1 Introduction↩︎

We show in this paper that if \(P\) is a group-embeddable or right LCM cancellative semigroup, then every representation of \(P\) that extends to a completely contractive representation of the non-selfadjoint operator algebra generated by the left-regular representation \({\mathcal{A}}_\lambda(P)\), dilates to a representation of the reduced boundary quotient \(C^\star\)-algebra \(\partial C^\star_\lambda(P)\). The beauty of our technique is that it is quite a general dilation theoretic idea that we use. While the first mentioned property (complete contractivity) might be difficult to verify in general, it turns out to be an appropriate generalisation of the matricial von Neumann’s inequality for \(\mathbb{N}^k\). And while the boundary quotient is sometimes difficult to compute, for Ore semigroups, it is just the \(C^\star\)-algebra of the enveloping group, and thus dilation to the boundary quotient turns out to be an appropriate generalisation of the notion of unitary dilation. This allows us to bring existing well-known dilation theorems like those of B. Sz.-Nagy [1], T. Ando [2], and J. A. Holbrook [3] under the same umbrella, provides necessary and sufficient conditions for unitary dilations of Ore semigroup representations, and demonstrates that they are all manifestations of the same dilation phenomenon. This is considerably more general than existing generalisations of Sz.-Nagy’s dilation theorem, like the results by S. Brehmer [4].

The general goal of dilation theory is to view maps between spaces as parts of nicer maps between potentially larger spaces. For example, contractions on Hilbert spaces may be dilated to unitaries on larger Hilbert spaces, upto powers [1]: for any contraction \(T\) on a Hilbert space \({\mathcal{H}}\), there exists a unitary \(U\) on a bigger Hilbert space \({\mathscr K}\supseteq {\mathcal{H}}\) such that \(T^n = \mathscr{P}_{\mathcal{H}}U^n \vert_{\mathcal{H}}\) for all \(n \in \mathbb{N}\) . This can be done for commuting pairs of contractions as well [2]. However, such a result fails to hold more generally for arbitrary commuting triples of contractions, as noted in [5], unless they are \(2 \times 2\) complex matrices, in which case power dilations work for all \(k\)-tuples [3]. This leads one to the following question, the answer to which is well-known:
Question A: Under what conditions may commuting \(k\)-tuples of contractions be dilated to commuting \(k\)-tuples of unitaries upto powers?
Answer A: A contractive \(k\)-tuple of commuting contractions has a commuting unitary dilation upto powers if and only if it satisfies a matricial von Neumann inequality.
Since \(k\) commuting contractions induce a contractive representation of \(\mathbb{N}^k\), we realise that perhaps this question is answered in greater generality by viewing the representations of semigroups (in this case \(\mathbb{N}^k\), but potentially a lot more general) as (non-selfadjoint) operator algebras, and their dilation theory. For example, one may also formulate the following question:
Question B: Under what conditions may a contractive representation of a semigroup be dilated to a unitary representation? Or, more generally, what might be the counterpart of a unitary dilation?

As it turns out, the boundary quotient is the correct choice for generalising unitary dilations. We briefly recall some of the motivation behind dilation to the “boundary”. Consider the disk algebra \({\mathcal{A}}({\mathbb{D}})\), holomorphic functions on the unit disk \({\mathbb{D}}\) which are bounded and hence extend to continuous functions on its closure \(\bar{{\mathbb{D}}}\). They form a non-selfadjoint algebra. Their closure under involution \(C(\bar{{\mathbb{D}}})\) is a \(C^\star\)-algebra, but it is not the “smallest” \(C^\star\)-algebra into which \({\mathcal{A}}({\mathbb{D}})\) embeds isometrically. The right candidate for the latter would be all continuous functions on the unit circle, \(C(\mathbb{T})\), and one may verify this by observing that by the maximum modulus principle, the map that sends a function in \({\mathcal{A}}({\mathbb{D}})\) to its restriction on \(\mathbb{T}\) is in fact an isometry. One notes that \(\mathbb{T}\) is the (Šilov) boundary, and this phenomenon generalises classically to all homogeneous algebras. \(C^\star\)-algebras. This was one of the motivations behinds the development of the theory of \(C^\star\)-envelopes. For a (possibly noncommutative and non-selfadjoint) operator algebra \({\mathcal{A}}\), one may consider \(C^\star_{env}({\mathcal{A}})\) to be the “smallest” \(C^\star\)-algebra into which \({\mathcal{A}}\) embeds completely isometrically. \(C^\star\)-envelopes were first defined by W. Arveson [6], and he conjectured their existence in general. In case of unital operator algebras, the existence of the \(C^\star\)-envelope was proved by M. Hamana through injective envelopes [7] and M. Dritschel and S. A. McCullough through dilation-theoretic techniques [8]. The non-unital case follows from R. Meyer’s work [9]. The Dritschel-McCullough dilation theorem shows that every completely contractive representation of \({\mathcal{A}}\) has a maximal dilation, and the \(C^\star\)-algebra generated by a maximal dilation of a completely isometric representation of \({\mathcal{A}}\) is \(C^\star_{env}({\mathcal{A}})\), which is in general “smaller” than \(C^\star({\mathcal{A}})\) (in fact, a quotient), in a similar fashion as \(C(\mathbb{T})\), the \(C^\star\)-envelope of \({\mathcal{A}}({\mathbb{D}})\), is a quotient of \(C(\bar{{\mathbb{D}}})\). Moreover, ever completely contractive representation of \({\mathcal{A}}\) dilates to a representation of \(C^\star_{env}({\mathcal{A}})\).

