_{(u,v)\in U\times U}\!\mathbb B(u,v)\to U\!\!\times\!U\] over the pair groupoid.
Actually we are interested in the Fell Bundle \[\begin{aligned} \mathscr B^\mathscr H\!&:=\!\bigsqcup_{x\in\Xi}\mathfrak B_x\overset{p}{\to}\Xi\,, \end{aligned}\] with fibres \(\mathfrak{B}_x:=\mathbb{B}\big(\textsf{s}(x),\textsf{r}(x)\big)\) and the structure \[(A_x,B_y)\in \left(\mathscr{\mathscr B^\mathscr H}\right)^{(2)} \quad \quad \text{ if and only if }\quad \quad s(x)=r(y),\] where we can see \(A_x\cdot B_y\in \mathfrak{B}_{xy}=\mathbb{B}(s(y),r(x))\). The Fell Bundle \(\mathscr B^\mathscr H\) is called the pull-back Fell bundle of \(\mathbb B^\mathscr H\) through the groupoid morphism \((\textsf{s},\textsf{r}):\Xi\to U\!\times\!U\) which it is also denoted by \((\textsf{s},\textsf{r})^*(\mathbb B^\mathscr H)\) (see (Kumjian 1998 Rem. 2.6)).
Definition 1.2
The discrete groupoid \(\Xi\) is called symmetric if the convolution Banach \(^*\)-algebra \(\ell^{\infty,1}(\Xi)\) is symmetric.
The discrete groupoid \(\Xi\) is called rigidly symmetric if, given any Hilbert bundle \(\mathscr H\!=\bigsqcup_{x\in X}\mathcal{H}_x\) over its unit space, the Banach \(^*\)-algebra \(\ell^{\infty,1}\big(\Xi\,\big\vert\,\mathscr B^\mathscr H\big)\) is symmetric.
The discrete groupoid \(\Xi\) is called hypersymmetric if given any Fell bundle \(\mathscr C\!=\bigsqcup_{x\in\Xi}\mathfrak C _x\) , the Banach \(^*\)-algebra \(\ell^{\infty,1}(\Xi\!\mid\!\mathscr C)\) is symmetric.
It is clear that rigid symmetry implies symmetry; just take the constant field \(\mathcal{H}_u\!:=\mathbb C\) for every \(u\) . As said in the Introduction, even for discrete groups it is still unknown if the two notions coincide.
::: {.theorem #principala} For a discrete groupoid, rigid symmetry and hypersymmetry coincide. :::
Proof. The fact that hypersymmetry implies rigid symmetry is obvious, since \(\mathscr B^\mathscr H\) is a particular type of discrete groupoid Fell bundle. So we need to show the converse implication.
We will first prove a result concerning the existence of isometric representations of such Fell bundles. We rely on the integrated form of such a representation. In general the connections between representations and their integrated forms is an intricate issue (see (Muhly and Williams 2008 Sect. 3,4,5) for instance), especially at the level of ‘disintegration’, but for discrete \(\Xi\) this simplifies a lot. We will sketch the constructions we need without saying how they fit the general case. But just a hint: the counting measure on \(U\) is invariant with respect to the Haar system composed of counting measures on the fibers of the discrete groupoid. So one is given an arbitrary Fell bundle \(\mathscr C=\bigsqcup_{x\in\Xi}\mathfrak C _x\overset{q}{\to}\Xi\) . Let \[\pi:\mathscr C\to\mathscr B^\mathscr H\!=\bigsqcup_{x\in\Xi}\mathbb B\big(\mathcal{H}_{\textrm{ s}(x)},\mathcal{H}_{\textrm{ r}(x)}\big)\] be a representation, where \(\mathscr H=\bigsqcup_{u\in U}\mathcal{H} _u\) is a Hilbert bundle over \(U\) . In our context this just means that \(\pi\) is a morphism of Fell bundles. Its integrated form \(\Pi:\ell^{\infty,1}(\Xi\!\mid\!\mathscr C)\to\mathbb B\Big(\bigoplus_{u\in U}\mathcal{H}_u\big)\) is defined by \[\begin{equation} \big[\Pi(\Phi)h\big](u):=\!\sum_{\textrm{ r}(x)=u}\!\pi\big[\Phi(x)\big]h\big[\textrm{ s}(x)\big]\,,\quad\forall\,h\in\bigoplus_{u\in U}\mathcal{H}_u\,. \tag{4.4} \end{equation}\] We embed \(\mathfrak C_v\) into \(C_\textrm{ c}(\Xi\!\mid\!\mathscr C)\subset\ell^{\infty,1}(\Xi\!\mid\!