Relationship to a construction by Sims and Yeend

Let us restrict our attention to compactly aligned product systems over positive cones of quasi-lattice orders. In [55], Sims and Yeend constructed a C^∗-algebra N O_E from acompactly aligned product system E = (Ep)p∈P so that it generalises constructions such as C^∗-algebras associated to finitely aligned higher rank graphs and Katsura’s Cuntz–Pimsner algebra of a single correspondence. The universal representation ofE inN O_E is quite often faithful, but Example 3.16 of [55] shows that it may fail to be injective even if (G, P) is totally ordered andA acts by compact operators onEp for allpinP. In this subsection, we will see thatN O_E coincides withA×_EP when either the universal representation ofE inN OE is faithful andP is directed orE is a faithful product system. In both casesN OE satisfies an analogue of (C3) [15, Proposition 3.7]. This subsection is based on [15] and [55].

We first recall the definitions from [55] of Cuntz–Nica–Pimsner covariance and Cuntz–Nica–Pimsner algebra. Fix a quasi-lattice ordered group (G, P) and letE = (Ep)p∈P be a compactly aligned product system overP. Let ¯I_e:=A and, for eachp∈P\ {e}, set A representationψof E in a C^∗-algebra B isCuntz–Pimsner covariant according to [55, Definition 3.9] if TheCuntz–Nica–Pimsner algebraassociated toE, denoted byN O_E, is then the universal C^∗-algebra for Cuntz–Nica–Pimsner covariant representations (see [55, Proposition 3.2] for further details). The requirement thateι^p_e be faithful for allp∈P implies that the representation ofE inN O_E is faithful.

Sims and Yeend proved in [55, Lemma 3.15] that this is satisfied wheneverP has the following property:

given a non-empty setF ⊆P that is bounded above, in the sense that there isp∈P withs≤pfor alls∈F, thenF has a maximal elementr. That is,r6≤sfor alls∈F\ {r}.

The next example of a product system is given by Sims and Yeend in [55, Example 3.16]. It consists of a compactly aligned product system for which not alleι^p_e’s are injective. We recall their example here and describe its associated covariance algebra.

Example 6.3.1. Let Z×Zbe equipped with the lexicographic order and let P be its positive cone.

SoP = (N\ {0})×Z

∪ {0} ×Nande={0} × {0}. Define a product system overP as follows:

letA:=C² and, for eachp∈P, letEp :=C² be regarded as a HilbertA-module with right action given by coordinatewise multiplication and usualC²-valued inner-product. Following the notation of [55], we setS := {0} ×N and for all p∈ S, we let C² act on Ep on the left by coordinatewise

For allq∈P\S, define µp,q:Ep⊗_C2Eq ∼=Epq by

(z1, z2)⊗(w1, w2)7→(z1w1, z1w2).

This is a proper product systemE= (Ep)p∈P overC². Thus it is also compactly aligned. SinceP is totally ordered, all representations ofE are Nica covariant. Sims and Yeend proved that such a product system has no injective Cuntz–Nica–Pimsner covariant representation. Their argument is the following: for allp6=e, ¯I_p= kerϕ_(0,1) ={0}. Hence, ifq∈P\S,eι^q_e=ϕ_q is not injective and any Cuntz–Nica–Pimsner covariant representation ofE vanishes on kerϕ_q ={0} ×C.

Let us now describe the associated covariance algebraA×_E P. We will show that (A×_E P)^δ is

ofEinB(`²(N×Z)). We claim thatφis strongly covariant and preserves the topologicalZ×Z-grading ofA×EP. First, for each finite setF ⊆P,

So in order to prove thatφis strongly covariant, it suffices to verify that, given a finite setF ⊆P, one

has X

So by taking finite setsF⁰⊆P withF⁰⊇F, we conclude from the definition of strong covariance that X

for allr= (r₁, r₂)> pwithr₁> p₁. In case p₂=p₁+ 1, then Repeating this argument for all of thepi’s and observing that

φ^(p)(Tp)(f)(q) =

we conclude thatφis indeed strongly covariant. The associated representation ofA×_EP on`<sup>2</sup>(N×Z) is faithful on (A×<sub>E</sub> P)<sup>δ</sup> because it is injective onC<sup>2</sup>. Its image in B(`²(N×Z)) is the C^∗-algebra generated byφe(C²) and the family of isometries{vp|p∈P}.

