Interpolation and non-dilatable families of $\mathcal{C}_{0}$-semigroups

We generalise a technique of Bhat and Skeide (2015) to interpolate commuting families $\{S_{i}\}_{i \in \mathcal{I}}$ of contractions on a Hilbert space $\mathcal{H}$, to commuting families $\{T_{i}\}_{i \in \mathcal{I}}$ of contractive $\mathcal{C}_{0}$-semigroups on $L^{2}(\prod_{i \in \mathcal{I}}\mathbb{T}) \otimes \mathcal{H}$. As an excursus, we provide applications of the interpolations to time-discretisation and the embedding problem. Applied to Parrott's construction (1970), we then demonstrate for $d \in \mathbb{N}$ with $d \geq 3$ the existence of commuting families $\{T_{i}\}_{i=1}^{d}$ of contractive $\mathcal{C}_{0}$-semigroups which admit no simultaneous unitary dilation. As an application of these counter-examples, we obtain the residuality wrt. the topology of uniform wot-convergence on compact subsets of $\mathbb{R}_{\geq 0}^{d}$ of non-unitarily dilatable and non-unitarily approximable $d$-parameter contractive $\mathcal{C}_{0}$-semigroups on separable infinite-dimensional Hilbert spaces for each $d \geq 3$. Similar results are also developed for $d$-tuples of commuting contractions. And by building on the counter-examples of Varopoulos--Kaijser (1973--74), a 0--1-result is obtained for the von Neumann inequality. Finally, we discuss applications to rigidity as well as the embedding problem, \textit{viz.} that `typical' pairs of commuting operators can be simultaneously embedded into commuting pairs of $\mathcal{C}_{0}$-semigroups, which extends results of Eisner (2009--10).


Introduction
Dilation theory concerns itself with the problem of embedding single operators on Hilbert or Banach spaces (as well as operator-theoretic entities such as linear maps between trace-class operators on Hilbert spaces) or parameterised families of operators (such as C 0 -semigroups) into larger systems which are in some suitable sense 'reversible'.One can also consider the problem of finding non-dilatable systems.In the discrete setting, the existence of non-dilatable commuting families of three or more contractions on a Hilbert space is well-known thanks to Parrott's construction [33].This raises the natural question, whether a continuous analogue for families of semigroups exists.The present paper answers this positively as follows: We generalise a relatively recent technique developed by Bhat and Skeide [3] to interpolate families of contractions by families of contractive C 0 -semigroups.Applying this to Parrott's counterexamples, we obtain the existence of commuting families of at least 3 contractive C 0 -semigroups which admit no dilations in the sense of Sz.-Nagy [43,44].
In addition to looking at the properties of individual families of contractions or contractive C 0 -semigroups, we are concerned with spaces thereof, topologised in a certain weak sense (defined more precisely below).In this framework, various residuality results about 1-parameter C 0semigroups and spaces of contractions on Hilbert spaces have been developed, e.g.[17,18], [12,Theorem 2.2], [31,Corollary 3.2], [15,Theorem 4.1] (which builds on the density result established in [36,37]), [6,Theorem 1.3].Residuality results in the Banach space setting have also been studied in connection with the invariant subspace problem and hypercyclicity (see [22,23,21]).Continuing this pursuit, the present paper develops a general 0-1-dichotomy, which we apply to our non-dilatable constructions to prove within the space of d-parameter contractive C 0 -semigroups on a separable infinite-dimensional Hilbert space, the residuality of dilatable and unitarily approximable (resp.non-dilatable and non-unitarily approximable) semigroups for d P t1, 2u (resp.d ě 3).
In the discrete setting an analogous split is obtained for spaces of tuples of commuting contractions.Furthermore, going beyond the question of dilatability, stronger 0-1-results are possible.
Based on numerical experimentation, Orr Shalit has intuited via concentration of measure phenomena that it must be the case that either most tuples satisfy von Neumann polynomial inequalities or that most do not.Our methods appear to confirm this intuition.Indeed for d ě 3, building on the results of Varopoulos, Kaijser, et al. [45,46,4,40], we prove that the negative conclusion holds.
