1 Introduction

Given two \(C^*\)-algebras \({{\mathfrak {A}}}\) and \({{\mathfrak {B}}}\), their algebraic tensor product \({{\mathfrak {A}}}\odot {{\mathfrak {B}}}\) can easily be endowed with a natural structure of \(*\)-algebra. However, it is a well-known fact that in general \({{\mathfrak {A}}}\odot {{\mathfrak {B}}}\) can be completed to a \(C^*\)-algebra in several different ways. Phrased differently, more than one \(C^*\)-norm can be introduced on \({{\mathfrak {A}}}\odot {{\mathfrak {B}}}\). As is known, among the possible norms, the so-called maximal and minimal norms play a privileged role in that any other \(C^*\)-norm is bounded between this two, see e.g. [13]. However, when at least one of the two given \(C^*\)-algebras is nuclear, there is by definition precisely one way to complete \({{\mathfrak {A}}}\odot {{\mathfrak {B}}}\), namely all \(C^*\)-norms on it agree with each other. For instance, commutative \(C^*\)-algebras and approximately finite-dimensional \(C^*\)-algebras are all examples of nuclear algebras, see [13]. Tensor products of \(C^*\)-algebras is an accomplished theory where virtually anything is known, although further pieces of information have been added until very recent times, see e.g. [9,10,11].

When one starts adding more structure, such as a grading, possibly novel aspects may and will occur. A case in point is given by \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras, which in the physicists’ parlance are often referred to as superalgebras, cf. [8]. In particular, graded tensor products of \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras have recently been given a good deal of attention in [3], where a definition of quantum detailed balance for product systems endowed with a \({{\mathbb {Z}}}_2\)-grading has been proposed. Among other things, the aforementioned paper provides an in-depth analysis of the maximal norm on the graded tensor products of \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras, as well as showing that a product state on the algebraic Fermi product of two \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras is well defined whenever just one of the two states is even, namely it is invariant with respect to the grading. This last result is actually a generalization of the analogous result obtained in [1] for the pivotal example of the CAR algebra, see also [2] and [7].

In this paper, instead, much of the attention is lavished on the so-called spatial norm. In particular, we prove that it is the smallest of all compatible norms. To make this statement more precise, we need to single out the notion of a compatible norm on the algebraic Fermi product. This is by definition a norm such that the grading can be extended to its completion. As a matter of fact, the problem of deciding whether any \(C^*\)-norm on a \({{\mathbb {Z}}}_2\)-graded product has revealed delicate to handle, not least because producing counterexamples turns out to be as delicate. Indeed, norms other than the maximal and the minimal one are typically obtained by means of abstract methods, and therefore are difficult to explicitly compute with.

Our analysis of compatible norms also requires to make intensive use of extreme even states, namely the extreme points of the (weakly*) compact convex set of all even states. Extreme even states for \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras parallel pure states for general \(C^*\)-algebras. Indeed, they are sufficiently many to separate the elements of the \(C^*\)-algebras. Furthermore, they return the set of all pure states when the grading is trivial.

The paper is organized as follows. In Sect. 2 we set the notation and recall the basic notions on graded \(C^*\)-algebras and their graded tensor products, which we also refer to as Fermi tensor products, as is done in [3]. We then focus on even states on a given graded \(C^*\)-algebra. In particular, we also provide a description of the extreme even states of a commutative graded \(C^*\)-algebra. We end the section by showing that a product state is even if and only if both of its marginal states are.

In Sect. 3 we consider \({{\mathbb {Z}}}_2\)-graded Hilbert spaces and their Fermi tensor product as a spatial counterpart of the construction for abstract \(C^*\)-algebras. We introduce a notion of Fermi product of grading-equivariant representations and show that the GNS representation of the product state of two even states is nothing but the Fermi product of the two GNS representations.

Finally, in Sect. 4, after recalling the definition of the spatial norm, we first show that it is a cross norm, as is the maximal norm. We then move on to prove that the maximal and minimal norms are compatible with the grading. In Proposition 4.10 we prove that the Fermi product of two \(C^*\)-algebras can be endowed with only one \(C^*\)-norm when one of the two is commutative. In Theorem 4.12 we prove that the spatial norm is actually minimal among all compatible norms.

As an outlook for the foreseeable future, we believe the present study might be a first step towards addressing distributional symmetries on probability spaces based on graded \(C^*\)-algebras, as has already been done for the CAR algebra in [4,5,6], and for any \(C^*\)-algebra (with trivial grading) in [12], where it is the spatial norm to play a key role.

2 On tensor products of \(*\)-algebras

In this section, we collect some results on \({{\mathbb {Z}}}_2\)-graded algebraic structures obtained as tensor product of graded \(*\)-algebras. We first observe that, if not otherwise specified, throughout the paper all the structures we deal with will be taken unital.

If \({{\mathfrak {B}}}\subset {{\mathfrak {A}}}\) is an inclusion of unital \(C^*\)-algebras with a common unity, the unital linear mapping \(E:{{\mathfrak {A}}}\rightarrow {{\mathfrak {B}}}\) is called a projection if \(E(b)=b\) for all \(b\in {{\mathfrak {B}}}\), and is said to be a \({{\mathfrak {B}}}\)-bimodule map if \(E(ab)=E(a)b\), and \(E(ba)=bE(a)\) for all \(a\in {{\mathfrak {A}}}, b\in {{\mathfrak {B}}}\). A positive \({{\mathfrak {B}}}\)-bimodule projection E is called a conditional expectation.

Consider the \(C^*\)-algebras \({{\mathfrak {A}}}_1\) and \({{\mathfrak {A}}}_2\), and denote by \({{\mathfrak {A}}}_1\otimes {{\mathfrak {A}}}_2\) the algebraic tensor product \({{\mathfrak {A}}}_1\odot {{\mathfrak {A}}}_2\) with the product and involution given by

$$\begin{aligned} (a_1\otimes a_2)(a_1'\otimes a_2'):=a_1a_1'\otimes a_2a_2',\quad (a_1\otimes a_2)^*:=a_1^*\otimes a_2^*, \end{aligned}$$

for all \(a_1,a_1'\in {{\mathfrak {A}}}_1\), \(a_2,a_2'\in {{\mathfrak {A}}}_2\). Let us denote by \({{\mathfrak {A}}}_1\otimes _{\max } {{\mathfrak {A}}}_2\) and \({{\mathfrak {A}}}_1\otimes _{\min } {{\mathfrak {A}}}_2\) the completion of \({{\mathfrak {A}}}_1\otimes {{\mathfrak {A}}}_2\) with respect to the maximal and minimal \(C^*\)-cross norm, respectively [13].

Let \({{\mathcal {S}}}({{\mathfrak {A}}})\) be the weak*-compact collecting the states on a \(C^*\)-algebra \({{\mathfrak {A}}}\). If one takes \(\omega _1\in {{\mathcal {S}}}({{\mathfrak {A}}}_1)\) and \(\omega _2\in {{\mathcal {S}}}({{\mathfrak {A}}}_2)\), their product state \(\psi _{\omega _1,\omega _2}\in {{\mathcal {S}}}({{\mathfrak {A}}}_1\otimes _{\min } {{\mathfrak {A}}}_2)\) is well defined also on \({{\mathfrak {A}}}_1\otimes _{\max } {{\mathfrak {A}}}_2\), and consequently the notation \(\psi _{\omega _1,\omega _2}\in {{\mathcal {S}}}({{\mathfrak {A}}}_1\otimes {{\mathfrak {A}}}_2)\) will be used in the sequel.

Consider now \({{\mathbb {Z}}}_2=\{1,-1\}\) with the product as the group operation, and a \(*\)-algebra \({{\mathfrak {A}}}\). The latter is called an involutive \({{\mathbb {Z}}}_2\)-graded algebra if it decomposes as

$$\begin{aligned} {{\mathfrak {A}}}={{\mathfrak {A}}}_1\oplus {{\mathfrak {A}}}_{-1} \end{aligned}$$

and

$$\begin{aligned} ({{\mathfrak {A}}}_i)^*=({{\mathfrak {A}}}^*)_i,\,\, {{\mathfrak {A}}}_i{{\mathfrak {A}}}_j\subset {{\mathfrak {A}}}_{ij},\quad i,j=1,-1. \end{aligned}$$

The subspaces \({{\mathfrak {A}}}_i\), \(i=1,2\) are called the homogeneous components of \({{\mathfrak {A}}}\), and correspondingly any element of \({{\mathfrak {A}}}_i\) is called a homogeneous element of \({{\mathfrak {A}}}\). For any homogeneous element \(x\in {{\mathfrak {A}}}_{\pm 1}\) we denote its grade by

$$\begin{aligned} \partial (x)=\pm 1. \end{aligned}$$

Assigning a \({{\mathbb {Z}}}_2\)-grading on \({{\mathfrak {A}}}\) is equivalent to equipping \({{\mathfrak {A}}}\) with an involutive \(*\)-automorphism \(\theta\) (i.e. \(\theta ^2=\mathrm{id}_{{\mathfrak {A}}}\)). Indeed, from one hand for a given \({{\mathbb {Z}}}_2\)-graded \(*\)-algebra \({{\mathfrak {A}}}\) one takes

$$\begin{aligned} \theta \lceil _{{{\mathfrak {A}}}_1}=\mathrm{id}_{{{\mathfrak {A}}}_1},\quad \theta \lceil _{{{\mathfrak {A}}}_{-1}}=-\mathrm{id}_{{{\mathfrak {A}}}_{-1}}. \end{aligned}$$

On the other hand, if \(\theta \in \mathrm{aut}({{\mathfrak {A}}})\) is such that \(\theta ^2=\mathrm{id}_{{\mathfrak {A}}}\), after taking

$$\begin{aligned} \varepsilon _1:=\frac{1}{2}(\mathrm{id}_{{{\mathfrak {A}}}}+\theta ), \quad \varepsilon _{-1}:=\frac{1}{2}(\mathrm{id}_{{{\mathfrak {A}}}}-\theta ), \end{aligned}$$

and denoting

$$\begin{aligned} {{\mathfrak {A}}}_1:=\varepsilon _1({{\mathfrak {A}}}),\quad {{\mathfrak {A}}}_{-1}:=\varepsilon _{-1}({{\mathfrak {A}}}), \end{aligned}$$

one gives \({{\mathfrak {A}}}_1\cap {{\mathfrak {A}}}_{-1}=\{0\}\). As a consequence, their direct sum \({{\mathfrak {A}}}={{\mathfrak {A}}}_1\oplus {{\mathfrak {A}}}_2\) is a \({{\mathbb {Z}}}_2\)-graded \(*\)-algebra.