One makes a connection here, that \({\mathcal{A}}(\bar {\mathbb{D}})\) is in fact the operator algebra generated by a representation of the semigroup \(\mathbb{N}\) on the Hilbert space \(L_2(\bar {\mathbb{D}})\) that maps the generator \(1\) to \(z\). So, one might venture to ask what a suitable notion of boundary might be for representations of semigroups in general. We point out that in the context of representations of semigroups, one has a quotient object, the boundary quotient, which is in general smaller than the \(C^\star\)-algebra generated by the left-regular representation of the semigroup. The representation theory of semigroups is considerably more involved than their counterpart for groups, in the same fashion as non-selfadjoint operator algebras often offer more flexibility (and hence, hide greater complexity) in their representations than their selfadjoint counterparts. In constructing both universal and reduced \(C^\star\)-algebras of semigroups (\(C^\star_s(P)\) and \(C^\star_\lambda(P)\) respectively), one must account for their ideal structure [10]. Even then, one looks for “smaller” \(C^\star\)-algebras which encode complete information about \(P\), and this leads us to the boundary quotient. For \(\mathbb{N}^k\), the (full and reduced) boundary quotient is isomorphic to \(C(\mathbb{T}^k) \cong C^\star(\mathbb{Z}^k)\), which is a quotient of \(C^\star(\mathbb{N}^k)\), a much bigger \(C^\star\)-algebra (the Toeplitz algebra in the \(k=1\) case). Similar phenomena are seen for free semigroups \(\mathbb{F}_k^+\), where the \(C^\star\)-algebra of the semigroup is the Cuntz-Toeplitz algebra \(\mathcal{TO}_k\), while the boundary quotient is the Cuntz algebra \(\mathcal{O}_k\), a quotient of \(\mathcal{TO}_k\). This phenomenon was generalised to right-angled Artin monoids by J. Crisp and M. Laca in [11]. Their construction was further extended to more general semigroups in two directions, one through full and reduced partial crossed products (e.g. [12], [13]), the other through quotienting these \(C^\star\)-algebras by products of complements of projections of ideals which intersect every ideal non-trivially (called the boundary quotient relations) [14]. If \(P\) embeds into a group \(G\), one has the co-universal property that reduced boundary quotient \(\partial C^\star_\lambda(P)\) is the “smallest” \(C^\star\)-algebra generated by a \(G\)-equivariant representation of \(C^\star_s(P)\) [13]. C. Sehnem proved in [15] that \(C^\star_{env}({\mathcal{A}}_\lambda(P))\) is canonically isomorphic to \(\partial C^\star_\lambda(P)\) if \(P\) is group-embeddable, and the same result was proved without the group-embeddability assumption by K. Brix, C. Bruce, and A. Dor-On in [16] in the right LCM setting, assuming cancellation.

And this finally leads us to our dilation result. If a representation of a semigroup \(P\) extends to a completely contractive representation of \({\mathcal{A}}_\lambda(P)\), then we may use the Dritschel-McCullough dilation machine to dilate it to a representation of \(C^\star_{env}(P)\). Then, if the criterion of [15], [16], are met, we may dilate it to a representation of \(\partial C^\star_\lambda(P)\), and thus we have our main theorem:

This immediately allows us to answer question \(B\) posed above. By Theorem 31, we have that for a semigroup \(P\) with enveloping group \(G\), \(\partial C^\star_\lambda(P)\) is isomorphic to \(C^\star_\lambda(G)\) if and only if \(P\) is right Ore. Given that group representations are unitary, this allows us to answer Question B:
Answer B: If \(P\) is Ore, then every contractive representation that extends to a completely contractive representation of \({\mathcal{A}}_\lambda(P)\), has a unitary dilation.
Sufficient conditions for representations to dilate to \(\mathcal{Q}(P)\) in the right LCM setting were obtained by M. Laca and B. Li in [17] through constructive methods, but their conditions are very strict, and are not satisfied by many representations, including the classical ones in [2], [3]. But the conditions we require for the Dritschel-McCoullough dilation machinery to operate for the dilation to \(\partial C^\star_\lambda(P)\) are more general than theirs, and not just sufficient but also necessary. This allows one to reconcile with some existing operator-theoretic dilation results like the ones mentioned above. The only drawback is that it might be difficult to check complete contractivity in specific cases, and may still need heavy operator-theoretic machinery. But we show that this condition is equivalent to the matricial von Neumann inequality in the case of \(\mathbb{N}^k\), and hence is not expected to require techniques considerably more difficult than existing ones, and is in fact more generally applicable.