1.1 Definitions and main problems.Let H be a Hilbert space, and LpHq, CpHq, U pHq denote the spaces of bounded, contractive, and unitary operators on H respectively.Let I be a non-empty index set, tS i u iPI Ď CpHq, and tT i u iPI be a commuting family of contractive C 0 -semigroups on H.We say that tS i u iPI has a power-dilation if there is a commuting family tV i u iPI Ď U pH 1 q of unitaries on a Hilbert space H 1 as well an isometry r P LpH, H 1 q, such that holds for all n " pn i q iPI P ś iPI N 0 with supppnq " ti P I | n i ‰ 0u finite.a We say that tT i u iPI has a simultaneous unitary dilation if there exists a commuting family tU i u iPI Ď ReprpR : H 1 q of sot-continuous unitary representations of R on a Hilbert space H 1 as well as an isometry r P LpH, H 1 q, such that ź iPI T i pt i q " r ˚´ź iPI U i pt i q ¯r holds for all t " pt i q iPI P ś iPI R ě0 with suppptq " ti P I | t i ‰ 0u finite.Generalising slightly, if pG, ¨, eq is a topological group and M Ď G a topological submonoid, we can consider continuous contractive/unitary homomorphisms T : M Ñ LpHq over M on H, i.e.T is an sot-continuous map such that T peq " I, T pxyq " T pxqT pyq for x, y P M , and T pxq is contractive/unitary for x P M .In the case of pG, M q " pZ d , N d 0 q, d P N, there is a natural 1:1correspondence b between contractive/unitary homomorphisms S over N d 0 on H and commuting families tS i u d i"1 of contractions/unitaries on H, and in the unitary case, S extends uniquely c to a unitary representation of G " Z d on H.In the case of pG, M q " pR d , R d ě0 q, d P N, there is a natural 1:1-correspondence d between continuous contractive/unitary homomorphisms T over R d ě0 on H and commuting families tT i u d i"1 of d-parameter contractive/unitary C 0 -semigroups on H, and in the unitary case, T extends uniquely e to a continuous unitary representation of G " R d on H. Putting these observations together, we can define unitary dilations à la Sz.-Nagy in a more uniform manner: Let G be a topological group and M Ď G a submonoid.A continuous contractive homomorphism T : M Ñ LpHq is said to have a unitary dilation if there exist a continuous unitary representation U P ReprpG : H 1 q of G on a Hilbert space H 1 , as well an isometry r P LpH, H 1 q, such that T pxq " r ˚U pxq r for all x P M .In this case, we refer to the tuple pH 1 , U, rq as the unitary dilation.One can readily check that this coincides with the above definitions in the cases of pG, M q " pR d , R d ě0 q or pZ d , N d 0 q, d P N. For any topological space X and Hilbert space H, the topology of uniform wot-convergence on compact subsets of X is given as follows: Let pT pαq q αPΛ be a net of operator-valued functions T pαq : X Ñ LpHq, and let T : X Ñ LpHq be a further operator-valued function.Then

and only if
a throughout we use the convention S 0 :" I H for any bounded operator S on a Hilbert space H.
b via Spnq " i for all n " pn i q d i"1 P N d 0 and S i " Sp0, 0, . . ., 1 i , . . ., 0q for i P t1, 2, . . ., du and all n P N 0 .c via Spnq " Sppmaxt´n i , 0uq d i"1 q ˚Sppmaxtn i , 0uq d i"1 q for n " pn i q d i"1 P Z d .d via T ptq " ś d i"1 T i pt i q for all t " pt i q d i"1 P R d ě0 and T i ptq " T p0, 0, . . ., t i , . . ., 0q for i P t1, 2, . . ., du and all t P R ě0 .e via T ptq " T ppmaxt´t i , 0uq d i"1 q ˚T ppmaxtt i , 0uq d i"1 q for t " pt i q d i"1 P R d .for all ξ, η P H and compact K Ď X.We refer to this as the k wot -topology.
Let M be a topological monoid.We shall consider the space pS c s pM, Hq, k wot q of continuous contractive homomorphisms T : M Ñ LpHq over M on H, and endowed with the k wot -topology.We shall also consider the subspace pS u s pM, Hq, k wot q of continuous unitary homomorphisms T : M Ñ LpHq over M on H.