It turns out that a \({{\mathbb {Z}}}_2\)-graded \(*\)-algebra is a pair \(({{\mathfrak {A}}},\theta )\), where \({{\mathfrak {A}}}\) is a \(*\)-algebra, and \(\theta\) an involutive \(*\)-automorphism on \({{\mathfrak {A}}}\).

Following [3], we say that \(\theta\) is a \({{\mathbb {Z}}}_2\)-grading of \({{\mathfrak {A}}}\). Moreover, we denote the \(*\)-subalgebra \({{\mathfrak {A}}}_+:={{\mathfrak {A}}}_1\) the even part, and the subspace \({{\mathfrak {A}}}_-:={{\mathfrak {A}}}_{-1}\) the odd part of \({{\mathfrak {A}}}\), respectively. Note that \(\varepsilon _1\) is a conditional expectation. Thus, for any \(a\in {{\mathfrak {A}}}\), we can write \(a=a_++a_-\), with \(a_+\in {{\mathfrak {A}}}_+\), \(a_-\in {{\mathfrak {A}}}_-\), and this decomposition is unique. In addition, one gets \(\theta (a_+)=a_+\), \(\theta (a_-)=-a_-\).

Taking \(\theta =\mathrm{id}_{{{\mathfrak {A}}}}\), one sees that any \(*\)-algebra \({{\mathfrak {A}}}\) is equipped with a \({{\mathbb {Z}}}_2\) trivial grading. Here, \({{\mathfrak {A}}}_+={{\mathfrak {A}}}\) and \({{\mathfrak {A}}}_-=\{0\}\).

A simple example of \({{\mathbb {Z}}}_2\)-graded \(*\)-algebra is obtained by taking an Hilbert space \({{\mathcal {H}}}\), and a bounded self-adjoint unitary U on \({{\mathcal {H}}}\).Footnote 1 The adjoint action \(\mathrm{ad}_{U}(\cdot ):=U \cdot U^*\) is an involutive \(*\)-automorphism which induces a \({{\mathbb {Z}}}_2\)-grading on \({{\mathcal {B}}}({{\mathcal {H}}})\). Another example of paramount importance for its applications to Physics is of course the CAR algebra, see e.g. [2, 3].

Let \(\big ({{\mathfrak {A}}}_i,\theta _i\big )\), \(i=1,2\), be a pair of \({{\mathbb {Z}}}_2\)-graded \(*\)-algebras. The map \(T:{{\mathfrak {A}}}_1\rightarrow {{\mathfrak {A}}}_2\) is said to be even if it is grading-equivariant:

$$\begin{aligned} T\circ \theta _1=\theta _2\circ T. \end{aligned}$$

When \(\theta _2=\mathrm{id}_{{{\mathfrak {A}}}_2}\), the map \(T:{{\mathfrak {A}}}_1\rightarrow {{\mathfrak {A}}}_2\) is even if and only if it is grading-invariant, that is \(T\circ \theta _1=T\). If T is \({{\mathbb {Z}}}_2\)-linear, then it is even if and only if \(T\lceil _{{{\mathfrak {A}}}_{1,-}}=0\). When \(\big ({{\mathfrak {A}}}_2,\theta _2\big )=\big ({{\mathbb {C}}},\mathrm{id}_{{\mathbb {C}}}\big )\), a functional \(f:{{\mathfrak {A}}}_1\rightarrow {{\mathbb {C}}}\) is even if and only if \(f\circ \theta =f\).

In the sequel, we will denote by \({{\mathcal {S}}}_+({{\mathfrak {A}}})\) the convex subset of even states. Even states play a role in giving a \({{\mathbb {Z}}}_2\)-grading on their GNS structures.

More in detail, suppose that \(({{\mathfrak {A}}},\theta )\) is a \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebra, and \(\varphi \in {{\mathcal {S}}}_+({{\mathfrak {A}}})\). Let \(({{\mathcal {H}}}_\varphi ,\pi _\varphi , \xi _\varphi , V_{\theta ,\varphi })\) be the GNS covariant representation of \(\varphi\), where the unitary self-adjoint \(V_{\theta ,\varphi }\) fixes \(\xi _\varphi\) and verifies

$$\begin{aligned} \pi _\varphi (\theta (a))=V_{\theta ,\varphi }\pi _\varphi (a)V_{\theta ,\varphi }, \quad a\in {{\mathfrak {A}}}. \end{aligned}$$

Then, \(({{\mathcal {B}}}({{\mathcal {H}}}),\mathrm{ad}_{V_{\theta ,\varphi }})\) is a \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebra.

It is well known that in the case of a \(C^*\)-algebra with trivial grading, the pure states separate the points and are exactly the extreme elements of the convex of the states. For similar results in presence of a nontrivial \({{\mathbb {Z}}}_2\)-grading, the notion of grading-invariant functionals is central. To this aim, take a \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebra \(({{\mathfrak {A}}}, \theta )\), and denote by \({{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}))\) the set of the extreme points of the weakly* compact convex set \({{\mathcal {S}}}_+({{\mathfrak {A}}})\) of all even states of \({{\mathfrak {A}}}\).

Proposition 2.1

For any given \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebra \(({{\mathfrak {A}}}, \theta )\), the states of \({{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}))\) separate \({{\mathfrak {A}}}\).

Proof

We first observe that \({{\mathcal {S}}}_+({{\mathfrak {A}}})\) separate \({{\mathfrak {A}}}\), since the conditional expectation \(\varepsilon _1\) from \({{\mathfrak {A}}}\) to \({{\mathfrak {A}}}_+\) is faithful. The statement is then an application of the Krein–Milman theorem.

Indeed, let a be a positive element in \({{\mathfrak {A}}}\) such that \(\omega (a)=0\) for any \(\omega \in {{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}))\). Then, for any convex combination \(\varphi\) of states in \({{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}))\), one has \(\varphi (a)=0\). Now, given any even state \(\eta\), there exists a net \(\{\eta _i\}_{i\in I}\), I being a directed set, for which \(\eta _i\) is a convex combination of states in \({{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}))\), and \(\eta _i\rightarrow _i \eta\) in the weak* topology. But then we have

$$\begin{aligned} \eta (a)=\lim _i \eta _i(a)=0. \end{aligned}$$

As the equality holds for all even states, it follows that \(a=0\). \(\square\)

Corollary 2.2

Given a \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebra \(({{\mathfrak {A}}}, \theta )\), for every \(a\in {{\mathfrak {A}}}\) one has

$$\begin{aligned} \Vert a\Vert =\sup \{\Vert \pi _\omega (a)\Vert :\omega \in {{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}))\}. \end{aligned}$$

Proof

Thanks to Proposition 2.1, the representation \(\oplus _{\omega \in {{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}))} \pi _\omega\) is faithful and thus isometric. This ends the proof. \(\square\)

Lemma 2.3

Let \(({{\mathfrak {A}}}, \theta )\) be a \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebra. Any state \(\omega\) on the even subalgebra \({{\mathfrak {A}}}_+\) admits exactly one even extension to \({{\mathfrak {A}}}\).

Proof

Clearly, \(\omega \circ \varepsilon _1\) extends \(\omega\) and is even by construction. The uniqueness of such an extension follows from the fact that even states vanish on \({{\mathfrak {A}}}_-\). \(\square\)

Proposition 2.4

Let \(({{\mathfrak {A}}}, \theta )\) be a \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebra. An even state \(\omega\) belongs to \({{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}))\) if and only if the restriction \(\omega \lceil _{{{\mathfrak {A}}}_+}\) is a pure state.

Proof

In light of Lemma 2.3 the map \(S: {{\mathcal {S}}}_+({{\mathfrak {A}}})\rightarrow {{\mathcal {S}}}({{\mathfrak {A}}}_+)\), given by \(S(\omega )=\omega \lceil _{{{\mathfrak {A}}}_+}\) for every \(\omega \in {{\mathcal {S}}}_+({{\mathfrak {A}}})\), establishes an affine bijection between the compact convex sets \({{\mathcal {S}}}_+({{\mathfrak {A}}})\) and \({{\mathcal {S}}}({{\mathfrak {A}}}_+)\). Therefore, it also establishes a bijection between the extreme points of the former set and the extreme points of the latter. \(\square\)

For commutative \(C^*\)-algebras, one can completely determine the extreme even states.

Proposition 2.5

For a (locally) compact Hausdorff space X, the extreme even states of the commutative \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebra \((C(X),\theta )\) are given by the set \(\big \{\frac{1}{2}(\delta _x+\delta _{\theta (x)})\mid x\in X\big \}\).

Proof

Consider the equivalence relation \(\sim\) on X for which \(x\sim y\) if and only if \(f(x)=f(y)\) for any \(f\in C(X)\) such that \(f= f\circ \theta\). If \(X\!/\sim\) is the quotient space, the even \(C^*\)-subalgebra \(C(X)_+:=\{f\in C(X)\mid f= f\circ \theta \}\) clearly identifies with \(C(X_+)\). Indeed, if \(\pi : X\rightarrow X_+\) is the natural projection, and \(\iota : C(X_+)\rightarrow C(X)\) is the isometric \(*\)-homomorphism given by \(\iota (f)=f\circ \pi\) for any \(f\in C(X_+)\), one has that \(C(X)_{+}= \iota (C(X_+))\). Thus, for any given \(x\in X\), the restriction of the even state \(\frac{1}{2}(\delta _x+\delta _{\theta (x)})\) to \(C(X)_+\) is pure, as it is nothing but \(\delta _{\pi (x)}\).

