### CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: the Multi-Parameter Case and Beyond

Michael Skeide and I have recently uploaded our new paper to the arxiv: CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond. In this gigantic (219 pages) paper, we propose a framework for studying the dilation theory of CP-semigroups parametrized by rather general monoids (i.e., semigroups with unit), and we use this framework for obtaining new results regarding the possibility or impossibility of constructing or having a dilation, we use it also for obtaining new structural results on the “mechanics” of dilations, and we analyze many examples using our tools. We present results that we have announced long ago, as well as some surprising discoveries.

This is an exciting moment for me, since we have been working on this project for more than a decade.

“Excuse me, did you really say decade?”

Yes, I know, it’s hard to believe that one can work on a single paper for ten years. But it’s true. To be precise, we started some email discussions in 2007, and kicked off this as an official project in 2009. The problem of characterizing precisely which CP-semigroups have a dilation turned out to be extremely difficult, but Michael did not feel like giving up, so we continued working and improving our results, and also finding counter-examples to things that we could not prove, which raised new questions, which were extremely difficult, but Michael did not feel like giving up, and so on, etc. I am grateful for this collaboration, because Michael, besides being a genius, has also a very strong dedication for the quality of work. Honestly, if I was in this alone I would have given up long ago (meaning that I would either discard it or publish the wishy-washy results that we could obtain within a year or two).

So, aren’t you curious to hear what we do in this paper? Though the paper has an abstract and an introduction, which were carefully written, I will try here to give a more informal account, allowing myself to be less precise, and to put emphasis on one particular key novelty of our paper: the superproduct system of a dilation. I will be a bit sloppy with details, and if the following makes you curious you can find details and/or references in our paper.

#### 1. CP-semigroups and dilations

Let $S$ be a commutative semigroup with unit; for simplicity one may consider a sub-semigroup of $\mathbb{R}_+^d$ (in the paper, we treat general monoids, and in some cases we restrict attention to Ore monoids, and there are interesting aspects that arise from noncommutativity of the semigroup. But here I will ignore noncommutative monoids to make the presentation a bit more readable. In fact, some of the novelties of our paper are interesting even in the one-parameter case).

A CP-semigroup is a family $T =(T_s)_{s\in S}$ of completely positive maps acting on a unital C*-algebra $B$. We make a standing assumption that all our CP-semigroups are contractive, that is $\|T_s\| \leq 1$ for all $s \in S$. A CP-semigroup is said to be a Markov semigroup if it consists of unital maps: $T_s(1) = 1$; it is said to be an E-semigroup if it consists of *-endomorphisms, and it is said to be an E${}_0$-semigroup if it consists of unital *-endomorphisms. When the semigroup $S$ is simply the half line $\mathbb{R}_+ = [0,\infty)$, then we say that we are in the “one parameter case”. The study of one parameter semigroups of CP maps and *-endomorphisms is motivated by mathematical physics, and began already in the 1970s.

A dilation of $T$ is a triple $(A,\theta,p)$, where $A$ is a unital C*-algebra containing $B$, $p$ is a projection such that $B = pAp$, and $\theta = (\theta_s)_{s \in S}$ is an E-semigroup such that

(*) $T_s(b) = p(\theta_s(b))p$

for all $b \in B$ and $s \in S$. Sometimes, we just say that $\theta$ is a dilation of $T$. We say that a dilation is strong if (*) is replaced by

(**) $T_s(pap) = p\theta_s(a)p$

for all $a \in A$ and $s \in S$. The existence of a dilation means that we can understand a CP-semigroup as a “part of” an E-semigroup, which is an object in a more restrictive category, hence presumably better understood. Thus, this notion obeys the general philosophy of “dilation theory”, which was the topic of the survey that I advertised in the previous blog post. (Some applications of dilations are discussed in Section 8.3 of the survey).

It is known that if $S = \mathbb{N}$ or $S = \mathbb{R}_+$ then there always exists a dilation, in fact there always is a minimal strong dilation. (I will say something about minimality below). Our mission in this paper was ambitious: to develop dilation theory for general multi-parameter semigroups on general C*-algebras (and beyond), and in particular to completely understand when does there exist a dilation, and of what kind.

#### 2. CP-semigroups and subproduct systems

Our methods rely on the machinery of Hilbert C*-modules and Hilbert C*-correspondences. Recall, that a Hilbert C*-module (over a C*-algebra $B$) is a right $B$-module $E$ that carries a $B$-valued “inner product” that satisfies the usual properties, e.g., $\langle x, y + zb \rangle = \langle x, y \rangle + \langle x, z \rangle b$ for all $x,y,z \in E$ and all $b \in B$. Of course, we also require $\langle x, x \rangle \geq 0$ (an inequality taking place in $B$) and $\langle x, x \rangle = 0$ if and only if $x = 0$.