Given the above topological definitions, consider the following problem: Question 1.2 Let M be a topological monoid and H a Hilbert space.Is S u s pM, Hq residual in pS c s pM, Hq, k wot q?
Suppose now that H is separable and infinite-dimensional and that M is locally compact Polish (e.g.M P tR d ě0 , N d 0 | d P Nu).Let C be the space of wot-continuous contractive-valued functions of the form T : M Ñ CpHq.Further let X :" S c s pM, Hq and A :" S u s pM, Hq.It is known that pC, k wot q and pA, k wot q are Polish spaces (see [6,Proposition 1.16 and Proposition 1.18]).In particular, the subspace X of pC, k wot q is a second-countable metrisable space, and by Alexandroff's lemma (see [29,Theorem 3.11], [1,), A is a G δ -subset in pX, k wot q.Under these assumptions on M and H, the above problem, viz.whether A is residual in pX, k wot q, is thus equivalent to whether A is dense inside pX, k wot q.Now, we often consider topological monoids M as submonoids of topological groups, G.In case unitary semigroups extend uniquely to unitary representations (this holds e.g. for pG, M q P tpR d , R d ě0 q, pZ d , N d 0 q | d P Nu), recent work in [8] has shown that the density question is closely related to the existence of unitary dilations (this will be made clear in Proposition 2.19 below).This leads to a related problem, which is interesting in and of itself: 3 Let G be a topological group, M Ď G a submonoid, and H a Hilbert space.Does every T P S c s pM, Hq have a unitary dilation?
1.2 Summary of main results.This paper addresses the above two problems for the concrete cases of pG, M q " pZ d , N d 0 q and pR d , R d ě0 q for d P N and (separable) infinite-dimensional H.In discrete-time, Parrott's construction (see [33, §3], [44, §I.6.3])showed that there always exists non-power dilatable families of d ě 3 contractions, provided the dimension of the Hilbert spaces was suitably large.For a long time no counter-examples to simultaneously unitarily dilatable families seemed to be be known in the continuous setting.In light of Andô's power-dilation theorem for pairs of contractions (see [2]) and S lociński's theorem on simultaneous unitary dilations of pairs of commuting contractive C 0 -semigroups (see [41], [42,Theorem 2], and [38,Theorem 2.3]), the bound d " 3 is optimal in the discrete setting and could also be postulated to be optimal in the continuous setting.We shall show that this is indeed the case.To achieve this, our first goal is to reduce non-dilatability in the continuous-time setting to non-dilatability in the discrete-time setting.This in turn is achieved via interpolation.
Let T denote the unit circle tz P C | |z| " 1u endowed with the unique Haar-probability measure.For any non-empty index set I, the product space ś iPI T is equipped with the product (probability) measure.f We consider the Hilbert space tensor product L 2 p ś iPI Tq b H, g viewing L 2 p ś iPI Tq as an 'auxiliary space'.Our first main result extends a recent construction due to Bhat and Skeide [3]: Lemma 1.4 (Generalised Bhat-Skeide Interpolation).Let I be a non-empty index set and tS i u iPI be a commuting family of bounded operators (resp.contractions) on a Hilbert space H.There exists a commuting family tT i u iPI of C 0 -semigroups (resp.contractive for all n " pn i q iPI P ś iPI N 0 with supppnq " ti P I | n i ‰ 0u finite. As an excursus we apply the Bhat-Skeide interpolation to the embedding problem in §2.3.And we discuss in §2.4 a time-discretised version and show that the interpolation and its discretisations enjoy a certain natural property (see Proposition 2.13).Then in §2.5 we use the interpolation to answer Question 1.3 negatively for d-parameter semigroups with d ě 3: Theorem 1.5 (Non-dilatable families of semigroups).Let I be a non-empty index set and H be an infinite-dimensional Hilbert space.If |I| ě 3, there exists a commuting family tT i u iPI of contractive C 0 -semigroups on H, which admit no simultaneous unitary dilation.Furthermore, the T i can be chosen to have bounded generators.