Conversely, if \(\omega \in {{\mathcal {E}}}({{\mathcal {S}}}_+(C(X))\), then there exists \(x\in X\) such that \(\omega \lceil _{C(X_+)}=\delta _{\pi (x)}\), which means \(\omega =\frac{1}{2}(\delta _x+\delta _{\theta (x)})\). \(\square\)

Suppose that \(({{\mathfrak {A}}}_1,\theta _1)\) and \(({{\mathfrak {A}}}_2,\theta _2)\) are \({{\mathbb {Z}}}_2\)-graded \(*\)-algebras, and consider the linear space \({{\mathfrak {A}}}_1\odot {{\mathfrak {A}}}_2\). In what follows, we recall the definition of the involutive \({{\mathbb {Z}}}_2\)-graded tensor product, which will be henceforth denoted by \({{\mathfrak {A}}}_1\, \textcircled {{F}}\, {{\mathfrak {A}}}_2\), as in [3]. For homogeneous elements \(a_1\in {{\mathfrak {A}}}_1\), \(a_2\in {{\mathfrak {A}}}_2\) and \(i,j\in {{\mathbb {Z}}}_2\), we set

$$\begin{aligned}\varepsilon (a_1,a_2):&=\left\{ \begin{array}{rl} -1 &{}\quad \text {if}\,\, \partial (a_1)=\partial (a_2)=-1,\\ 1 &{}\quad \text {otherwise}. \end{array} \right. \\\epsilon (i,j):&=\left\{ \begin{array}{rl} -1 &{}\quad \text {if}\,\, i=j=-1,\\ 1 &{}\quad \text {otherwise}. \end{array} \right. \end{aligned} $$

Given \(x,y\in {{\mathfrak {A}}}_1\odot {{\mathfrak {A}}}_2\) with

$$\begin{aligned} \begin{aligned}&x:=\oplus _{i,j\in {{\mathbb {Z}}}_2}x_{i,j}\in \oplus _{i,j\in {{\mathbb {Z}}}_2}({{\mathfrak {A}}}_{1,i}\odot {{\mathfrak {A}}}_{2,j}),\\&y:=\oplus _{i,j\in {{\mathbb {Z}}}_2}y_{i,j}\in \oplus _{i,j\in {{\mathbb {Z}}}_2}({{\mathfrak {A}}}_{1,i}\odot {{\mathfrak {A}}}_{2,j}), \end{aligned} \end{aligned}$$

the involution, which by a minor abuse of notation we continue to denote by \(^*\), and the multiplication on \({{\mathfrak {A}}}_1\, \textcircled {{F}}\, {{\mathfrak {A}}}_2\) are defined as (see also e.g. [3])

$$\begin{aligned} \begin{aligned} x^*:=&\sum _{i,j\in {{\mathbb {Z}}}_2}\epsilon (i,j)x_{i,j}^*\, ,\\ xy:=&\sum _{i,j,k,l\in {{\mathbb {Z}}}_2}\epsilon (j,k)x_{i,j}y_{k,l}. \end{aligned} \end{aligned}$$
(2.1)

The \(*\)-algebra thus obtained also carries a \({{\mathbb {Z}}}_2\)-grading, which is induced by the \(*\)-automorphism \(\theta =\theta _1\, \textcircled {{F}}\, \theta _2\) given on the elementary tensors by

$$\begin{aligned} (\theta _1\, \textcircled {{F}}\, \theta _2)(a_1\, \textcircled {{F}}\, a_2):=\theta _1(a_1)\, \textcircled {{F}}\, \theta _2(a_2),\quad a_1\in {{\mathfrak {A}}}_1,\,\, a_2\in {{\mathfrak {A}}}_2. \end{aligned}$$

where \(a_1\, \textcircled {{F}}\, a_2\) is nothing but \(a_1\otimes a_2\) thought of as an element of the \({{\mathbb {Z}}}_2\)-graded \(*\)-algebra \({{\mathfrak {A}}}_1\, \textcircled {{F}}\, {{\mathfrak {A}}}_2\), since \({{\mathfrak {A}}}_1\, \textcircled {{F}}\, {{\mathfrak {A}}}_2={{\mathfrak {A}}}_1\odot {{\mathfrak {A}}}_2\,\) as linear spaces. As of now, we will use \(a_1\otimes a_2\) and \(a_1\, \textcircled {{F}}\, a_2\) interchangeably when no confusion can occur.

The even and odd part of the Fermi product are respectively

$$\begin{aligned} \begin{aligned} \big ({{\mathfrak {A}}}_1\, \textcircled {{F}}\, {{\mathfrak {A}}}_2\big )_+&:=\big ({{\mathfrak {A}}}_{1,+}\odot {{\mathfrak {A}}}_{2,+}\big )\oplus \big ({{\mathfrak {A}}}_{1,-}\odot {{\mathfrak {A}}}_{2,-}\big ),\\ \big ({{\mathfrak {A}}}_1\, \textcircled {{F}}\, {{\mathfrak {A}}}_2\big )_-&:=\big ({{\mathfrak {A}}}_{1,+}\odot {{\mathfrak {A}}}_{2,-}\big )\oplus \big ({{\mathfrak {A}}}_{1,-}\odot {{\mathfrak {A}}}_{2,+}\big ). \end{aligned} \end{aligned}$$

For \(\omega _i\in {{\mathcal {S}}}({{\mathfrak {A}}}_i)\), \(i=1,2\), the state \(\psi _{\omega _1,\omega _2}\) has a counterpart in \({{\mathfrak {A}}}_1\, \textcircled {{F}}\, {{\mathfrak {A}}}_2\) by means of the product functional \(\omega _1\times \omega _2\), defined as usual by

$$\begin{aligned} \omega _1\times \omega _2\bigg (\sum _{j=1}^n a_{1,j}\textcircled {{F}}\, a_{2,j}\bigg ):=\sum _{j=1}^n \omega _1(a_{1,j})\omega _2(a_{2,j}), \end{aligned}$$

for all \(\sum _{j=1}^n a_{1,j} \textcircled {{F}}\, a_{2,j}\in {{\mathfrak {A}}}_1\, \textcircled {{F}}\, {{\mathfrak {A}}}_2\). Contrarily to the case of trivial grading, the map defined above is not necessarily positive. The following proposition, which generalizes the results obtained in [1] for the CAR algebra, gives a necessary and sufficient condition for the positivity.

Proposition 2.6

Let \(({{\mathfrak {A}}}_1,\theta _1), ({{\mathfrak {A}}}_2,\theta _2)\) be \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras, and \(\omega _1\in {{\mathcal {S}}}({{\mathfrak {A}}}_1)\), \(\omega _2\in {{\mathcal {S}}}({{\mathfrak {A}}}_2)\). Then \(\omega _1\times \omega _2\) is positive on \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) if and only if at least one between \(\omega _1\) and \(\omega _2\) is even. Moreover, \(\omega _1\times \omega _2\) is even if and only if both \(\omega _1\) and \(\omega _2\) are even.

Proof

The “if” part follows from [3, Proposition 7.1]. For the “only if” part we shall argue by contradiction, exactly as is done in the proof of [1, Theorem 1]. Suppose neither \(\omega _1\) nor \(\omega _2\) is even. Then there exist odd \(a_1\in {{\mathfrak {A}}}_1\) and \(a_2\in {{\mathfrak {A}}}_2\) such that \(\omega _1(a_1)\ne 0\) and \(\omega _2(a_2)\ne 0\). Furthermore, there is no loss of generality if we also assume that \(a_1\) and \(a_2\) are both self-adjoint. Now, on the one hand we have \(\omega _1\times \omega _2\,(a_1\,\textcircled {{F}}\, a_2)=\omega _1(a_1)\omega _2(a_2)\) is a real number different from zero. On the other hand, we also have

$$\begin{aligned} \overline{\omega _1\times \omega _2}\, (a_1\,\textcircled {{F}}\, a_2)&=\omega _1\times \omega _2\, ((a_1\,\textcircled {{F}}\, a_2)^*)\\&=-\omega _1\times \omega _2(a_1^*\otimes a_2^*)\\&=-\omega _1\times \omega _2(a_1\,\textcircled {{F}}\, a_2) \end{aligned}$$

from which we see that \(\omega _1\times \omega _2\, (a_1\,\textcircled {{F}}\, a_2)\) must be zero, and a contradiction is thus arrived at.

If \(\omega _1\) and \(\omega _2\) are both even, then the product is seen at once to be even as well. Conversely if \(\omega _1\times \omega _2\) is even, then \(\omega _1\) and \(\omega _2\) must be both even. For instance, for \(\omega _1\) we have

$$\begin{aligned} \omega _1(\theta _1(a_1))=\omega _1\times \omega _2\, (\theta _1\,\textcircled {{F}}\, \theta _2 (a_1\otimes {\mathbb {1}}_{{{\mathfrak {A}}}_2}))=\omega _1\times \omega _2\, (a_1\otimes {\mathbb {1}}_{{{\mathfrak {A}}}_2})=\omega _1(a_1), \end{aligned}$$

for every \(a_1\in {{\mathfrak {A}}}_1\). \(\square\)

We would like to remark that the above proposition holds for products of an arbitrary number \(n\ge 2\) of states \(\omega _1, \omega _2, \ldots , \omega _n\). More precisely, the product functional \(\omega _1\times \omega _2\times \ldots \times \omega _n\) will be positive if and only if \(n-1\) of the marginal states are even, and it will be even if and only if all the marginal states are so.

3 Fermi tensor product of Hilbert spaces

A \({{\mathbb {Z}}}_2\)-graded Hilbert space is a pair \(({{\mathcal {H}}}, U)\), where \({{\mathcal {H}}}\) is a (complex) Hilbert space and U a self-adjoint unitary acting on \({{\mathcal {H}}}\).

Note that \({{\mathcal {H}}}\) decomposes into a direct sum

$$\begin{aligned} {{\mathcal {H}}}={{\mathcal {H}}}_+\oplus {{\mathcal {H}}}_- \end{aligned}$$

where \({{\mathcal {H}}}_+:=\mathrm{Ker}(I-U)\), \({{\mathcal {H}}}_-:=\mathrm{Ker}(I+U)\), and I is the identity operator. As usual, vectors belonging to \({{\mathcal {H}}}_+\) (\({{\mathcal {H}}}_-\)) are referred to as even (odd) vectors, and elements belonging to any of these subspaces are collectively referred to as homogeneous vectors. The grade \(\partial (\xi )\) of any homogeneous vector \(\xi\) is 1 or \(-1\), according to whether it belongs to \({{\mathcal {H}}}_+\) or \({{\mathcal {H}}}_-\), respectively. Recalling that \(({{\mathcal {B}}}({{\mathcal {H}}}),\mathrm{ad}_U)\) is a \({{\mathbb {Z}}}_2\)-graded \(*\)-algebra, after applying a homogeneous element of \({{\mathcal {B}}}({{\mathcal {H}}})\) to a homogeneous vector in \({{\mathcal {H}}}\) we get a homogenous vector whose grade can be easily determined, as the following result shows.

Lemma 3.1

Let \(({{\mathcal {H}}}, U)\) be a \({{\mathbb {Z}}}_2\)-graded Hilbert space. If \(T\in {{\mathcal {B}}}({{{\mathcal {H}}}})\) and \(\xi \in {{\mathcal {H}}}\) are both homogeneous, then \(T\xi\) is still homogeneous and

$$\begin{aligned} \partial (T\xi )=\partial (T)\partial (\xi ). \end{aligned}$$

Proof

There are four cases to deal with, which can be treated in the same way. For instance, if T is even and \(\xi\) is odd, then we have \(UT\xi =(UTU^*)U\xi =T(-\xi )=-T\xi\), that is \(T\xi\) is odd. \(\square\)

Let \(({{\mathfrak {A}}}, \theta )\) be a \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebra, and let \(\omega\) be an even state of \({{\mathfrak {A}}}\), whose covariant GNS representation is \(({{\mathcal {H}}}_\omega , \pi _\omega , \xi _\omega , V_{\theta , \omega })\). Thus, \(({{\mathcal {H}}}_\omega , V_{\theta , \omega })\) is a \({{\mathbb {Z}}}_2\)-graded Hilbert space. The following result allows us to realize the even part of \({{\mathcal {H}}}_\omega\).