An operator $t : E \to E$ is said to be adjointable if there exists another operator $t^* : E \to E$, such that $\langle t x, y \rangle = \langle x, t^* y \rangle$ for all $x,y \in E$. We denote by $B^a(E)$ the algebra of all adjointable operators on $E$. It is a C*-algebra (with the *-operation $t \mapsto t^*$ and the operator norm).

A Hilbert C*-correspondence is a Hilbert C*-module $E$ that is in fact a bimodule over $B$ where the left action is given by adjointable operators. If we have two correspondences $E$ and $F$ over $B$, then we can form their tensor product $E \odot F$ which is the Hilbert C*-correspondence, obtained from the tensor product $E \otimes F$ when it is endowed with the natural bimodule operations and the inner product

$\langle x \otimes y, x' \otimes y' \rangle = \langle y, \langle x, x' \rangle y' \rangle$.

We then write $x \odot y$ for the image of $x \otimes y$ in $E \odot F$.

A fundamental tool that we use is Paschke’s GNS construction. Given a CP map $\Phi : B \to B$, there exists a Hilbert C*-module $E$ and a vector $\xi \in E$ such that

(GNS) $\Phi(b) = \langle \xi, b\xi \rangle$

for all $b \in B$. One can construct such a pair $(E, \xi)$ as follows: first, one constructs the tensor product $B \otimes B$, and defines on it a semi-inner product by

$\langle a \otimes b , a' \otimes b' \rangle = a^*\Phi(b^*b')a'$,

and then extending linearly. One quotients out the space of null vectors and completes, and this gives rise to the C*-module $E$. The vector $\xi$ is defined to be the image of $1 \otimes 1$ in $E$, and it is easy to check that it satisfies (GNS). We give the structure of a correspondence on $E$ by the obvious $B$-actions. Then $E$ is generated by $\xi$, and the pair $(E,\xi)$ is determined uniquely as the unique (up to isomorphism) correspondence generated by a vector that satisfy equation (GNS).

Now suppose that we have a CP-semigroup $T = (T_s)_{s \in S}$ on a C*-algebra $B$. Then for every $s \in S$, we can construct the GNS representation $(E_s, \xi_s)$. If $s, t \in S$, we can make the following computation:

$\langle \xi_s \odot \xi_t, b \xi_s \odot \xi_t \rangle = \langle \xi_t, \langle \xi_s, b \xi_s \rangle \xi_t \rangle = T_t(T_s(b)) = T_{t+s}(b)$.

Note that here we made use of the semigroup property of $T$. This means that if we write

$F = \overline{span}\{b \xi_s \odot \xi_t b' :b,b' \in B\}$

then $(F,\xi_s \odot \xi_t)$ is a GNS representation for $T_{t+s}$, and by uniqueness we find that there exists an isomorphism of C*-correspondence $E_{s+t} \to F \subset E_s \odot E_t$ that maps $\xi_{s+t}$ to $\xi_s \odot \xi_t$. We can summarize this a bit sloppily by saying that

(I) $E_{s+t} \subset E_s \odot E_t$

and

(II) $\xi_{s+t} = \xi_s \odot \xi_t$.

We call a family $(E_s)_{s \in S}$ satisfying (I) a subproduct system (we also require that the identifications are associative – this also follows from the semigroup properties of $T$) and a family $(\xi_s)_{s\in S}$ satisfying (II) a unit. The pair $((E_s)_{s \in S}, (\xi_s)_{s\in S})$ is called the GNS subproduct systems and unit (or simply the GNS system) of the semigroup $T$.

To recap, with every CP-semigroup $T = (T_s)_{s \in S}$ we have a subproduct system and unit, from which one can completely recover the semigroup. The subproduct system can be considered as a kind of algebraic invariant attached to the semigroup, and we will see below that in principle it contains information about the possibility of constructing a dilation for $T$.

#### 3. Dilations and superproduct systems

We have just seen that with every CP-semigroup, there is a family $(E_s)_{s \in S}$ of C*-correspondences satisfying equation (I) above (with associative identifications), and we called this a subproduct system. If the family satisfies the stronger requirement

$E_{s+t} = E_s \odot E_t$

then we say it is a product system. Product systems of Hilbert spaces were introduced by Arveson as the definitive tool for studying and classifying E${}_0$-semigroups on $B(H)$. Afterwards, product systems of C*-correspondences appeared, and they were shown by Michael Skeide (my co-author in this current paper) to be a complete invariant for E${}_0$-semigroups on $B^a(E)$.

In a previous work with Baruch Solel, we showed that for a CP-semigroup to have a full normal dilation, a necessary condition is that the subproduct system can be embedded into a product system. (To be precise, we did not work with the GNS-subproduct system, but rather with another construction, called the Arveson-Stinepring subproduct system; see the appendix of the new paper for the connection). Moreover, we showed that for Markov semigroups the embedability of the subproduct system in a product system is a sufficient condition for the existence of a dilation. However, those results only held for normal CP-semigroups on von Neumann algebras (and the monoids parametrizing the semigroups were assumed to be subsemigroups of $\mathbb{R}_+^d$). A more dramatic deficiency, was that it could not rule out the existence of non-full dilations.