This relation between matricial norm properties of a representation of a semigroup and the existence of a dilation to its boundary quotient might seem unexpected, and makes sense only through reflection into the nature of the Dritschel-McCullough dilation machine, and the equivalence of the \(C^\star\)-envelope and the boundary quotient. Meanwhile, one may note that though the semigroup and its boundary quotient are generated by the same elements, the latter might have many more relations, simply due to the fact that the former does not have inverses in general while the latter has adjoints which must respect the ideal structure of the former. These are precisely the foundation set relations. This leads us to the Corollary 52, that every representation that extends to a completely contractive representation of \({\mathcal{A}}_\lambda(P)\), also admits a dilation with these additional relations among the generators.

The structure of the paper is the following: in section 2, we provide dilation-theoretic preliminaries, and introduce the notions of envelopes and boundaries. In section 3, we provide background on semigroups, while 4 and 5 are background on semigroup \(C^\star\)-algebras and boundary quotients respectively. Finally, in section 6, we demonstrate the dilation phenomenon.

Acknowledgements↩︎

We wish to thank Chris Bruce, Adam Dor-On, and Xin Li for various insightful comments. We also wish to thank Joachim Zacharias, Boyu Li, Mike Whittaker, Charles Starling, and Runlian Xia for helpful comments on the draft.

2 Preliminaries on Dilation Theory, Envelopes, and Boundaries↩︎

The general goal of dilation theory is to view an operator (or a map into a space of operators) as a part of something simpler on a larger space. Its origins can be traced to operator dilation theory stemming from B. Szőkefalvi-Nagy’s celebrated dilation theorem [1]. In this paper, we shall be dealing principally with dilations of representations. For us, an operator algebra \({\mathcal{A}}\) will be a norm-closed subalgebra of \({\mathcal{B}}({\mathcal{H}})\), not necessarily self-adjoint. Furthermore, we shall assume our operator algebras to be unital, and our representations to be unital as well. We refer the reader to K. R. Davidson’s recent textbook [18] for a detailed account on some of the topics we briefly mention in this section, and O. M. Shalit’s notes [19] for a detailed introduction to dilation theory. For a general introduction to completely bounded maps we refer to V. I. Paulsen’s celebrated textbook [20].

We should mention that there is a high amount of non-uniformity in terminology regarding the terms introduced in the rest of this section. What we refer to as “maximal” dilations, are occasionally called “extremal”, or \(\partial\)-representations. We refer the reader to [8], [18], [21], [22] for further reading.

The second condition in Theorem 3 is called the unique extension property.

This theorem also proved that the \(C^\star\)-envelope exists for every operator algebra. We are yet to properly define what a boundary means in this context. The basic examples of \(C^\star\)-envelopes, as well as motivation for boundaries, arise in the context of uniform algebras.

The result in Example 5 is not an isolated one. The \(C^\star\)-envelope of any uniform algebra is the \(C^\star\)-algebra of continuous functions on its Šilov boundary. The unconditional existence of the \(C^\star\)-envelope (Theorem 4) provides a noncommutative analogue of this result.

The existence of the Šilov ideal can be seen from the existence of \(C^\star\)-envelope: \(C^\star_{env}({\mathcal{A}})\) is canonically isomorphic to \(C^\star({\mathcal{A}})/\mathcal{I}_s\).

3 Preliminaries on Semigroups↩︎

A semigroup is a set with an associative binary operation, traditionally referred to as a multiplication. In this paper that by semigroup we mean a semigroup with an identity element, that is, a monoid. For example, we shall in this spirit assume \((\mathbb{N},+)\) to mean the additive natural numbers with \(0\). All semigroups under consideration in this paper will be discrete and countable. We begin with the following essential property:

We denote the set of invertible elements (also called units) of \(P\) by \(P^\star\). Every group is (trivially) a semigroup, and it is interesting to consider which conditions on a semigroup would ensure that it is (faithfully) embeddable in a group. A necessary but not sufficient condition for this is the cancellative property. A. Mal’cev gave an infinite list of conditions which are necessary and sufficient for group-embeddability of a semigroup, and showed that no finite list of those conditions suffices. We refer the reader to Section 12 of [23] for further details. We mention one particular situation in which \(P\) is group-embeddable, which will be useful later. We denote \(pP := \{{px \; \vert \; x \in P}\}\) and define \(Pp\) in a similar fashion.