Proposition 3.2

The even part of \({{\mathcal {H}}}_\omega\) is given by \({{\mathcal {H}}}_{\omega , +}=\overline{\pi _\omega ({{\mathfrak {A}}}_+)\xi _\omega }\).

Proof

From the obvious inclusion \({\pi _\omega ({{\mathfrak {A}}}_+)\xi _\omega }\subset {{\mathcal {H}}}_{\omega , +}\), one also has \(\overline{{\pi _\omega ({{\mathfrak {A}}}_+)\xi _\omega }}\subset {{\mathcal {H}}}_{\omega , +}.\)

For the converse inclusion, pick \(\eta \in {{\mathcal {H}}}_{\omega , +}\), that is \(V_{\theta , \omega }\eta =\eta\). By cyclicity of \(\xi _\omega\), there exists a sequence \(\{a_n\}_{n\in {{\mathbb {N}}}}\) such that \(\eta =\lim _{n\rightarrow \infty }\pi _\omega (a_n)\xi _\omega\). But then \(V_{\theta , \omega }\eta =V_{\theta , \omega }\big (\lim _{n\rightarrow \infty }\pi _\omega (a_n)\xi _\omega \big )=\lim _{n\rightarrow \infty }\pi _\omega (\theta (a_n))\xi _\omega\). From the equality \(\eta =\frac{1}{2}(\eta +V_{\theta , \omega }\eta )\) we see that \(\eta =\lim _{n\rightarrow \infty }\pi _\omega (\varepsilon _1(a_n))\xi _\omega\), and the thesis follows since \(\varepsilon _1(a_n)\in {{\mathfrak {A}}}_+\). \(\square\)

The tensor product \({{\mathcal {H}}}_1\otimes {{\mathcal {H}}}_2\) of two \({{\mathbb {Z}}}_2\)-graded Hilbert spaces \(({{\mathcal {H}}}_1, U_1)\) and \(({{\mathcal {H}}}_2, U_2)\) can be endowed with the natural grading induced by the self-adjoint unitary \(U_1\otimes U_2\). Note that a simple tensor \(\xi _1\otimes \xi _2\) in \({{\mathcal {H}}}_1\otimes {{\mathcal {H}}}_2\) is even precisely when \(\xi _1\) and \(\xi _2\) are both even or when they are both odd. In the sequel, most of the times the Hilbert space \({{\mathcal {H}}}_1\otimes {{\mathcal {H}}}_2\) will be thought of as a graded Hilbert space, with grading \(U_1\otimes U_2\). Rather than write \(({{\mathcal {H}}}_1\otimes {{\mathcal {H}}}_2, U_1\otimes U_2 )\), we will simply denote this graded Hilbert space by \({{\mathcal {H}}}_1\,\textcircled {{F}}\,{{\mathcal {H}}}_2\).

Given \(T_1\in {{\mathcal {B}}}({{\mathcal {H}}}_1)\) and a homogeneous operator \(T_2\in {{\mathcal {B}}}({{\mathcal {H}}}_2)\), we define \(T_1\odot T_2\) as the linear operator acting on \({{\mathcal {H}}}_1\odot {{\mathcal {H}}}_2\) as

$$\begin{aligned} T_1\odot T_2 (\xi \odot \eta )=\varepsilon (T_2, \xi )(T_1 \xi \odot T_2 \eta ) \end{aligned}$$
(3.1)

for any homogeneous \(\xi \in {{\mathcal {H}}}_1\) and any \(\eta \in {{\mathcal {H}}}_2\). Clearly, by using (3.1) one can extend \(T_1\odot T_2\) to the whole vector space \({{\mathcal {H}}}_1\odot {{\mathcal {H}}}_2\). Furthermore, it is easy to see that \(T_1\odot T_2\) is a bounded operator, and thus it can be extended by continuity to \({{\mathcal {H}}}_1\textcircled {{F}}\,{{\mathcal {H}}}_2\). Its unique extension is denoted by \(T_1\,\textcircled {{F}}\, T_2\).

Let now \(({{\mathfrak {A}}}_1, \theta _1), ({{\mathfrak {A}}}_2, \theta _2)\) be \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras. If \(\pi _i: {{\mathfrak {A}}}_i\rightarrow {{\mathcal {B}}}({{\mathcal {H}}}_i)\) are grading-equivariant representations, it is possible to define a map, \(\pi _1\,\textcircled {{F}}\, \pi _2\), of \({{\mathfrak {A}}}_1\,\textcircled {{F}}\, {{\mathfrak {A}}}_2\) acting on the Fermi tensor product \({{\mathcal {H}}}_1\,\textcircled {{F}}\, {{\mathcal {H}}}_2\) as

$$\begin{aligned} \pi _1\,\textcircled {{F}}\, \pi _2 (a_1\textcircled {{F}}\, a_2):=\pi _1(a_1)\,\textcircled {{F}}\, \pi _2(a_2) \end{aligned}$$

for every \(a_1\in {{\mathfrak {A}}}_1\) and \(a_2\in {{\mathfrak {A}}}_2\).

Proposition 3.3

Under the above assumptions, one has that \(\pi _1\,\textcircled {{F}}\, \pi _2\) is a \(*\)-representation of \({{\mathfrak {A}}}_1\,\textcircled {{F}}\, {{\mathfrak {A}}}_2\) acting on the Fermi tensor product \({{\mathcal {H}}}_1\,\textcircled {{F}}\, {{\mathcal {H}}}_2\) .

Proof

All we have to do is ascertain that \(\pi _1\,\textcircled {{F}}\, \pi _2\) preserves both product and involution of \({{\mathfrak {A}}}_1\,\textcircled {{F}}\, {{\mathfrak {A}}}_2\). To this end, it is enough to verify the involved equalities only on homogeneous elements.

Let \(a_i\in {{\mathfrak {A}}}_i\) be homogeneous elements, \(i=1, 2\), \(\xi , \xi '\in {{\mathcal {H}}}_1\) \(\eta , \eta ' \in {{\mathcal {H}}}_2\) be homogeneous vectors, and observe that \(\pi _i(a_i)\) has the same grade of \(a_i\), since \(\pi _i\) is grading-equivariant.

On the one hand, from (2.1) and (3.1), we have

$$\begin{aligned}&\langle \pi _1\,\textcircled {{F}}\, \pi _2\big ((a_1\otimes a_2)^*\big ) (\xi \otimes \eta ), \xi '\otimes \eta ' \rangle \\&\quad =\varepsilon (a_1, a_2)\langle \pi _1(a_1^*)\,\textcircled {{F}}\,\pi _2(a_2^*)\, (\xi \otimes \eta ), \xi '\otimes \eta ' \rangle \\&\quad =\varepsilon (a_1, a_2)\varepsilon (a_2, \xi ) \langle \pi _1(a_1^*)\xi \otimes \pi _2(a_2^*)\eta , \xi '\otimes \eta '\rangle \\&\quad =\varepsilon (a_1, a_2)\varepsilon (a_2, \xi ) \langle \xi , \pi _1(a_1) \xi ' \rangle \langle \eta , \pi _2(a_2) \eta ' \rangle . \end{aligned}$$

On the other hand, from (3.1) we have

$$\begin{aligned}&\langle (\pi _1\,\textcircled {{F}}\, \pi _2(a_1\otimes a_2))^* (\xi \otimes \eta ), \xi '\otimes \eta ' \rangle \\&\quad =\langle \xi \otimes \eta ,\pi _1\,\textcircled {{F}}\, \pi _2(a_1\otimes a_2) (\xi '\otimes \eta ' )\rangle \\&\quad =\varepsilon (a_2, \xi ') \langle \xi \otimes \eta , \pi _1(a_1)\xi '\otimes \pi _2(a_2)\eta '\rangle \\&\quad =\varepsilon (a_2, \xi ')\langle \xi , \pi _1(a_1) \xi ' \rangle \langle \eta , \pi _2(a_2) \eta ' \rangle , \end{aligned}$$

and the two expressions equal each other because

$$\begin{aligned} \varepsilon (a_1, a_2)\varepsilon (a_2, \xi )=\varepsilon (a_2, \xi ') \end{aligned}$$

whenever \(\langle \xi , \pi _1(a_1) \xi ' \rangle\) is different from 0, as a painstaking inspection of the signs shows.

As for the product, pick homogeneous \(a_1, b_1\in {{\mathfrak {A}}}_1\) and \(a_2, b_2\in {{\mathfrak {A}}}_2\). With homogeneous \(\xi \in {{\mathcal {H}}}_1, \eta \in {{\mathcal {H}}}_2\) by applying (2.1) and (3.1) we have

$$\begin{aligned}&\pi _1\,\textcircled {{F}}\, \pi _2 \big ((a_1\otimes a_2)(b_1\otimes b_2)\big )(\xi \otimes \eta )\\&\quad =\varepsilon (a_2, b_1)\pi _1\,\textcircled {{F}}\, \pi _2(a_1b_1\otimes a_2b_2)(\xi \otimes \eta )\\&\quad =\varepsilon (a_2, b_1)\varepsilon (a_2 b_2, \xi )\pi _1(a_1b_1)\xi \otimes \pi _2(a_2b_2)\eta \\&\quad =\varepsilon (a_2, b_1)\varepsilon (a_2 b_2, \xi )\pi _1(a_1)\pi _1(b_1)\xi \otimes \pi _2(a_2)\pi _2(b_2)\eta . \end{aligned}$$

On the other hand, by applying (3.1) we have

$$\begin{aligned}&\pi _1\,\textcircled {{F}}\, \pi _2 (a_1\otimes a_2)[ \pi _1\,\textcircled {{F}}\, \pi _2 (b_1\otimes b_2)(\xi \otimes \eta )]\\&=\varepsilon (b_2, \xi )\pi _1\,\textcircled {{F}}\, \pi _2 (a_1\otimes a_2) (\pi _1(b_1)\xi \otimes \pi _2(b_2)\eta )\\&=\varepsilon (b_2, \xi )\varepsilon (a_2, \pi _1(b_1)\xi )\pi _1(a_1)\pi _1(b_1)\xi \otimes \pi _2(a_2)\pi _2(b_2)\eta . \end{aligned}$$

Again, a painstaking inspection of the signs shows that

$$\begin{aligned} \varepsilon (a_2, b_1)\varepsilon (a_2 b_2, \xi )=\varepsilon (b_2, \xi )\varepsilon (a_2, \pi _1(b_1)\xi ) \end{aligned}$$

and the proof is complete. \(\square\)

The following result says that \(\pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}\) is the GNS representation of the product state of two even states \(\omega _1\) and \(\omega _2\).