\mathscr C)\) setting for each \(a\in\mathfrak C_v\) \[(\theta_v a)(x):=a\ \ \textrm{ if}\ \ x=v\,,\quad(\theta_v a)(x):=0_{\mathfrak C_x}\ \ \textrm{ if}\ \ x\ne v\,\] and we get from ((4.4)) \[\big[\Pi(\theta_v a)h\big](u)=\pi(a)h(v)\ \ \textrm{ if}\ \ u=v\,,\quad\big[\Pi(\theta_v a)h\big](u)=0_{\mathcal{H}_u}\ \ \textrm{ if}\ \ u\ne v\,.\] It follows immediately that if \(\Pi\) is injective, then the restriction of \(\pi\) to \(\mathfrak C_v\) is also injective, i. e. isometric. Actually \(\Pi\) extends to the full \(C^*\)-algebra \(C^*(\Xi\!\mid\!\mathscr C)\) of the Fell bundle, which contains densely \(\ell^{\infty}(\Xi\!\mid\!\mathscr C)\) . Injective representations of \(C^*\)-algebras do exist; we conclude that the Fell bundle representation \(\pi\) is isometric on the \(C^*\)-algebras corresponding to the units. But then, by a standard argument, the isometry also propagates on all the Banach spaces of the Fell bundle: if \(b\in\mathfrak C_x\) the axioms allow us to write \[\,\big\Vert\,\!\pi(b)\!\,\big\Vert\,_{\mathfrak B_x}^2=\,\,\big\Vert\,\!\pi(b)^*\pi(b)\!\,\big\Vert\,_{\mathfrak B_{\textrm{ s}(x)}}=\,\,\big\Vert\,\!\pi\big(b^\bullet\bullet b\big)\!\,\big\Vert\,_{\mathfrak B_{\textrm{ s}(x)}}=\,\,\big\Vert\,\!b^\bullet\bullet b\!\,\big\Vert\,_{\mathfrak C_{\textrm{ s}(x)}} =\,\,\big\Vert\,\!b\!\,\big\Vert\,_{\mathfrak C_x}^2.\]
Now define \[\Upsilon_\pi\!:\ell^{\infty,1}(\Xi\!\mid\!\mathscr C)\to\ell^{\infty,1}\big(\Xi\,\big\vert\,\mathscr B^\mathscr H\big)\,,\quad\big(\Upsilon_\pi(\Phi)\big)(x)\!:=\pi\big(\Phi(x)\big)\,.\] It is a well-defined linear isometry: we compute for the range part of the norm ((4.1)) (using more detailed notations); the same is true for the source part. \[\begin{aligned} \,\big\Vert\,\!\Upsilon_\pi(\Phi)\!\,\big\Vert\,^{\textrm{ r}}_{\ell^{\infty,1}(\Xi\,\vert\,\mathscr B^\mathscr H)}\,&=\sup_u\!\sum_{\textrm{ r}(x)=u}\!\,\big\Vert\,\!\pi\big(\Phi(x)\big)\!\,\big\Vert\,_{\mathfrak B_x}\\ &=\sup_u\!\sum_{\textrm{ r}(x)=u}\!\,\big\Vert\,\!\Phi(x)\!\,\big\Vert\,_{\mathfrak C_x}=\,\,\big\Vert\,\!\Phi\!\,\big\Vert\,^{\textrm{ r}}_{\ell^{\infty,1}(\Xi\mid\mathscr C)}. \end{aligned}\] It is also an involutive morphism. For the multiplication, for instance, we have: \[\begin{aligned} \big[\Upsilon_\pi(\Phi)\star\Upsilon_\pi(\Psi)\big](\Xi)&=\sum_{yz=x}\big[\Upsilon_\pi(\Phi)\big](y)\big[\Upsilon_\pi(\Psi)\big](z)\\ &=\sum_{yz=x}\pi[\Phi(y)]\pi[\Psi(z)]\\ &=\pi\Big(\sum_{yz=x}\Phi(y)\bullet\Psi(z)\Big)\\ &=\big[\pi(\Phi\star\Psi)\big](x)\\ &=\big[\Upsilon_\pi(\Phi\star\Psi)\big](x)\,. \end{aligned}\]Thus we proved that \(\ell^{\infty,1}(\Xi\!\mid\!\mathscr C)\) can be embedded as a closed \(^*\)-algebra of the symmetric Banach \(^*\)-algebra \(\ell^{\infty,1}\big(\Xi\,\big\vert\,\mathscr B^\mathscr H\big)\) , so it is also symmetric, by (Palmer 2001, 2:Th.11.4.2). The proof is finished. ◻
Example 4.1 If \(\Xi\equiv\textsf{ H}\) is a discrete group with unit \(\textsf{e}\), one recovers the first part of Theorem (2.1) (the second one may be found in (Jauré and Măntoiu 2022)). In this case, since \(U=\{\textsf{e}\}\) , the \(\ell^{\infty,1}\)-algebras reduce to the usual \(\ell^1\)-algebras associated to Fell bundles with discrete groups (Fell and Doran 1988). The Hilbert bundle reduces to a single Hilbert space \(\mathcal{H}_{\textsf{e}}=:\mathcal{H}\) , the Fell bundle \(\mathscr B^\mathscr H\) is only composed of \(\mathbb B(\mathcal{H})\) and then \(\ell^{\infty,1}\big(\textsf{H}\,\big\vert\,\mathscr B^\mathscr H\big)\) becomes \(\ell^1\big(\textsf{H},\mathbb B(\mathcal{H})\big)\) , which is isomorphic to the projective tensor product \(\ell^1(\textsf{H})\!\otimes\!\mathbb B(\mathcal{H})\) . We recover in the case the classical notion of rigidly symmetric group; see (Poguntke 1992). This explains a posteriori our terminology.