Remark 1.6
In the non-classical setting of CP-semigroups (on unital C ˚-algebras), Shalit and Skeide [39,Corollary 18.10] have recently constructed contractive CP-resp.Markov semigroups over R 3 ě0 which admit no 'strong' resp.'weak' dilations (see [40, §8.1], [39,Definition 2.3]).We note here two things: These counter-examples and our ones in Theorem 1.5 do not appear to be derivable from one other.Moreover, the approach in [39] used to construct multi-parameter counter-examples is not based on the interpolation of Bhat and Skeide [3], rather other interpolation techniques are developed.
In §3 we proceed to develop a 0-1-dichotomy for the general setting of homomorphisms.Applying this to our counter-examples, we obtain the following answer to Question 1.2: Corollary 1.7 Let d P N and H be a separable infinite-dimensional Hilbert space.Consider the space X :" S c s pR d ě0 , Hq of d-parameter contractive C 0 -semigroups on H under the k wot -topology as well as the subspaces A :" S u s pR d ě0 , Hq of d-parameter unitary C 0semigroups on H and D of d-parameter semigroups admitting a unitary dilation.Then D " A and Consider the space X :" CpHq In the discrete setting, we shall apply similar reasoning to obtain an analogous 0-1dichotomy for d-tuples of commuting contractions satisfying the von Neumann inequality (see Remark 3.8).
• We write elements of product spaces in bold and denote their components in light face fonts with appropriate indices, e.g. the i th components of t P R d ě0 and z P ś iPI T are denoted t i and z i respectively.
• All Hilbert spaces shall be takten to have dimension at least 1.This is to avoid the issue that on 0-dimensional spaces both the identity and zero operators are equal, which unnecessarily complicates matters.
• For Hilbert spaces H, H • For a Hilbert space H and topological group G, the set ReprpG : Hq denotes the set of sot-continuous unitary representations U : G Ñ LpHq of G on H.
• For a Hilbert space H, we denote the identity operator as I H .If it is clear from the context, we simply write I.For example, in the context of H " H 0 b H 1 for Hilbert spaces H 0 , H 1 , given R P LpH 0 q, S P LpH 1 q, the expression pR b IqpI b Sq `I is to be interpreted as pR b I H 1 qpI H 0 b Sq `IH .
• k wot denotes the topology of uniform wot-convergence on compact subsets (see §1.1).
• pw denotes the weak polynomial topology on the space of tuples of contractions (see Remark 1.1).
• For n P N and i, j P t1, 2, . . ., nu, E ij P M n pCq denotes the matrix containing a 1 in entry pi, jq and 0's elsewhere.
In any context, given a choice pG, M q of a topological group and submonoid M Ď G, we shall repeatedly refer to the spaces A ˚Ď A Ď X Ď C and D Ď X, where • C :" the set of wot-continuous contraction-valued functions T : M Ñ CpHq; • X :" S c s pM, Hq, the set of sot-continuous contractive homomorphisms; • A :" S u s pM, Hq, the set of sot-continuous unitary homomorphisms; • A ˚:" tU | M | U P ReprpG : Hqu; • D :" tT P X | T has a unitary dilationu.
These spaces shall generally be equipped with the k wot -topology and A, A ˚shall always denote the closures of A, A ˚within pX, k wot q.Note further that C, X, A can be defined for topological monoids M which are not submonoids of topological groups.

Interpolation and non-dilatability
We first present the interpolation result of Bhat and Skeide, which extends a single contraction to a C 0 -semigroup.We then generalise this to extend arbitrary commuting families of contractions to commuting families of contractive C 0 -semigroups.In the following we use the notation t ttu u and ttu to denote the fractional and integer part of a number t P R, i.e. ttu " maxtn P Z | n ď tu and t ttu u " t ´ttu P r0, 1q.
2.1 Bhat-Skeide interpolation for contractions.In [3, Lemma 2.4 (1)] Bhat and Skeide developed an interpolation result for isometries over Hilbert C ˚-modules (a generalisation of Hilbert spaces).Their construction, however, also works for contractions over Hilbert spaces.We slightly modify the spaces and notation, but the constructions remain essentially the same.To start off, for t P R we let ϕ t : T Ñ T and p t : T Ñ R be defined via ϕ t pzq :" e ´ı2πt z " e ı2πps´tq p t pzq :" 1 1 r0, 1´t ttu uq psq for z " e ı2πs P T, s P r0, 1q.Since ϕ t is a measure-preserving bijection, the Koopman-operator U ptq P LpL 2 pTqq defined by U ptqf :" `T˘, and that ϕ s`t " ϕ s ˝ϕt for s, t P R. Thus U P ReprpR : L 2 pTqq.Since p t for each t P R is a t0, 1u-valued measurable function, the multiplication operator P ptq :" M pt P LpL 2 pTqq is a projection.For t 0 P R and pt 0 , 8q Q t ÝÑ t 0 we have t ttu u ÝÑ t tt 0 u u and hence P ptq " M 1 1 te ı2πs |sPr0, 1´t ttu uqu sot ÝÑ M 1 1 te ı2πs |sPr0, 1´t tt 0 u uqu " P pt 0 q.Thus P is right-sot-continuous.