Proposition 3.4

Let \(({{\mathfrak {A}}}_i, \theta _i)\) be \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras, \(i=1, 2\). If \(\omega _i\in {\mathcal {S}}( {{{\mathfrak {A}}}}_i)\) are even states for \(i=1, 2\) then

$$\begin{aligned} \pi _{\omega _1\times \omega _2}=\pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2} \end{aligned}$$

up to unitary equivalence, as representations of the Fermi tensor product \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\).

Proof

First note that \(\pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}\) is a cyclic representation of \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) with cyclic vector \(\xi _{\omega _1}\otimes \xi _{\omega _2}\), where \(\xi _{\omega _i}\) is the GNS vector of \(\omega _i\) for \(i=1, 2\). Therefore, to conclude it is enough to make sure the vector state associated with \(\xi _{\omega _1}\otimes \xi _{\omega _2}\) coincides with the product state \(\omega _1\times \omega _2\). To this aim, consider homogeneous \(a_i\in {{\mathfrak {A}}}_i\), \(i=1, 2\). After recalling that the cyclic vectors are even, by applying (3.1) one finds

$$\begin{aligned}&\langle \pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}(a_1\otimes a_2) \xi _{\omega _1}\otimes \xi _{\omega _2} , \xi _{\omega _1}\otimes \xi _{\omega _2}\rangle \\&\quad =\varepsilon (a_2, \xi _{\omega _1})\langle \pi _{\omega _1}(a_1)\xi _{\omega _1}\otimes \pi _{\omega _2}(a_2)\xi _{\omega _2}, \xi _{\omega _1}\otimes \xi _{\omega _2} \rangle \\&\quad = \langle \pi _{\omega _1}(a_1)\xi _{\omega _1}, \xi _{\omega _1}\rangle \langle \pi _{\omega _2}(a_2)\xi _{\omega _2}, \xi _{\omega _2}\rangle \\&\quad =\omega _1(a_1)\omega _2(a_2)=\omega _1\times \omega _2\,(a_1\otimes a_2) \end{aligned}$$

which ends the proof. \(\square\)

4 Norms on \({{\mathbb {Z}}}_2\)-graded tensor product of \(C^*\)-algebras

In this section we undertake a detailed study of the so-called spatial norm on the Fermi tensor product of \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras. We show that it is minimal besides being a cross norm, as is the maximal one already introduced in [3]. Given \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras \(({{\mathfrak {A}}}_1,\theta _1)\) and \(({{\mathfrak {A}}}_2,\theta _2)\), the latter norm is given by

$$\begin{aligned} \Vert x\Vert _{\max }:=\sup \{\Vert \pi (x)\Vert : \pi \,\,\text {is a representation}\}, \end{aligned}$$

for all \(x\in {{\mathfrak {A}}}_1\, \textcircled {{F}}\,{{\mathfrak {A}}}_2\), and it is obviously the biggest norm on \({{\mathfrak {A}}}_1\, \textcircled {{F}}\,{{\mathfrak {A}}}_2\).

The spatial norm is defined in terms of the GNS representations of products of even states. More precisely,

$$\begin{aligned} \Vert x\Vert _{\min }:=\sup \{\Vert \pi _{\omega _1\times \omega _2}(x)\Vert : \omega _1\in {{\mathcal {S}}}_+({{\mathfrak {A}}}_1),\,\, \omega _2\in {{\mathcal {S}}}_+({{\mathfrak {A}}}_2)\}, \end{aligned}$$

for all \(x\in {{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\). In principle, the above definition might provide only a seminorm. In fact, it is known that it actually defines a norm. For want of a reference, we nonetheless include a full proof of this fact.

Proposition 4.1

Under the above assumptions, the seminorm \(\Vert \cdot \Vert _{\min }\) is a \(C^*\)-norm on the Fermi product \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\).

Proof

As the seminorm \(\Vert \cdot \Vert _{\min }\) obviously satisfies the \(C^*\)-equality, we only need to make sure it also separates the elements of \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\). Let \(c=\sum _{i=1}^n a_i\otimes b_i\) be such an element with \(c\ne 0\), where \(a_i\in {{\mathfrak {A}}}_1\) and \(b_i\in {{\mathfrak {A}}}_2\), \(i=1, 2, \ldots , n\). Take now

$$\begin{aligned} \widetilde{{{\mathfrak {A}}}_1}:= C^*(\{a_i, \theta _1(a_i)\mid i=1, 2, \ldots , n\}) \end{aligned}$$

and

$$\begin{aligned} \widetilde{{{\mathfrak {A}}}_2}:= C^*(\{b_i, \theta _2(b_i)\mid i=1, 2, \ldots , n\} ). \end{aligned}$$

By definition, \(\widetilde{{{\mathfrak {A}}}_1}\) and \(\widetilde{{{\mathfrak {A}}}_2}\) inherit the \({{\mathbb {Z}}}_2\)-grading from \({{\mathfrak {A}}}_1\) and \({{\mathfrak {A}}}_2\), respectively. Since \(\widetilde{{{\mathfrak {A}}}_1}\) and \(\widetilde{{{\mathfrak {A}}}_2}\) are separable \(C^*\)-algebras, there exist faithful states \(\widetilde{\omega _1}\) and \(\widetilde{\omega _2}\) on \(\widetilde{{{\mathfrak {A}}}_1}\) and \(\widetilde{{{\mathfrak {A}}}_2}\), respectively. If \(\widetilde{\varepsilon }_i\) is the canonical conditional expectation from \(\widetilde{{{\mathfrak {A}}}_i}\) onto \(\widetilde{{{\mathfrak {A}}}}_{i,+}\), \(i=1,2\), then \(\widetilde{\omega _i}\circ \widetilde{\varepsilon _i}\) are still faithful. But then by [13, Theorem IV.4.9 (iii)], the product state \(\psi _{\widetilde{\omega _1}\circ \widetilde{\varepsilon _1},\widetilde{\omega _2}\circ \widetilde{\varepsilon _2}}\) is faithful on \(\widetilde{{{\mathfrak {A}}}_1}\otimes _{\mathrm{min}}\widetilde{{{\mathfrak {A}}}_2}\). Therefore, we have

$$\begin{aligned} \widetilde{\omega _1}\circ \widetilde{\varepsilon _1}\times \widetilde{\omega _2}\circ \widetilde{\varepsilon _2}\, (c^*c)=\psi _{\widetilde{\omega _1}\circ \widetilde{\varepsilon _1},\widetilde{\omega _2}\circ \widetilde{\varepsilon _2}}(c^* c)>0. \end{aligned}$$
(4.1)

The conclusion now readily follows by considering any extensions \(\omega _1, \omega _2\) of \(\widetilde{\omega _1}, \widetilde{\omega _2}\) to the whole \({{\mathfrak {A}}}_1\) and \({{\mathfrak {A}}}_2\), respectively. Indeed, from (4.1)

$$\begin{aligned} \Vert \pi _{\omega _1\circ \varepsilon _1\times \omega _2\circ \varepsilon _2}(c)\Vert ^2&\ge \Vert \pi _{\omega _1\circ \varepsilon _1\times \omega _2\circ \varepsilon _2}(c)\xi _{\omega _1\circ \varepsilon _1\times \omega _2\circ \varepsilon _2}\Vert ^2\\&=\omega _1\circ \varepsilon _1 \times \omega _2\circ \varepsilon _2\,(c^*c)>0. \end{aligned}$$

\(\square\)

Here, we present the definition of a \(C^*\)-cross norm in the presence of a \({{\mathbb {Z}}}_2\)-grading, which reduces to the usual one when the grading is trivial.

Definition 4.2

For any given \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras \(({{\mathfrak {A}}}_1,\theta _1)\) and \(({{\mathfrak {A}}}_2,\theta _2)\), a \(C^*\)-norm \(\Vert \cdot \Vert _{\beta }\) on \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) is said to be cross if

$$\begin{aligned} \Vert a_1\textcircled {{F}}\,a_2\Vert _{\beta }=\Vert a_1\Vert \Vert a_2\Vert , \quad \text {for homogeneous }a_1\in {{\mathfrak {A}}}_1, a_2\in {{\mathfrak {A}}}_2. \end{aligned}$$

As one could expect, the norms introduced above are cross:

Proposition 4.3

Under the above assumptions, both \(\Vert \cdot \Vert _{\mathrm{max}}\) and \(\Vert \cdot \Vert _{\min }\) are cross norms on \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\).

Proof

We start by observing that if \(\Vert \cdot \Vert _\beta\) is any \(C^*\)-norm on the Fermi product \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\), then \(\Vert a\otimes b\Vert _\beta \le \Vert a\Vert \Vert b\Vert\) for any \(a\in {{\mathfrak {A}}}_1\), \(b\in {{\mathfrak {A}}}_2\). Indeed, one has

$$\begin{aligned} \Vert a\otimes b\Vert _\beta =\Vert (a\otimes {\mathbb {1}}_{{{\mathfrak {A}}}_2})({\mathbb {1}}_{{{\mathfrak {A}}}_1}\otimes b)\Vert _\beta \le \Vert a\otimes {\mathbb {1}}_{{{\mathfrak {A}}}_2}\Vert _\beta \Vert {\mathbb {1}}_{{{\mathfrak {A}}}_1}\otimes b\Vert _\beta =\Vert a\Vert \Vert b\Vert \end{aligned}$$

because both

$$\begin{aligned} {{\mathfrak {A}}}_1\ni a\mapsto a\otimes {\mathbb {1}}_{{{\mathfrak {A}}}_2}\in {{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2, \quad {{\mathfrak {A}}}_2\ni b\mapsto {\mathbb {1}}_{{{\mathfrak {A}}}_1}\otimes b\in {{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2 \end{aligned}$$

are injective \(^*\)-homomorphisms.