We direct the reader to Theorem 1.24 in [24] or Section 1.1 of [25] for further details on Theorem 9. Not all group-embeddable semigroups are Ore: for example, see [26] for the case of Artin monoids. It is also a pertinent question as to which group we choose to embed \(P\) into, if there are multiple options. Here, we may carry out a universal construction following [23]. If a semigroup \(P\) is group-embeddable, there exists a universal group embedding \(\iota_{univ} : P \hookrightarrow G_{univ}(P)\) such that for any other group embedding \(\iota : P \hookrightarrow G\), we have a unique homomorphism \(\Phi:G_{univ}(P) \to G\) such that \(\iota = \Phi \circ \iota_{univ}\), that is, the following diagram commutes:

Hence, it will suffice for us to just state that that \(P\) embeds into some group \(G\) without specifying which group it is.

In the group-embeddable setting, A. Nica first defined quasi-lattice ordered semigroups in [27], and studied their isometric covariant representations. The following generalisation, which was developed in [14], removed the group-embeddability assumption and the reference to the order.

This \(r\) may possibly be non-unique, but one observes directly that if \(r\) and \(r'\) both satisfy \(pP \cap qP = rP = r'P\), then there exists \(u \in P^\star\) such that \(r' = r u\). We denote the set of right LCM elements of \(p,q\) by \(p \vee q = \{{r \in P \textrm{ } \; \vert \; \textrm{ }pP \cap qP = rP}\}\). For a finite subset \(F \subset P\), we denote \(\vee F := \{{r \in P \textrm{ } \; \vert \; \textrm{ }\bigcap_{p \in F} pP = rP}\}\). These semigroups have been referred to and studied under various names in the past: for examples, M. Lawson considers the dual definition in [28] as CRM monoids (after Clifford, Reilly, and McAlister), and M. D. Norling refers to them as semigroups satisfying Clifford’s condition in [29].

We clarify a further bit of notation for the subsequent considerations. For all \(p \in P\) and \(X \subseteq P\), we denote \(p^{-1}X := \{{x \in P \; \vert \; px \in X}\}\). Further, if \(P\) embeds in \(G\), \(p^{-1}X = \{{p^{-1}x \; \vert \; x \in G}\} \cap P\). For \(p_1,q_1, \dots , p_n, q_n \in P\), the set \(p_1^{-1}q_1p_2^{-1}q_2 \dots p_n^{-1}q_nP\) does not depend on the actual product of these elements, but is equal to \(p_1^{-1}(q_1(p_2^{-1}(q_2( \dots (p_n^{-1}(q_nP))))))\), which is equal to \(p_1^{-1}q_1P \cap p_1^{-1}q_1p_2^{-1}q_2P \cap \dots \cap p_1^{-1}q_1p_2^{-1}q_2 \dots p_n^{-1}q_nP\). Finally, we conclude this section with some examples:

4 Semigroup \(C^\star\)-algebras and Amenability↩︎

In recent years, there has been significant interest in the study of \(C^\star\)-algebras and operator algebras of semigroups [32]. The representation theory of semigroups is much more nuanced than their counterpart for groups. While it is a perfectly good idea to define the reduced semigroup \(C^\star\)-algebra of a semigroup \(P\) as the norm-closed subalgebra of \({\mathcal{B}}(\ell_2(P))\) generated by the (isometric) left-regular representation and closed under involution, unfortunately, considering the universal algebra generated by these isometries yields something that is far too large. In fact, it was proved by G. J. Murphy [33] that such a \(C^\star\)-algebra for even \(\mathbb{N}^2\), the universal \(C^\star\)-algebra generated by two commuting isometries, is non-nuclear! The problem was addressed by X. Li in [10] by insisting that the representation keep track of the ideal structure of the semigroup as well, and this led to proper notions of universal \(C^\star\)-algebras of semigroups. X. Li also studied the notion of amenability in the context of semigroups in the same paper.

4.1 Semigroup \(C^\star\)-algebras↩︎

Consider a discrete countable left cancellative semigroup \(P\), and its left-regular representation \(V: P \to {\mathcal{B}}(\ell_2(P))\), \(V_p(\delta_q) = \delta_{pq}\). It is straightforward to verify that \(V\) is an isometric representation (i.e., \(V_p\) is an isometry for every \(p \in P\)). This allows us to define the following reduced semigroup algebras.

Some authors also refer to \(C^\star_\lambda(P)\) as the Toeplitz algebra of the semigroup, \(\mathcal{T}_\lambda(P)\). A proper candidate for a universal semigroup \(C^\star\)-algebra was proposed by X. Li in [10], using the structure of the space of right ideals of the semigroup. We say a subset \(X \subseteq P\) is a right (resp. left) ideal of \(P\) if \(Xp \subseteq X\) (resp. \(pX \subseteq X\)) for all \(p \in P\). Groups lack non-trivial ideals, but for semigroups such spaces can be interesting. We further restrict ourselves to the space of constructible right ideals, which is defined in the following fashion:

One might observe here that the set \(\mathcal{J}\) is actually a semilattice, i.e. an abelian semigroup of idempotents with the operation being intersection of ideals. There is a duality between semilattices and totally disconnected locally compact Hausdorff spaces. We refer the reader to [32] for further details on this duality. It is proved in Lemma 3.3 of [10] that (I1) and (I2) imply (I3), that is, one may state: \[\begin{align} \label{eq:32form32of32Ideals} \mathcal{J}= \left\{ q_1^{-1} p_1 \dots q_n^{-1} p_n P \; \vert \; n \in \mathbb{N}, \textrm{ and }p_i,q_i \in P \textrm{ for all } i\in\{1, \dots n\} \right\} \cup \{\emptyset\}. \end{align}\tag{1}\] We further need to define the following independence condition on \(\mathcal{J}\), from [10]:

If \(P\) is right LCM (as in Definition 10), then all constructible right ideals are principal, that is, \(\mathcal{J}= \{ pP \; \vert \; p \in P \}\). The independence condition on \(\mathcal{J}\) is thus automatically satisfied if \(P\) is right LCM. In [10], X. Li formulated the following two universal \(C^\star\)-algebras, Definitions 15 and 16, which capture the ideal structure of the semigroup:

By Lemma 2.8 of [10], these definitions imply that \(v_p^\star e_X v_p = e_{p^{-1}X}\), and thus \[\begin{align} \label{eq:32e-v32relations} e_{q_1^{-1} p_1 \dots q_n^{-1} p_n P} = v_{q_1}^\star v_{p_1} \dots v_{q_n}^\star v_{p_n} v_{p_n}^\star v_{q_n} \dots v_{p_1}^\star v_{q_1} \end{align}\tag{2}\]

By equation 1 , this proves that \(C^\star(P)\) is generated by the isometries \(\{v_p \; \vert \; p \in P\}\). The canonical surjective \(\star\)-homomorphism, the left-regular representation \(\lambda:C^\star(P) \to C^\star_\lambda(P)\), can thus be affected by mapping \(v_p\) to \(V_p\) for every \(p \in P\). We define the diagonal subalgebra \[D(P) := C^\star(e_X \; \vert \; X \in \mathcal{J}) \subseteq C^\star(P),\] that is, the universal \(C^\star\)-algebra generated by the projections corresponding to elements in \(\mathcal{J}\). By (F2iii), one may observe that this is a commutative \(C^\star\)-algebra. For every \(X \in \mathcal{J}\), one has the projection \(E_X \in C^\star_\lambda(P)\) (on the subspace \(\ell_2(X) \subseteq \ell_2(P)\)) which may be obtained from equation 2 by applying the map \(\lambda\). This allows one to define \[D_\lambda(P) := C^\star(E_X \; \vert \; X \in \mathcal{J}) = \lambda(D(P)) \subseteq {\mathcal{B}}(\ell_2(P)).\]

X. Li further defines the following universal \(C^\star\)-algebra when \(P\) is group-embeddable, which is instrumental in most considerations:

It can be shown that conditions (F1ii), (F2i), (F2iii) can be deduced to hold from (S3), and hence there exists a canonical surjective \(\star\)-homorphism \(\pi_s:C^\star(P) \to C^\star_s(P)\) that maps \(v_p\in C^\star(P)\) to \(v_p\in C^\star_s(P)\) and \(e_X\in C^\star(P)\) to \(e_X\in C^\star_s(P)\). \(C^\star_s(P)\) is also generated by \(\{v_p \; \vert \; p \in P\}\), and one has a a canonical \(\star\)-homomorphism, the left-regular representation, \(\lambda_s:C^\star_s(P) \to C^\star_\lambda(P)\), which maps \(v_p\) to \(V_p\). Furthermore, one defines \[D_s(P) := C^\star(e_X \; \vert \; X \in \mathcal{J}) \subseteq C^\star_s(P),\] and \(\pi_s\vert_{D(P)}:D(P) \to D_s{(P)}\) is a surjective \(\star\)-homomorphism.

That is, one has the following commutative diagrams:

One should be careful that unlike the reduced case, the term “universal Toeplitz \(C^\star\)-algebra” \(\mathcal{T}_u(P)\) means something different from the universal \(C^\star\)-algebras defined here. M. Laca and C. Sehnem defined and studied these universal algebras. \(\mathcal{T}_u(P)\) is in general a quotient of \(C^\star_s(P)\), and they are isomorphic if and only if \(\mathcal{J}\) satisfies the independence condition. We refer the reader to [12] for details.

4.2 Amenability of Semigroups↩︎

We briefly look at how amenability of the semigroup manifests itself in the relation between its semigroup \(C^\star\)-algebras. We state the definition used by X. Li in [10]:

It is also possible to define right-amenability in a similar fashion. Amenability of semigroups is in general a broad and interesting topic: for example, X. Li showed in [10] that one may obtain for semigroups conditions analogous to (1)-(7), Theorem 6.8, [34], as stated for groups. However, we restrict ourselves to only two of these conditions for the moment, and refer the interested reader to Section 4 of [10], or to A. Paterson’s celebrated monograph [35] for further details.