Note also that ϕ and p and hence U and P are periodic, viz.U ptq " U pt ttu uq and P ptq " P pt ttu uq for all t P R. In particular P pnq " P p0q " M 1 1 T " I for n P Z.The operator-valued functions U and P also enjoy the following property: for s, t P R. We shall refer to this as the Bhat-Skeide commutation relation (or simply: BSCR).
Proof.If s P Z, then due to periodicity, P psq " P p0q " I and P ps`tq " P ptq.Since t tsu u`t ttu u " t ttu u ă 1, the right-hand expression is equal to U ptq ¨pI ´pP ptq ´P ps `tqqq " U ptq ¨pI ´pP ptq Ṕ ptqqq " U ptq " P psqU ptq.If t P Z, then due to periodicity, P ptq " P p0q " I, U ptq " U p0q " I, and P ps `tq " P psq.Since t tsu u `t ttu u " t tsu u ă 1, the right-hand expression is equal to U ptq ¨pI ´pP ptq ´P ps `tqqq " I ¨pI ´pI ´P psqqq " P psq " P psqU ptq.If s, t P R z Z, then the proof of (BSCR) is essentially the same as in [3,Lemma 2.4 (1)].(The visualisation in Figure 1 of the effect of U ptq ˚P psqU ptq on elements of L 2 pTq also depicts this computation.)Lemma 2.2 (Bhat-Skeide, 2015).Let S P LpHq be a bounded operator (resp.contraction) on a Hilbert space H. Then T : R ě0 Ñ LpL 2 pTq b Hq given by Proof (Sketch).First observe that T has bounded growth: Since P is projection valued and U is a unitary semigroup, one has T ptq ď P ptq S ttu ` I´P ptq S ttu`1 ď 2 maxt1, S u ttu`1 ď Ce ωt for all t P R ě0 , where C " 2 maxt1, S u und ω " logpmaxt1, S uq.
Secondly, observe that if S is a contraction, then T is a contraction-valued function: Letting t P R ě0 , since P ptq a projection, it is clear that P ptq b I `pI ´P ptqq b S is a contraction.And since U ptq is unitary, it follows that T ptq " pU ptq b IqpP ptq b I `pI ´P ptqq b SqpI b S ttu q is contractive.We now demonstrate the remaining properties.

Interpolation property:
For n P N 0 we have T pnq " pU pnqbIqpP pnqbS tnu `pI´P pnqqb Here t 1 " 1 ´t ttu u and t 2 " 1 ´t ts `tu u.

Semigroup law:
We have T p0q " I b S 0 " I b I " I. Let s, t P R ě0 be arbitrary.We show that T psqT ptq " T ps `tq.By Proposition 2.1, P psqU ptq " U ptqQps, tq, where Qps, tq " I ´pP ptq ´P ps `tqq if t tsu u `t ttu u ă 1 and Qps, tq " P ps `tq ´P ptq otherwise.This yields Relying entirely on the relations between projections P ptq and P ps `tq, the product of the final two expressions in parentheses simplifies to P ps `tq b S ts`tu `pI ´P ps `tqq b S ts`tu`1 , and thus the final expression is equal to T ps `tq.For details of this computation we refer the reader to the proof of [3,Lemma 2.4 (1)].Note in particular that the properties of S do not play any role in this simplification.