Since \(\Vert a\otimes b\Vert _{\mathrm{max}}\ge \Vert a\otimes b\Vert _{\min }\), for any \(a\in {{\mathfrak {A}}}_1\) and \(b\in {{\mathfrak {A}}}_2\), it is enough to prove that \(\Vert a\otimes b\Vert _{\mathrm{min}}\ge \Vert a\Vert \Vert b\Vert\) for homogeneous a, b. To this aim, note that a straightforward application of Proposition  3.4 gives \(\Vert a\otimes b\Vert _{\min }\ge \Vert \pi _{\omega _1}(a)\Vert \,\Vert \pi _{\omega _2}(b)\Vert\) for any even states \(\omega _i\) on \({{\mathcal {S}}}_+({{\mathfrak {A}}}_i)\), \(i=1,2\). The thesis then follows from Corollary 2.2. \(\square\)

Henceforth the completion of \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) with respect to the spatial norm \(\Vert \cdot \Vert _{\min }\) will always be denoted by \({{\mathfrak {A}}}_1\,\textcircled {{F}}_{\min }\,{{\mathfrak {A}}}_2\), whereas the completion with respect to the maximal norm \(\Vert \cdot \Vert _{\mathrm{max}}\) will be denoted by \({{\mathfrak {A}}}_1\,\textcircled {{F}}_{\mathrm{max}}\,{{\mathfrak {A}}}_2\). We next show that the tensor product of faithful GNS representations of even states is still a faithful representation on the \(C^*\)-algebra \({{\mathfrak {A}}}_1\,\textcircled {{F}}_{\min }\,{{\mathfrak {A}}}_2\). To this aim, we first need a technical lemma on normal states. We say that a state \(\omega\) of a given \(C^*\)-algebra \({{\mathfrak {A}}}\) is normal in a representation \(\pi :{{\mathfrak {A}}}\rightarrow {{\mathcal {B}}}({{\mathcal {H}}})\) if \(\omega (a)=\mathrm{Tr}(\pi (a)T),\,\,a\in {{\mathfrak {A}}}\), for some positive trace-class operator T with \(\mathrm{Tr}(T)=1\).

Lemma 4.4

Let us take \(i=1,2\), and \(\omega _i\in {\mathcal {S}}_+({{\mathfrak {A}}}_i)\). Suppose for each \(i=1, 2\) we are given a normal state in \(\pi _{\omega _i}\), say \(\varphi _i(\cdot ):=\mathrm{Tr} (\pi _{\omega _i}(\cdot )T_i)\), where \(T_i\in {{\mathcal {B}}}({{\mathcal {H}}}_{\omega _i})\) is a positive trace-class operator with unit trace. Then the product state \(\varphi _1\times \varphi _2\) on \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) is a normal state in \(\pi _{\omega _1\times \omega _2}\). More precisely, one has

$$\begin{aligned} \varphi _1\times \varphi _2(\cdot )= \mathrm{Tr}(\pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}(\cdot )\,\, T_1\,\textcircled {{F}}\,T_2). \end{aligned}$$

Proof

We first point out that no loss of generality occurs if we suppose that each \(T_i\) is even for \(i=1, 2\). Indeed, if \(T_i\) is not even, and \(({{\mathcal {H}}}_{\omega _1}, \pi _{\omega _i}, \xi _{\omega _i}, V_{\theta _i,\omega _i})\) is the GNS covariant representation of \(\omega _i\), then it is enough to pass to \(T_i':= \frac{1}{2}(T_i+V_{\theta _i,\omega _i}TV_{\theta _i,\omega _i})\), since one can check that \(\varphi _i(\cdot )=\mathrm{Tr}(\pi _{\omega _i}(\cdot )\, T_i')\).

With homogeneous \(a_i\in {{\mathfrak {A}}}_i\) and homogeneous vectors \(\xi _i\in {{\mathcal {H}}}_{\omega _i}\), by using (2.1) and (3.1) one can easily see that

$$\begin{aligned} (\pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}(a_1\otimes a_2))\,\, T_1\,\textcircled {{F}}\,T_2\, (\xi _1\otimes \xi _2)=\varepsilon (a_2, T_1\xi _1)\pi _{\omega _1}(a_1)T_1\xi _1\otimes \pi _{\omega _2}(a_2)T_2\xi _2 \end{aligned}$$
(4.2)

In order to compute the trace of \(A:=(\pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}(a_1\otimes a_2))\,\, T_1\,\textcircled {{F}}\,T_2\), we need to single out a convenient orthonormal basis of the Hilbert space \({{\mathcal {H}}}_{\omega _1}\textcircled {{F}}\,{{\mathcal {H}}}_{\omega _2}\). To this end, note that \(T_1\) and \(T_2\) preserve the homogeneous subspaces of \({{\mathcal {H}}}_{\omega _1}\) and \({{\mathcal {H}}}_{\omega _2}\), respectively. Therefore, one can exhibit a orthonormal basis of \({{\mathcal {H}}}_{\omega _i}\) made up of homogeneous eigenvectors of \(T_i\), \(i=1, 2\). If we take I, J, L, K as set of indices, we denote by \(\{e_i\}_{i\in I}\cup \{e'_j\}_{j\in J}\) the basis of the eigenvectors of \(T_1\), where the \(e_i\)’s are even and the \(e'_j\)’s are odd. Likewise, we denote by \(\{f_l\}_{l\in L}\cup \{f'_k\}_{k\in K}\) the basis of the eigenvectors of \(T_2\), where the \(f_l\)’s are even and the \(f'_k\)’s are odd. As is easily verified, the set

$$\begin{aligned} \{e_i\otimes f_l\}_{i\in I, l\in L}\cup \{e_i\otimes f'_k\}_{ i\in I, k\in K}\cup \{e'_j\otimes f_l\}_{j\in J, l\in L}\cup \{e'_j\otimes f'_k\}_{ j\in J, k\in K} \end{aligned}$$

is an orthonormal basis of the Fermi Hilbert space \({{\mathcal {H}}}_{\omega _1}\,\textcircled {{F}}\, {{\mathcal {H}}}_{\omega _2}\). Consequently, by (4.2) and taking into account the orthogonality relations one has:

$$\begin{aligned} Tr (A)&= \sum _{i\in I, l\in L} \langle A\,e_i\otimes f_l, e_i\otimes f_l\rangle \\&\quad +\sum _{i\in I, k\in K} \langle A\,e_i\otimes f'_k, e_i\otimes f'_k\rangle \\&\quad + \sum _{j\in J, l\in L} \langle A\,e'_j\otimes f_l, e'_j\otimes f_l\rangle \\&\quad +\sum _{j\in J, k\in K} \langle A\,e'_j\otimes f'_k, e'_j\otimes f'_k\rangle \\&=\sum _{i\in I, l\in L}\langle \pi _{\omega _1}(a_1)T_1e_i, e_i \rangle \langle \pi _{\omega _2}(a_2)T_2f_l, f_l \rangle \\&\quad +\sum _{i\in I, k\in K}\langle \pi _{\omega _1}(a_1)T_1e_i, e_i \rangle \langle \pi _{\omega _2}(a_2)T_2f'_k, f'_k \rangle \\&\quad +\sum _{j\in J, l\in L}\langle \pi _{\omega _1}(a_1)T_1e'_j, e'_j \rangle \langle \pi _{\omega _2}(a_2)T_2f_l, f_l \rangle \\&\quad +\sum _{j\in J, k\in K}\langle \pi _{\omega _1}(a_1)T_1e'_j, e'_j \rangle \langle \pi _{\omega _2}(a_2)T_2f'_k, f'_k \rangle \\&= Tr(\pi _{\omega _1}(a_1)T_1)Tr(\pi _{\omega _2}(a_2)T_2)\\&= \varphi _1\times \varphi _2(a_1\otimes a_2). \end{aligned}$$

\(\square\)

Proposition 4.5

Let us take \(i=1,2\), \(({{\mathfrak {A}}}_i,\theta _i)\) a \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebra, and \(\omega _i\in {{\mathcal {S}}}_+({{\mathfrak {A}}}_i)\). If \(\pi _{\omega _i}\) is a faithful representation, then \(\pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}\) is a faithful representation of \({{\mathfrak {A}}}_1 \,\textcircled {{F}}_{\min }\,{{\mathfrak {A}}}_2\).

Proof

First observe that \(\pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}\) can be extended to a representation of \({{\mathfrak {A}}}_1 \,\textcircled {{F}}_{\min }\,{{\mathfrak {A}}}_2\). Indeed, by definition of \(\Vert \cdot \Vert _{\min }\) one has \(\Vert \pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}\, (x)\Vert \le \Vert x\Vert _{\min }\) for every \(x\in {{\mathfrak {A}}}_1 \,\textcircled {{F}}\,{{\mathfrak {A}}}_2\).

Therefore, by density we need only prove that \(\Vert x\Vert _{\min }\le \Vert \pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}\, (x)\Vert\) for every \(x\in {{\mathfrak {A}}}_1 \,\textcircled {{F}}\,{{\mathfrak {A}}}_2\).

Let \(\varphi _i\) be even state on \({{\mathfrak {A}}}_i\), \(i=1, 2\). By cyclicity we have

$$\begin{aligned}&\Vert \pi _{\varphi _1\times \varphi _2}\,(x)\Vert =\sup _{y:\,\varphi _1\times \varphi _2\,(y^*y)\ne 0}\frac{ \varphi _1\times \varphi _2\,( y^*x^*xy)^\frac{1}{2}}{\varphi _1\times \varphi _2\,(y^*y)^\frac{1}{2}}. \end{aligned}$$

Since \(\pi _{\omega _i}\) is faithful, \(\varphi _i\) is a weak* limit of a net of states that are normal in the representation \(\pi _{\omega _i}\), for \(i=1, 2\), cf. the proof of [13, Theorem 4.9 (iii)]. In other terms, there exist two nets \(\{T_k\}_{k\in K}\subset {\mathcal {B}}({{\mathcal {H}}}_{\omega _1})\) and \(\{S_k \}_{k\in K}\subset {\mathcal {B}}({{\mathcal {H}}}_{\omega _2})\), of normalized positive trace-class operators such that for every \(a_i\in {{\mathfrak {A}}}_i\) one has

$$\begin{aligned} \varphi _1(a_1)=\lim _k Tr (\pi _{\omega _1}(a_1)T_k), \quad \varphi _2(a_2)=\lim _k Tr(\pi _{\omega _2}(a_2)S_k). \end{aligned}$$

By virtue of Lemma 4.4, the product state \(\varphi _1\times \varphi _2\) is seen at once to be the weak* limit of the net \(\{\eta _k\}_{k\in K}\) with

$$\begin{aligned} \eta _k(x):=Tr (\pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}\,(x) T_k\textcircled {{F}}\, S_k), \quad x\in {{\mathfrak {A}}}_1 \,\textcircled {{F}}\,{{\mathfrak {A}}}_2. \end{aligned}$$