In the new paper, we obtain a generalization of the above result on normal Markov semigroups as follows (this appears as Theorem 12.9 in our paper):

Theorem: Let $T$ be a Markov semigroup over the opposite of an Ore monoid. Then $T$ admits a strict full dilation if and only if the GNS-subproduct system of $T$ embeds into a product system.

Explanations: An “Ore monoid” is a monoid that is right reversible and cancellative (any cancellative abelian monoid is an Ore monoid). A “full dilation” is a dilation $(A, \theta, p)$ where $A$ is of the form $A = B^a(E)$ for some $B$-module $E$. Here by “strict” we mean that $\theta$ is continuous in appropriate topology on $B^a(E)$.

We used the above theorem to construct – for every pair of commuting UCP maps – a dilation consisting of commuting unital *-endomorphisms (recovering and somewhat sharpening an earlier result of Solel).

The above theorem is interesting, but we had expected it and it was obtained early on in our research. Let me say imprecisely (from a historical point of view), that such a result allowed us to construct examples of Markov semigroups over $\mathbb{N}^3$ that have no full dilation (this was a somewhat surprising counter example – read the intro of that paper to see why). However, we were not satisfied with the fact that our theorem only treated full dilations. We did not know whether or not there exists examples of Markov semigroup $T$ acting on a C*-algebra $B$ such that there exists no dilation whatsoever $(A,\theta,p)$. The fact that a dilation is full means that $A$ is Morita equivalent to $B$; there is no apparent reason to restrict the notion of dilations only to Morita equivalent algebras.

To treat non full dilations, we introduced the notion of a superproduct system. In fact, superproduct product system have appeared in the literature without being called that (at least in the work of Claus Kostler, who suggested the terminology), and have appeared under that name, but for a completely different purpose, in the work of Margetts and Srinivasan (I blogged about their work several years ago). We discovered that they arise naturally in the context of dilations.

Let $(A,\theta,p)$ be a dilation (of some CP-semigroup $T$ on $B = pAp$). We can define a family $(F_s)_{s \in S}$ of C*-correspondences over $B$ as follows:

$F_s = \theta_s(p)Ap$

with a left action $b.x_s = \theta_s(b)x_s$ and a trivial right action. We can define maps $v_{s,t}: F_s \odot F_t \to F_{s+t}$ by

$v_{s,y} : x_s \odot y_t \mapsto \theta_t(x_s)y_t$.

Some straightforward calculations show that $v_{s,t}$ are all isometries, and that they compose associatively (see Theorem 9.1 in our paper). Thus, roughly, we have

$F_s \odot F_t \subseteq F_{s+t}$.

We call such a family of correspondences a superproduct system. Thus, with every dilation, there is a superproduct system. We obtain the following result (which is a simplification of Theorem 9.3 in our paper):

Theorem: If $(A,\theta,p)$ is a strong dilation of a CP-semigroup $T$, then the superproduct system of the dilation contains the GNS-subproduct system of the $T$.

It is noteworthy that every dilation of a Markov semigroup is strong; thus we have a necessary condition for the existence of any dilation for a Markov semigroup: the embeddability of the GNS-subproduct system into superproduct system. We don’t know – it is an open problem – whether this condition is also sufficient. We constructed subproduct systems that cannot be embedded into superproduct systems, thereby exhibiting examples of Markov semigroups that have no dilation whatsoever.

#### 4. Superproduct subsystems and minimality

The machinery of superproduct systems can be used not only to rule out existence of dilations, but also for studying the structure of dilations.

A dilation $(A,\theta,p)$ of a CP-semigroup $T$ is said to be compressible if there exists a projection $P\geq p$ in $A$ such that $(PAP,P\theta_s(P\bullet P)P, p)$ is still a dilation. We then say that $(PAP,P\theta(P\bullet P)P,p)$ is a compression of $(A,\theta,p)$.

It is clear that if a dilation is compressible, then it cannot really be considered to be “minimal”. There are several notions of minimality that have been studied in the literature or that are natural to consider. In Chapter 21 of our paper we study and compare them.

One of our most difficult results is the connection between the compressions of a (full) dilation and the various (super)product subsystems that exist between the GNS-subproduct system of the dilated semigroup and the (super)product system of the dilation. These results are bit too much to give the details of here.

One thing that I do want to tell you, and that I am very pleased about, is that we were able to use this machinery of superproduct systems to study and to obtain new results even about the one parameter case (where dilations are always known to exist). We used it to show that several different notions of minimality are equivalent in the one-parameter case, recovering and improving on earlier results of Arveson.

Finally, another one of our difficult achievements, is an analysis of the discrete two parameter case, in which we show that the different notions of minimality that have been shown to be equivalent in the one-parameter case, are not equivalent in general.