Continuity of of T :
Since T satisfies the semigroup law, it suffices to prove that T is continuous in 0. Since U is sot-continuous and P is right-sot-continuous, one has for t P p0, 1q that T ptq " Before proceeding, we take note of some basic but interesting properties of the Bhat-Skeide interpolation.Let S P LpHq be a bounded operator on a Hilbert space H and T : R ě0 Ñ LpL 2 pTq b Hq its interpolation as defined in Lemma 2.2.As in the proof of this result, if S is a contraction, then so is T ptq for all t P R ě0 .Conversely, if T is contractionvalued, then since I b S " T p1q, it follows that 1 ě T p1q " I b S " I S , whence S is a contraction.Whilst our focus in this paper is on contractions and contractive C 0 -semigroups, it is worth noting that the interpolation preserves other operator-theoretic properties.For example, if S is an isometry, then relying on the orthogonality of the projections tP ptq, I ´P ptqu for each t P R ě0 , one obtains i.e.T ptq is an isometry for all t P R ě0 .Conversely, if T is isometry-valued, then I b S ˚S " pI b Sq ˚pI b Sq " T p1q ˚T p1q " I, from which it follows that S is an isometry.In an analogous manner one obtains that T is unitary-valued if and only if S is a unitary.
The interpolation also preserves properties in the stochastic setting.Considering L p -spaces over probability spaces, it has been established that the properties of operators (or their duals) preserving the constant 1 1-function and a. e.-positive functions provide operator-theoretic counterparts to the stochastic notions of transition kernels.For details on these connections, see e.g.[9, §0.9, (9.6-7)], [10, §2.3 and Theorem 2.1], [47, §1.1-1.2],[14, Chapter 13].Here we simply focus on the operator-theoretic properties in and of themselves, in the L 2 -setting.
So consider a Hilbert space L 2 pX, µq for some probability space pX, µq, where X is a locally compact Hausdorff space.Let L 2 pX, µq `denote the closed convex subspace of f P L 2 pX, µq which are a.e.-positive, i.e. f pxq P R ě0 for µ-a.e. x P X. Say that an operator S P LpL 2 pX, µqq preserves a. e.-positive functions if SL 2 pX, µq `Ď L 2 pX, µq `.And say that S is unity-preserving if S1 1 X " 1 1 X .Now, the tensor product L 2 pTq b L 2 pX, µq. may be identified with L 2 pT ˆXq in the canonical way, h where T is endowed with the Haar measure and T ˆX is endowed with the product probability measure.Under this identification one further has that With this in mind, we now consider a bounded operator S P LpL 2 pX, µqq and its Bhat-Skeide interpolation T : R ě0 Ñ LpL 2 pTq b L 2 pX, µqq.If S is unity-preserving, then Similarly, S ˚is unity-preserving if and only if T ptq ˚is unity-preserving for all t P R ě0 .
Suppose now that S preserves a. e.-positive functions.For t P R ě0 , since U ptq is a Koopmanoperator and P ptq a projection and multiplication operator, one obtains T ptq ˚pf b gq, f b g " f b g, T ptq p f b gq h For f P L 8 pTq and g P L 8 pX, µq let f d g P L 8 pT ˆXq be defined by pθ, xq Þ Ñ f pθqgpxq.

Considering the algebraic tensor product C
`T˘b C `X˘, one can readily observe that the linear map By density, it follows that this map extends uniquely to a unitary map between L 2 pTq b L 2 pX, µq and L 2 pT ˆXq.
" f, U ptqP ptq f g, S ttu g ` f, U ptqpI ´P ptqq f g, S ttu`1 g ě 0 for all g, g P L 2 pTq `and f, f P L 2 pX, µq `, which implies that T ptq ˚pf b gq ě 0 a. e.From this it follows that T ptqF, f b g " F, T ptq ˚pf b gq ě 0 for all F P L 2 pT ˆXq `, f P L 2 pTq `, and g P L 2 pX, µq `, which in turn implies that T ptqF ě 0 a. e.It thus follows that T ptq preserves a. e.-positive functions for all t P R ě0 .Conversely, if this holds, then Sf, f " pIbSqp1 1 T bf q, 1 1 T b f " T p1qp1 1 T bf q, 1 1 T b f ě 0 for a. e.-positive functions f, f P L 2 pX, µq, from which it follows that S preserves a. e.-positive functions.Similarly, S ˚preserves a. e.positive functions if and only if T ptq ˚preserves a. e.-positive functions for all t P R ě0 .