Now, fix \(\varepsilon >0\) and let \({\widetilde{y}}\in {{\mathfrak {A}}}_1 \,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) with \(\varphi _1\times \varphi _2\, ({\widetilde{y}}^*{\widetilde{y}})\ne 0\) be such that:

$$\begin{aligned} \sup _{y:\, \varphi _1\times \varphi _2(y^*y)\ne 0}\frac{ \varphi _1\times \varphi _2( y^*x^*xy)^\frac{1}{2}}{\varphi _1\times \varphi _2(y^*y)^\frac{1}{2}}\le \frac{ \varphi _1\times \varphi _2( {\widetilde{y}}^*x^*x{\widetilde{y}})^\frac{1}{2}}{\varphi _1\times \varphi _2({\widetilde{y}}^*{\widetilde{y}})^\frac{1}{2}}+\frac{\varepsilon }{2}. \end{aligned}$$

Let \(k_o\in K\) be such that

$$\begin{aligned} \frac{ \varphi _1\times \varphi _2( {\widetilde{y}}^*x^*x{\widetilde{y}})^\frac{1}{2}}{\varphi _1\times \varphi _2({\widetilde{y}}^*{\widetilde{y}})^\frac{1}{2}}\le \frac{\eta _{k_o}( {\widetilde{y}}^*x^*x{\widetilde{y}})^\frac{1}{2}}{\eta _{k_o}({\widetilde{y}}^*{\widetilde{y}})^\frac{1}{2}}+\frac{\varepsilon }{2}. \end{aligned}$$

We have

$$\begin{aligned} \sup _{y:\, \varphi _1\times \varphi _2(y^*y)\ne 0}\frac{ \varphi _1\times \varphi _2( y^*x^*xy)^\frac{1}{2}}{\varphi _1\times \varphi _2(y^*y)^\frac{1}{2}}&\le \frac{\eta _{k_o}( {\widetilde{y}}^*x^*x{\widetilde{y}})^\frac{1}{2}}{\eta _{k_o}({\widetilde{y}}^*{\widetilde{y}})^\frac{1}{2}}+\varepsilon \\&\le \Vert \pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}\,(x) \Vert +\varepsilon . \end{aligned}$$

Since \(\varepsilon >0\) is arbitrary, we find that

$$\begin{aligned} \sup _{y:\, \varphi _1\times \varphi _2(y^*y)\ne 0}\frac{ \varphi _1\times \varphi _2( y^*x^*xy)^\frac{1}{2}}{\varphi _1\times \varphi _2(y^*y)^\frac{1}{2}}\le \Vert \pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}\,(x) \Vert \end{aligned}$$

for any pair of even states \(\varphi _i\), \(i=1, 2\). Therefore, we finally see that \(\Vert x\Vert _{\min }\le \Vert \pi _{\omega _1}\,\textcircled {{F}}\,\pi _{\omega _2}\,(x)\Vert\), and the proof is complete. \(\square\)

In the next proposition, we will see that for both norms introduced, the grading on \({{\mathfrak {A}}}_1 \textcircled {{F}}{{\mathfrak {A}}}_2\) extends to the \(C^*\)-completion. Whenever this happens, we call the norm compatible.

Definition 4.6

A \(C^*\)-norm \(\Vert \cdot \Vert _\beta\) on \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) is said to be compatible if the natural grading \(\theta _1\,\textcircled {{F}}\,\theta _2\) extends to a (necessarily involutive) \(*\)-automorphism of the completion \({{\mathfrak {A}}}_1\,\textcircled {{F}}_\beta \,{{\mathfrak {A}}}_2\).

Proposition 4.7

The maximal norm \(\Vert \cdot \Vert _{\mathrm{max}}\) and the spatial norm \(\Vert \cdot \Vert _{\min }\) are compatible.

Proof

We have to show that \(\theta :=\theta _1\,\textcircled {{F}}\, \theta _2\) can be extended to a \(*\)-automorphism of both \({{\mathfrak {A}}}_1\,\textcircled {{F}}_{\mathrm{max}}\,{{\mathfrak {A}}}_2\) and \({{\mathfrak {A}}}_1\,\textcircled {{F}}_{\min }\,{{\mathfrak {A}}}_2\).

As for the maximal norm, for \(x\in {{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) we have

$$\begin{aligned} \Vert \theta (x)\Vert _{\mathrm{max}}&=\sup \{\Vert \pi (\theta (x))\Vert : \pi \,\text { is a}\, *\text {-representation of}\,{{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2 \}\\&=\sup \{\Vert \pi '(x) \Vert : \pi '\,\text { is a}\, *\text {-representation of}\,{{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2 \}\\&=\Vert x\Vert _{\mathrm{max}}, \end{aligned}$$

since any representation \(\pi '\) of \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) can be written as \(\pi \circ \theta\).

As for the spatial norm, we start by recalling that for any pair of even states \(\omega _i\in {{\mathcal {S}}}_+({{\mathfrak {A}}}_i)\), \(i=1, 2\), the \(*\)-automorphism \(\theta =\theta _1\,\textcircled {{F}}\,\theta _2\) is unitarily implemented in the representation \(\pi _{\omega _1\times \omega _2}\). More precisely, in light of Proposition 3.4 one has

$$\begin{aligned} \pi _{\omega _1\times \omega _2}(\theta (x))= V_{\theta _1, \omega _1}\,\textcircled {{F}}\, V_{\theta _2, \omega _2} \left( \pi _{\omega _1\times \omega _2} (x)\right) V_{\theta _1, \omega _1}\,\textcircled {{F}}\, V_{\theta _2, \omega _2}, \, \, x\in {{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2 \end{aligned}$$

where \(V_{\theta _i, \omega _i}\in {{\mathcal {U}}}({{\mathcal {H}}}_{\omega _i})\) is the self-adjoint unitary that implements \(\theta _i\), \(i=1, 2\). But then for \(x\in {{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) we have

$$\begin{aligned}&\Vert \theta (x)\Vert _{\min }\\&\quad =\sup \{ \Vert \pi _{\omega _1\times \omega _2}(\theta (x))\Vert : \omega _i\in {{\mathcal {S}}}_+({{\mathfrak {A}}}_i),\, i=1, 2 \}\\&\quad =\sup \{\Vert V_{\theta _1, \omega _1}\,\textcircled {{F}}\, V_{\theta _2, \omega _2} \left( \pi _{\omega _1\times \omega _2} (x)\right) V_{\theta _1, \omega _1}\,\textcircled {{F}}\, V_{\theta _2, \omega _2}\Vert : \omega _i\in {{\mathcal {S}}}_+({{\mathfrak {A}}}_i),\, i=1, 2 \}\\&\quad =\sup \{ \Vert \pi _{\omega _1\times \omega _2} (x) \Vert : \omega _i\in {{\mathcal {S}}}_+({{\mathfrak {A}}}_i),\, i=1, 2 \}\\&\quad = \Vert x\Vert _{\min } \end{aligned}$$

and the proof is complete. \(\square\)

Recall that, for a trivial grading, any abelian \(C^*\)-algebra is nuclear, namely there is just one way to complete its algebraic tensor product with another \(C^*\)-algebra. This still holds for nontrivial \({{\mathbb {Z}}}_2\)-gradings. For the proof, we first consider a couple of technical results.

Lemma 4.8

For a (locally) compact Hausdorff space X, let \((C(X),\theta )\) be a \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebra with non-trivial grading. If K is a proper closed subset of \({\mathcal {E}}({\mathcal {S}}_+(C(X)))\), then there exists a non-null positive \(f\in C(X)\) such that \(\omega (f)=0\) for every \(\omega \in K\).

Proof

By Proposition 2.5, there exists a proper closed subset F of X such that \(K=\bigg \{\frac{1}{2}(\delta _x+\delta _{\theta (x)})\mid x\in F\bigg \}\). Since \(\theta ^2=\mathrm{id}_X\), in order for K to be a proper subset, it is necessary that \(G:=F\cup \theta (F)\) is still a proper subset of X. But then it is enough to consider a non-null positive function \(f\in C(X)\) that vanishes on G to have the thesis. \(\square\)

Lemma 4.9

Let \(({{\mathfrak {A}}}_1,\theta _1)\), \(({{\mathfrak {A}}}_2,\theta _2)\) be \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras with one of the two being commutative. Let \(\beta\) be any compatible \(C^*\)-norm on \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\), and take an extreme even state \(\omega\) on \({{\mathfrak {A}}}_1\,\textcircled {{F}}_\beta \,{{\mathfrak {A}}}_2\). Then there exist \(\omega _i\) extreme even states on \({{\mathfrak {A}}}_i\), \(i=1, 2\), such that \(\omega\) is the unique extension of \(\omega _1\times \omega _2\).

Proof

Suppose \({{\mathfrak {A}}}_1\) is commutative, and denote \({{\mathfrak {A}}}_\beta :={{\mathfrak {A}}}_1\,\textcircled {{F}}_\beta \,{{\mathfrak {A}}}_2\). If we set \(\widetilde{{{\mathfrak {A}}}}_1:=\{a_1\otimes {\mathbb {1}}_{{{\mathfrak {A}}}_2}: a_1\in {{\mathfrak {A}}}_1\}\), we see at once that \(\widetilde{{{\mathfrak {A}}}}_1\) lies in the centre \({\mathcal {Z}}({{\mathfrak {A}}}_\beta )\) of \({{\mathfrak {A}}}_\beta\). We claim that

$$\begin{aligned} \omega (yx)=\omega (y)\omega (x), \quad y\in {\mathcal {Z}}({{\mathfrak {A}}}_\beta ),\,\,x\in {{\mathfrak {A}}}_\beta . \end{aligned}$$

The thesis follows immediately from the claim if we define \(\omega _1(a_1):= \omega (a_1\otimes {\mathbb {1}}_{{{\mathfrak {A}}}_2})\), \(a_1\in {{\mathfrak {A}}}_1\), and \(\omega _2(a_2):=\omega ({\mathbb {1}}_{{{\mathfrak {A}}}_1}\otimes a_2)\), \(a_2\in {{\mathfrak {A}}}_2\).

Indeed, one has that \(\omega \lceil _{{{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2}=\omega _1\times \omega _2\), which means both \(\omega _1\) and \(\omega _2\) are even thanks to Proposition 2.6. Furthermore, they must also be extreme among all even states: if one of them, say \(\omega _1\), fails to be extreme, then from \(\omega _1=\gamma \varphi +(1-\gamma )\psi\) where \(0<\gamma <1\) and \(\varphi \ne \psi\) with \(\varphi , \psi\) even states, one finds \(\omega =\gamma \varphi \times \omega _2+(1-\gamma )\psi \times \omega _2\), contrary to the assumption that \(\omega\) is extreme among even states.