To see thatφbis faithful on A×EP, consider the canonical unitary representation of the torusT² on`²(N×Z). Explicitly, the unitaryU_zis given by

U_z(f)(q) =z^q₁¹z₂^q2f(q), q= (q₁, q₂)∈N×Z,

wherez= (z₁, z₂)∈T². This produces a continuous action ofT² onφb(A×EP) byT 7→U T U^∗.Hence it carries a topologicalZ×Z-grading (see Section 2.2). The corresponding spectral subspace at (m, n) is determined by amenable,φbis then an isomorphism onto its image. Its restriction to (A×EP)^δ yields an isomorphism onto the C^∗-algebra of all convergent sequences

(ζn)n∈N∈`^∞(N)| ∃ lim

The task of verifying whether a given representation is strongly covariant or not is considerably simplified whenE is compactly aligned. The proof of the next proposition is taken from [15, Proposition 3.7] and adapted to our context.

Proposition 6.3.2. Let E= (Ep)p∈P be a compactly aligned product system. A representationψ ofE in aC^∗-algebraB is strongly covariant if and only if it is Nica covariant and satisfies

(C)’ P be viewed as a closed submodule of E_F⁺ (see Section 6.1 for further details). SoP

p∈Fj^(p)(Tp) = 0 inA×_E P and, in particular,P

p∈Fψ^(p)(T_p) = 0.

Conversely, assume thatψ is Nica covariant and satisfies (C)’. In order to prove thatψis strongly covariant, we use the ideas employed in [15]. LetP_fin^∨ denote the set of all finite subsets ofP that are

∨-closed. Precisely,F∈ P_fin^∨ if it is finite and for allp, q∈F withp∨q <∞, one hasp∨q∈F. For

Here we introduce no special notation to identify an element ofK(E_p) with its image in N T_E.We observe thatB_F is a C^∗-subalgebra ofN T_E^eand, in addition, (see, for example, [10]). The previous proposition combined with [53, Theorem 6.3] gives us the

following:

Corollary 6.3.3. Let(G, P)be a quasi-lattice orderd group and letE = (Ep)p∈P be a compactly aligned product system. Suppose thatGis amenable. If Ais nuclear, thenA×EP is nuclear.

We denote byqN the^∗-homomorphism fromN TE to A×EP induced byjE ={jp}p∈P. The proof of the next result is essentially identical to that of Proposition 6.3.2. This is inspired by [15, Proposition 3.7].

Proposition 6.3.4. Letψ be an injective Nica covariant representation ofE in aC^∗-algebraB and letψ_N denote the induced^∗-homomorphism. Then(kerψ_N)∩ N T^e_E ⊆kerq_N.

The following is [15, Example 3.9].

Example 6.3.5. Let F2 denote the free group on two generators a and b. Then F2 is quasi-lattice ordered and its positive coneF⁺2 is the unital semigroup generated byaandb. Define a product system overF⁺2 by settingA:=C,Ea:=CandEb:={0}, whereCis regarded as a Hilbert bimodule overCin the usual way. SoEaⁿ=Cfor alln∈N. A subset ofF⁺2 that is bounded above has a maximal element, so that the representation ofE inN OE is injective. However, in [15] this example illustrates the fact that the conclusion of Proposition 6.3.4 may fail forN OE ifP is not directed andE is non-faithful.

Define a representation of E inC by ψp(λp) = λp for all p∈ P andλp ∈ Ep. So ψe is faithful.

Proposition 6.3.6. Let(G, P) be a quasi-lattice ordered group and letE = (Ep)p∈P be a compactly aligned product system over P. Suppose either that E is faithful or that P is directed and the rep-resentation of E in N O_E is injective. Then N O_E and A×_E P are canonically isomorphic to each other.

Proof. Let ¯j_E denote the representation ofE inN O_E.By Proposition 6.3.4, ker ¯j_N∩ N T_E^e⊆kerq_N. In particular,j_E is an injective Cuntz–Nica–Pimsner covariant representation of E inA×_E P. Hence, [15, Proposition 3.7] implies that the induced^∗-homomorphismj:N O_E →A×_E P is faithful on the fixed-point algebraN O^e_E. Therefore, ¯j_N vanishes on kerq_N and it factors throughA×_EP. Thusb¯j_N is the inverse ofj