We thus see that the Bhat-Skeide interpolation preserves operator-theoretic properties relevant in the characterisation Markov and bi-Markov processes.In particular, by applying [14, Theorem 13.2 and Proposition 13.6] one has that S P LpL 2 pX, µqq is a 'bi-Markov' operator i if and only if T ptq is for all t P R ě0 .
2.2 Interpolation for families of contractions.Now consider a non-empty index set I and a commuting family tS i u iPI of contractions on a Hilbert space H.The Bhat-Skeide interpolation can be extended to arbitrary families by defining the interpolants in such a way, that the auxiliary parts act on independent co-ordinates.Lemma 1.4).Consider the probability space Z :" ś iPI T. For each i P I and t P R define ϕ ´"ϕ t pz i q : j " i z j : j ‰ i ¯jPI p piq t pzq :" p t pz i q for z " pz j q jPI P Z, where ϕ t , p t are defined as in §2.1.Clearly, ϕ piq t is a measure-preserving bijection and p piq t is measurable.As in §2.1, defining U i , P i : R Ñ LpL 2 pZqq via U i ptqf " f ˝ϕpiq t (Koopman-operator) and P i ptqf " M p piq t f " p piq t f for t P R, f P L 2 pZq, we have that U i P ReprpR : L 2 pZqq and P i is right-sot-continuous and projection-valued.Moreover U i , P i are periodic with U i pnq " U i p0q " P i pnq " P i p0q " I for n P N. Now, the commutation relation (BSCR) in Proposition 2.1 is clearly satisfied by P i , U i .The proof of Lemma 2.2 is thus clearly applicable to P i , U i , S i .This yields that the operator-valued function T i : R ě0 Ñ LpL 2 pZq b Hq given for t P R ě0 by

Proof (of
i for n P N 0 .Let i, j P I with i ‰ j.Since the operators act on independent co-ordinates, it is routine to show that each of the operators in tP i psq, U i ptq | s, t P Ru commutes with each of the operators in tP j psq, U j ptq | s, t P Ru.Since S i and S j commute, it follows from the above construction that T i and T j commute.Hence tT i u iPI is a commuting family of C 0 -semigroups (resp.contractive C 0 -semigroups, if each of the S i are contractive) satisfying ś iPI T i pn i q " ś iPI pI b S n i i q for all n P ś iPI N 0 with supppnq finite.
i we refer the interested reader to the start of [14,Chapter 13] for the definition of bi-Markov operators.
Remark 2.3 (Non-commutative setting).Suppose tS i u iPI is a non-commuting family of contractions.We may still construct the interpolations tT i u iPI of tS i u iPI exactly as in the proof of Lemma 1.4, except that this will no longer be a commuting family.If we now impose a scheme of commutation relations on the discrete family tS i u iPI , one may consider whether an appropriate schema of commutation relations is satisfied by the continuous family.For example, consider the Weyl form of the canonical commutation relations (CCR) presented in [8,Example 1.8].The discrete counterpart of this can be presented as follows: Let d P N and tS i u d i"1 Ď LpHq be a family of contractions satisfying S j S i " V ij S i S j for i, j P t1, 2, . . ., du with i ‰ j.Here tV ij u d i,j"1 Ď LpHq is a family of invertible operators, each of which commutes with each of the S i .Now, by a simple induction argument one obtains the commutation relations for m, n P N 0 and i, j P t1, 2, . . ., du with i ‰ j.

Consider now the interpolation tT
3).Let i, j P t1, 2, . . ., du with i ‰ j.Let s, t P R ě0 and set m :" tsu and n :" ttu.Since tU i , U j u, tP i , P j u, tU j , P i u, and tU i , P j u are each commuting families, one may arrive at the following commutation relation for T i psq, T j ptq: where V ij ps, tq is completely determined by s, t, P i , P j , U i , U j , V ij .If V ij ps, tq " U p2B ij stq for each i, j P t1, 2, . . ., du with i ‰ j and each s, t P R ě0 , where B P M d pCq is some antisymmetric matrix and U an sot-continuous representation of pR, `, 0q on LpL 2 pT d q b Hq, then the semigroups tT i u d i"1 satisfy the CCR in Weyl form (cf. [8, Example 1.8]).From the above expression, it is however unclear whether V ij ps, tq can be expressed in this way.It would thus be nice to know if there is an appropriate alternative to our generalisation of the Bhat-Skeide interpolation, such that a family of contractions satisfying commutation relations such as (2.4) can be interpolated to a family of C 0 -semigroups satisfying the CCR.