All is left to do is prove the claim. This can be done by following the proof of Lemma 4.11 in [13] word by word when one considers a positive even \(y\in {\mathcal {Z}}({{\mathfrak {A}}}_\beta )\). If one takes a positive odd y, then \(\omega (y)=0\) and \(\omega (yx)\) vanishes as well by a standard application of the Cauchy–Schwarz inequality. Finally, the case of a possibly non-homogeneous y follows easily by linearity. \(\square\)

Proposition 4.10

If \(({{\mathfrak {A}}}_1,\theta _1)\), \(({{\mathfrak {A}}}_2,\theta _2)\) are \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras and one of the two is commutative, then every compatible \(C^*\)-cross norm \(\beta\) on \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) coincides with \(\Vert \cdot \Vert _{\min }\).

Proof

Suppose that \({{\mathfrak {A}}}_1\) is commutative. We start by showing that \(\Vert x\Vert _\beta \le \Vert x\Vert _{\min }\) for every \(x\in {{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\). Indeed, if \({{\mathfrak {A}}}_\beta :={{\mathfrak {A}}}_1\,\textcircled {{F}}_\beta \,{{\mathfrak {A}}}_2\), for any such x, Corollary 2.2 gives

$$\begin{aligned} \Vert x\Vert _\beta&= \sup \{\Vert \pi _\omega (x)\Vert : \omega \in {\mathcal {E}}({\mathcal {S}}_+({{\mathfrak {A}}}_\beta ))\}\\&\le \sup \{\Vert \pi _{\omega _1\times \omega _2}(x)\Vert : \omega _i\in {\mathcal {E}}({\mathcal {S}}_+({{\mathfrak {A}}}_i)),\,\, i=1, 2\}\\&=\Vert x\Vert _{\min }, \end{aligned}$$

where the inequality is a straightforward application of Lemma 4.9.

In order to prove the converse inequality, \(\Vert x\Vert _{\min }\le \Vert x\Vert _\beta\) for every \(x\in {{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\), we have to show that every product state \(\omega _1\times \omega _2\), with \(\omega _i\in {\mathcal {E}}({\mathcal {S}}_+({{\mathfrak {A}}}_i))\), \(i=1, 2\), is continuous w.r.t. \(\Vert \cdot \Vert _\beta\) and thus can be extended to \({{\mathfrak {A}}}_\beta\). To this end, define

$$\begin{aligned} E_\beta :=\{(\omega _1, \omega _2)\in {\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_1))\times {\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_2)): \omega _1\times \omega _2\, {\text{extends to}}\,\, {{\mathfrak {A}}}_\beta \}. \end{aligned}$$

Note that \(E_\beta\) is closed in \({\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_1))\times {\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_2))\), where the latter set is understood as being equipped with the product of the relative weak* topologies, cf. [13, Lemma 4.17]. We need to prove that \(E_\beta ={\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_1))\times {\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_2))\). We shall argue by contradiction. Suppose \(E_\beta\) is properly contained in \({\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_1))\times {\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_2))\). Then there exist (proper) open subsets \(U_i\subset {\mathcal {E}}({\mathcal {S}}_+({{\mathfrak {A}}}_i)\), \(i=1, 2\), such that \((U_1\times U_2) \cap E_\beta =\emptyset\). Consider \(K_i:={\mathcal {E}}({\mathcal {S}}_+({{\mathfrak {A}}}_i))\setminus U_i\), \(i=1, 2\). By Lemma  4.8\(K_1\) cannot separate the positive elements of \({{\mathfrak {A}}}_1\), that is there exists a non-zero \(a_1\ge 0\) in \({{\mathfrak {A}}}_1\) such that \(\omega _1(a_1)=0\) for every \(\omega _1\in K_1\). In particular, if now \(a_2\) is any element in \({{\mathfrak {A}}}_2\), we have \(\omega _1\times \omega _2(a_1\otimes a_2)=0\) for every \((\omega _1, \omega _2)\in E_\beta\). Thus, by Lemma 4.9, it follows that \(\omega (a_1\otimes a_2)=0\) for each \(\omega \in {{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}_\beta ))\), which contradicts Proposition 2.1. \(\square\)

We finally prove that \(\Vert \cdot \Vert _{\min }\) on the \({{\mathbb {Z}}}_2\)-graded tensor product of graded \(C^*\)-algebras is minimal.

Lemma 4.11

Let \(({{\mathfrak {A}}}_1,\theta _1)\), \(({{\mathfrak {A}}}_2,\theta _2)\) be \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras, and let \(\beta\) be any compatible \(C^*\)-norm on \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\). If \(\omega\) is an extreme even state on \({{\mathfrak {A}}}_1\,\textcircled {{F}}_\beta \,{{\mathfrak {A}}}_2\) such that the restriction of \(\omega\) to \({{\mathfrak {A}}}_2\) is an extreme even state, then there exist \(\omega _i\) extreme even states on \({{\mathfrak {A}}}_i\), \(i=1, 2\), such \(\omega\) is the unique extension of \(\omega _1\times \omega _2\).

Proof

The proof can be done in much the same way as in Lemma 4.9. \(\square\)

Theorem 4.12

The spatial norm \(\Vert \cdot \Vert _{\mathrm{min}}\) is minimal among all compatible norms on \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\).

Proof

Given a compatible norm \(\Vert \cdot \Vert _\beta\) on \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\), we need to show that for every \(x\in {{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) one has \(\Vert x\Vert _{\min } \le \Vert x\Vert _\beta\). This amounts to proving that for any \(\omega _i\in {{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}_i))\), \(i=1, 2\), the product state \(\omega _1\times \omega _2\) is bounded w.r.t. the norm \(\Vert \cdot \Vert _\beta\). As usual, denote by \({{\mathfrak {A}}}_\beta\) the completion of \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) under the norm \(\Vert \cdot \Vert _\beta\), and define

$$\begin{aligned} E_\beta :=\{(\omega _1, \omega _2)\in {\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_1))\times {\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_2)): \omega _1\times \omega _2\, \mathrm{extends\, to}\,\, {{\mathfrak {A}}}_\beta \}. \end{aligned}$$

Again, \(E_\beta\) is a closed subset of \({\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_1))\times {\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_2))\) in the product of the relative weak* topologies. By contradiction, suppose \(E_\beta\) is properly contained in \({\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_1))\times {\mathcal {E}}({\mathcal {S}}_{+}({{\mathfrak {A}}}_2))\).

Let us denote by \({{\mathcal {P}}}({{\mathcal {S}}}({{\mathfrak {C}}}))\) the pure states of a \(C^*\)-algebra \({{\mathfrak {C}}}\). If

$$\begin{aligned} T_i: {{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}_i))\rightarrow {{\mathcal {P}}}({{\mathcal {S}}}({{\mathfrak {A}}}_{i, +})), \quad i=1, 2, \end{aligned}$$

are the affine homeomorphisms given by the restriction map (as in Proposition 2.4), then

$$\begin{aligned} T_1\times T_2:{{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}_1))\times {{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}_2))\rightarrow {{\mathcal {P}}}({{\mathcal {S}}}({{\mathfrak {A}}}_{1, +}))\times {{\mathcal {P}}}({{\mathcal {S}}}({{\mathfrak {A}}}_{2, +})) \end{aligned}$$

is an affine homeomorphism. If we set \(S_\beta := (T_1\times T_2) (E_\beta )\), we clearly have that \(S_\beta\) is a proper closed subset of \({{\mathcal {P}}}({\mathcal {S}}({{\mathfrak {A}}}_{1, +}))\times {{\mathcal {P}}}({\mathcal {S}}({{\mathfrak {A}}}_{2, +}))\). We can now proceed as in the proof of [13, Lemma 4.18 ] to find non-zero positive elements \(a_i\in {{\mathfrak {A}}}_{i,+}\) such that \(\omega _1\times \omega _2(a_1\otimes a_2)=0\) for any \((\omega _1, \omega _2)\in E_\beta\). A contradiction will be arrived at if we show that the set of states \(\{\omega _1\times \omega _2: (\omega _1, \omega _2)\in E_\beta \}\) actually separates elements in \({{\mathfrak {A}}}_\beta\) of the form \(a_1\otimes a_2\) with \(a_1\) and \(a_2\) both even and positive. To this end, let \(\widetilde{{{\mathfrak {A}}}}_1\subset {{\mathfrak {A}}}_{1, +}\) be the unital \(C^*\)-subalgebra generated by \(a_1\). Let \(\rho \in {{\mathcal {P}}}({{\mathcal {S}}}(\widetilde{{{\mathfrak {A}}}}_1))\) such that \(\rho (a_1)\ne 0\) and let \(\varphi \in {{\mathcal {E}}}({{\mathcal {S}}}_+({{\mathfrak {A}}}_2))\) such that \(\varphi (a_2)\ne 0\). The product state \(\rho \times \varphi\) is bounded on \(\widetilde{{{\mathfrak {A}}}}_1\,\textcircled {{F}}{{\mathfrak {A}}}_2\) with respect to the norm \(\Vert \cdot \Vert _\beta\) because its restriction to \(\widetilde{{{\mathfrak {A}}}}_1\,\textcircled {{F}}{{\mathfrak {A}}}_2\) is just the spatial norm \(\Vert \cdot \Vert _{\mathrm{min}}\) by virtue of Proposition 4.10. Let now \(\omega\) be any extreme even extension of \(\rho \times \varphi\) to \({{\mathfrak {A}}}_\beta\). By Lemma 4.11\(\omega\) must be of the form \(\omega _1\times \omega _2\) with \(\omega _i\in {{\mathcal {E}}}({{{\mathcal {S}}}_+({{\mathfrak {A}}}_i)})\), \(i=1, 2\), and the proof is thus complete. \(\square\)

Corollary 4.13

Given \({{\mathbb {Z}}}_2\)-graded \(C^*\)-algebras \(({{\mathfrak {A}}}_i, \theta _i)\), \(i=1, 2\), any compatible norm on \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) is automatically cross.

Proof

Let \(\Vert \cdot \Vert _\beta\) be any norm on \({{\mathfrak {A}}}_1\,\textcircled {{F}}\,{{\mathfrak {A}}}_2\) as in the statement. For any \(a_i\in {{\mathfrak {A}}}_i\), \(i=1, 2\), one clearly has \(\Vert a_1\otimes a_2\Vert _\beta \le \Vert a_1\Vert \Vert a_2\Vert\). On the other hand, by Theorem 4.12 for homogeneous \(a_i\)’s we also have \(\Vert a_1\otimes a_2\Vert _\beta \ge \Vert a_1\otimes a_2\Vert _{\mathrm{min}}=\Vert a_1\Vert \Vert a_2\Vert\), where the last equality is due to Proposition 4.3. \(\square\)