One further has, for the full semigroup \(C^\star\)-algebra:

A. Nica developed the following notion of amenability for quasi-lattice ordered semigroups (generalisable to the right LCM case) in [27]:

We conclude this section with a couple of examples (and non-examples):

5 Boundary Quotients↩︎

Early studies into boundary quotients of semigroups were carried out by J. Crisp and M. Laca [11] for right-angled Artin monoids (\(\mathbb{F}_k^+\) and \(\mathbb{N}^k\) are both special cases of these), and they also investigated questions of pure infiniteness and simplicity. Their construction was generalised in two separate ways, which we briefly describe in this section. If \(P\) embeds into a group \(G\), there is a natural partial action of \(G\) on the (abelian) reduced diagonal subalgebra \(D_\lambda(P)\), and consequently its spectrum \(\Omega_P\). \(\Omega_P\) can be shown to have a unique smallest \(G\)-invariant subspace, which we refer to as \(\partial \Omega_P\). Then, the reduced crossed product \(C^\star\)-algebra \(\partial C^\star_\lambda(P) := C(\partial\Omega_P) \rtimes_rG\) is defined to be the boundary quotient \(C^\star\)-algebra of the semigroup. It satisfies certain desirable co-universal properties: it is the co-universal object in the category of \(G\)-equivariant representations of \(C^\star_s(P)\), and is isomorphic to the \(C^\star\)-envelope of the co-system of \(G\) acting on \({\mathcal{A}}_\lambda(P)\). We refer the reader to [12], [13], [37], [38] for details. For right LCM semigroups \(P\), one also has a characterisation of the boundary quotient \(\mathcal{Q}(P)\) via a maximality of relations picture. One defines foundation sets of \(P\) as those which generate principal ideals that intersect any other principal ideal non-trivially. A vanishing condition on the products of complement projections for the ideals corresponding to the foundation set elements, generates an ideal of the universal \(C^\star\)-algebra, and quotienting with respect to this ideal yields the boundary quotient \(C^\star\)-algebra. We refer the reader to [14], [17], [39], [40] for details. The connection between them has been briefly discussed in [12]. The core of the idea is that closed \(G\)-invariant subsets of the Nica spectrum (\(\Omega_P\)) correspond to sets of elementary relations on \(P\), and the unique smallest \(G\)-invariant subset \(\partial\Omega_P\) corresponds to the maximal such set of elementary relations.

5.1 Boundary Quotients via Partial Crossed Products↩︎

One way to define boundary quotients in the general setting is through partial actions, which are a generalisation of the notion of an action. The core of the idea is that instead of an action of a group by bijections of a set (or suitable isomorphisms, e.g. homeomorphisms in the topological case), we only ask for partial bijections on a set, that is, maps which are bijections on suitably restricted domains, and compositions / inverses that are defined on suitably restricted domains as well. It is also possible to define reduced and universal (partial) crossed product \(C^\star\)-algebras from these actions, and one has the canonical surjective \(\star\)-homomorphism from the latter to the former. We refer the interested reader to [41], [42] for further background.

Now, we consider the inverse semigroup for \(P\), constructed in the following fashion using the left-regular representation: \[\mathcal{I}_V = \left\{ V_{p_1}^\star V_{q_1} \dots V_{p_n}^\star V_{q_n} \;\vert\; \textrm{ for all } p_i,q_i \in P, \; 1\leq i \leq n, \; n \in \mathbb{N} \right\}.\] It is straightforward to verify that it is indeed an inverse semigroup (recall that an inverse semigroup is a semigroup satisfying that for every \(x\) there exists a unique \(y\) with \(xyx=x\)) with \(VV^\star V =V\) for all \(V \in \mathcal{I}_V\).

Assume that \(P\) embeds into a group \(G\). We denote \(\mathcal{I}_V^\times = \mathcal{I}_V \backslash\{0\}\), and define the homomorphism \[\sigma:\mathcal{I}_V^\times \to G, \;\;\;\;\;\;\;\;\;\;V_{p_1}^\star V_{q_1} \dots V_{p_n}^\star V_{q_n} \mapsto p_1^{-1} q_1 \dots p_n^{-1} q_n.\]

It can be proved [13] that \(D_\lambda(P) = \overline{span}\{\sigma^{-1}(e)\}\). We further define \[D_{g^{-1}} := \overline{span}\{V^\star V \;\vert\; V \in \mathcal{I}_V^\times, \; \sigma(V) = g\} \subseteq D_\lambda(P).\] It was shown in Section 3.3 [42], that \(D_{g^{-1}}\) is an ideal of \(D_\lambda(P)\) for every \(g \in G\). Noting that \(\ell_2(P)\) is a subspace of \(\ell_2(G)\), we may identify \(C^\star_\lambda(P) \subseteq {\mathcal{B}}(\ell_2(P))\). We observe that for all \(V \in \sigma^{-1}(g)\), denoting \(\lambda_g\) as the left-regular representation of \(G\) on \(\ell_2(G)\), \(V = \lambda_g V^\star V\), and hence it induces a \(\star\)-homomorphism \[\theta_g^\star: D_{g^{-1}} \to D_g, \;\;\;\;\;\;\;\;\;\; V^\star V \mapsto \lambda_g V^\star V \lambda_g^\star = VV^\star .\] By construction, \(D_\lambda(P)\) and \(D_{g^{-1}}\) are abelian \(C^\star\)-algebras, and if we denote their spectra as \(\Omega_P\) and \(\Omega_{g^{-1}}\) respectively, we may identify \(\Omega_{g^{-1}}\) as an open subset of \(\Omega_P\), and \(\theta_g^\star\) as described above induces \(\theta_{g^{-1}}\) to be a partial action of \(G\) on \(\Omega_P\) by \[\theta_g^\star(\chi) = \chi \circ \theta_{g^{-1}}, \;\;\;\;\textrm{ for all } \chi \in D_{g^{-1}} \cong C_0(\Omega_{g^{-1}}).\]

In [42] it was shown that \(\Omega_P\) has a unique smallest \(G\)-invariant subset, which we shall denote by \(\partial \Omega_P\). The space \(\partial \Omega_P\) can be identified using the semilattice of idempotents on \(P\).

One may define full boundary quotients as well, in an analogous fashion as done above, we refer the interested reader to [12] for further details. And now we conclude this subsection by stating a co-universality criterion for the boundary quotient.

In the right LCM cancellative setting situation, even if \(P\) is not group-embeddable, it is still possible to construct boundary quotients through normal coactions, as developed in [16].

5.2 Boundary Quotients via Foundation Sets↩︎

One has an alternate definition of the boundary quotient in terms of foundation sets. This definition was developed in the context of right LCM semigroups by [14], [17], [39], and extended to the general setting in [12], and further refined in the right LCM setting in [40]. C. Starling proved in [39] that this boundary quotient is in fact the tight \(C^\star\)-algebra of the inverse semigroup, and addressed the issues of pure infiniteness and simplicity.

That is, \(\mathcal{Q}(P)\) is a quotient of \(C^\star(P)\) by the ideal generated by relations of the form \(\Pi_{f \in F}(1-e_{fP}) = 0\) for every foundation set \(F\). As pointed out by B. Li and M. Laca in [17], denoting \(I_\infty\) to be the ideal of \(D(P) \cong D_\lambda(P) = C(\Omega_P)\) generated by projections of the form \(\Pi_{f \in F} (1-e_{fP}) = 0\) for every foundation set \(F\), one has \(C(\partial \Omega_P) \cong C(\Omega_P) / I_\infty\), and \(\mathcal{Q}(P) \cong C(\partial \Omega_P) \rtimes G \cong C(\partial \Omega_P) \rtimes P\), where the last equivalence (partial crossed product with respect to a semigroup action) is defined and studied in [17].

This notion may extended beyond the right LCM setting as well, as noted in Remark 5.5 of [14] by replacing finite subsets of \(P\) by finite subsets of \(\mathcal{J}\) which have non-trivial intersection with every ideal (since not every constructible ideal is principle in the general case). This was further studied in [12], where they used it in the concrete presentation of the covariance algebra, which is the full counterpart of the crossed-product version of the boundary quotient. We conclude this section by recognising that in practice it might be difficult to take the quotient with respect to every foundation set, but N. Brownlowe and N. Stammeier developed the notion of accurate refinement property in [40], where they reduced the computation by restricting to only certain types of foundation sets, and showed that this property holds in practice for a very large class of semigroups, in particular including directed semigroups and semigroups where principal right ideals corresponding to non-comparable elements are disjoint.

5.3 Computing Boundary Quotients↩︎

We begin this section with the following celebrated result by C. Sehnem.

Furthermore, when \(P\) is right-Ore (recall Definition 8: cancellative and left-reversible), then \(G=PP^{-1}\), \(\partial\Omega_P\) is just a single point, and thus \(\partial C^\star_\lambda(P) = C(\partial\Omega_P) \rtimes_r G = C^\star_\lambda (G)\). This brings us to the following theorem:

We state and prove the following proposition about \(\mathcal{Q}(P)\) below .

And now, we provide some explicit examples of computations below.

6 Dilating Semigroup Representations to the Boundary↩︎

In this section we address the problem of dilation of a representation of a semigroup to its boundary quotient. But first, we briefly discuss dilations of semigroup representations.

6.1 Dilations of Representations of Semigroups↩︎

Dilations of representations of semigroups are defined in a fashion identical to that for representations of operator algebras.

It can be easily seen that if \(T\) has an isometric dilation, then \(T\) must be a contractive representation (i.e., for each \(p\in P\), \(T(p)\) is a contraction). To the best of our knowledge, there is no general characterisation of constructible representations (Definition 27). However, in the right LCM setting, we do have the following notion of “covariant representations”, which was first formulated by A. Nica in [27] for quasi-lattice ordered semigroups and their \(C^\star\)-algebras, and generalised to right LCM semigroups in [32].

One may see that this is indeed well-defined: for any \(r,r' \in p \vee q\), there exists \(u \in P^\star\) such that \(r' = ru\), and since \(u\) is invertible, so must be \(V(u)\), i.e. \(V(u)\) must be a unitary. Hence \(V(r')V(r')^\star = V(ru)V(ru)^\star = V(r) V(u) V(u)^\star V(r)^\star = V(r)V(r)^\star\).

One might go a bit further and formulate conditions for contractive representations to admit isometric Nica-covariant dilations. We refer the reader to B. Li’s work on this [43]. Motivated by Neumark dilations [44], [45], B. Li introduced the concept of \(\star\)-regular dilation in [43], [46], which we shall not define here. We provide a necessary and sufficient condition for the existence of isometric Nica-covariant dilations:

6.2 Dilating to \(\mathcal{Q}(P)\) in the right LCM case↩︎

The following lemma is straightforward, but we still include it for the sake of completeness:

M. Laca and B. Li further extended this result to contractive representations using Theorem 38:

We now formulate a dilation theorem for representations of Ore semigroups, that provides a necessary and sufficient condition for unitary dilations.

And now we formulate the counterparts of these results for \(\partial C^\star_\lambda(P)\) using the Dritschel-McCullough dilation machine, and show that though our results have a very different flavour, they are stronger in many cases even when the full and reduced boundary quotients agree.

6.3 Dilating to \(\partial C^\star_\lambda(P)\)↩︎

The following are the main results of our paper. We begin with our generalisation of the matricial von Neumann’s inequality.

Note that since the algebra \(\{{V_p \; \vert \; p \in P}\}\) is dense in \({\mathcal{A}}_\lambda(P)\), there is only one way to extend a representation of \(P\) to one of \({\mathcal{A}}_\lambda(P)\) mapping \(V_p\) to \(T(p)\).

These two theorems very directly lead us to the following dilation results:

One must mention that extension property (A) is not automatically guaranteed, that is, not every representation has this property. An example of a contractive representation of \(\mathbb{N}^3\) that fails to satisfy extension property (A) was provided in [5], through a triple of commuting matrices which fail to satisfy a von Neumann inequality, and hence do not admit a unitary dilation. An interesting question now is to consider which semigroup representations have extension property (A).

We now provide an alternate characterisation of extension property (A) using von Neumann inequalities, and demonstrate that our extension property (A) may be viewed as a generalisation of matricial von Neumann inequalities.

We state and sketch a proof of the following theorem, which is now straightforward, and parts of which are somewhat well known.

Theorem 50 brings things full circle in our paper: we demonstrate that the matricial von Neumann inequality is equivalent to extension property (A) for \(\mathbb{N}^k\), and unitary dilation is equivalent to dilation to the reduced boundary quotient for amenable Ore semigroups (of which \(\mathbb{N}^k\) are an example). Hence, we present extension property (A) as a generalisation of the matricial von Neumann inequality, and dilation to the reduced boundary quotient as a generalisation of unitary dilations, for semigroups more general than \(\mathbb{N}^k\). And just as the matricial von Neumann inequality is equivalent to existence of unitary dilations for \(\mathbb{N}^k\), our extension property (A) is equivalent to the existence of dilations to the boundary quotient. In this sense, our work is a true generalisation of Sz.-Nagy’s dilation theorem and its counterparts.

Note that none of the examples stated above, except for the \(k=1\) case of (iv) actually satisfy either Nica-covariance or even Inequality ?? needed for a Nica-covariant dilation. This demonstrates our point that extension property (A) is in fact more general than the conditions required in [17]. However, it is not trivially satisfied by just any contractive representation, as is demonstrated by the failure to obtain an Ando-type dilation result with \(3\) or more contractions in general: S. Parrott [5] first gave an example of \(3\) contractions which fail to satisfy a von Neumann inequality, and hence do not have extension property (A). How one might formulate a necessary condition for extension property (A) in terms of the generators of the representation generally, is a very interesting question for which we do not yet have a concrete answer.

It is interesting to consider what the relations between the images of the generators of the dilation would be in the most general case. Note that one always has a canonical \(\star\)-homomorphism from the full crossed product to the reduced one, \(\mathcal{Q}(P) \cong C(\partial \Omega_P) \rtimes G \to C(\partial \Omega_P) \rtimes_r G \cong \partial C^\star_\lambda (P)\) that maps the image of \(v_p\) in the first to \(q(V_p)\) in the second. Thus, in the right LCM case, by Equation 3 , \(q(V_p) \in \partial C^\star_\lambda (P)\) are always going to satisfy the condition \(\Pi_{f \in F}\left(1 - q(V_f) q(V_f)^\star\right) = 0\) for every foundation set \(F\). This leads us to the following result:

References↩︎