Remark 2.4
If possible, it would be nice to find an interpolation technique similar to that of Bhat-Skeide (or our generalisation to commuting families) for the Banach space setting.

Application to the embedding problem.
In this section we consider the classical embedding problem in semigroup theory of finding for a given bounded operator S on Banach space E a (1-parameter) C 0 -semigroup T on E satisfying T p1q " S.Not all operators are embeddable (e.g.non-bijective Fredholm operators or non-zero nilpotent matrices).And in the literature various results provide either sufficient or necessary conditions or characterisations of embeddability for certain subclasses of operators (see e.g.[11,16]).To the best of our knowledge, no complete characterisation of the embeddability of bounded operators appears to be known.
In the context of Hilbert spaces, however, the Bhat-Skeide interpolation demonstrates that embeddability can always be achieved up to a certain modification.This leads us to the following problem: Problem 2.5 Given a Hilbert space H, what is the smallest dimension that an auxiliary Hilbert space H 1 can have, such that for each S P LpHq, the tensor product I H 1 b S P LpH 1 b Hq can be embedded into a C 0 -semigroup on H 1 b H? Note again, as mentioned in §1.3 that all Hilbert spaces are taken to be at least 1dimensional.If H and H 1 are Hilbert spaces with dimpH 1 q " 1, then H - where e P H 1 can be taken to be any unit vector.From this it is easy to observe for S P LpHq that I H 1 b S can be embedded into a C 0 -semigroup on H 1 b H if and only if S can be embedded into a C 0 -semigroup on H.If dimpH 1 q ą 1, then the 'if' direction continues to holds, whilst the 'only if' direction fails in general.Hence the above notion of embedding covers and also strictly extends the standard definition.

Proposition 2.6
The optimal dimension in Problem 2.5 for any Hilbert space is ℵ 0 .j Proof.Let H be a Hilbert space and let d H denote the optimal dimension in Problem 2.5 for H. Consider an arbitrary operator S P LpHq.Since the Bhat-Skeide interpolation of S (see Lemma 2.2) yields a C 0 -semigroup T on L 2 pTq b H satisfying T p1q " I L 2 pTq b S, it follows that d H is bounded above by dimpL 2 pTqq " ℵ 0 .To complete the proof, the possibility of d H being finite needs to be ruled out.So suppose to the contrary, there be a finite-dimensional Hilbert space H 1 for which I b S P LpH 1 b Hq can be embedded into a 1-parameter C 0 -semigroup for each S P LpHq.
Case 1. dimpHq is finite.Consider S :" 0 P LpHq.We show that I H 1 b S cannot be embedded into a C 0 -semigroup.So suppose to the contrary that T is a C 0 -semigroup on H 1 bH with T p1q " I H 1 b S " 0. The finite-dimensionality of H 1 b H implies norm-continuity of the semigroup.And since the subspace of invertible operators is norm-open and T ptq ÝÑ I in norm for t ÝÑ 0 `, it follows that T p 1 n q and thus T p1q " T p 1 n q n are invertible for sufficiently large n P N.This contradicts T p1q " 0.
Case 2. dimpHq is infinite.By infinite-dimensionality, there exists a non-bijective Fredholm operator S P LpHq, i.e. dimpkerpSqq and dimpkerpS ˚qq " dimpranpSq K q are finite with at least one these being non-zero.Now, one can readily verify that kerpI From these expressions and the assumptions on S and the auxiliary space, both dimpkerpI H 1 b Sqq and dimpranpI H 1 b Sq K q are finite, with at least one of these being non-zero.Thus I H 1 b S is a non-bijective Fredholm operator and thus by [11,Corollary 3.2] cannot be embedded into a C 0 -semigroup.In both cases, one sees that no finite-dimensional auxiliary space suffices.Hence d H must be at least ℵ 0 .
Remark 2. 7 We thank the anonymous referee for the suggested argument to prove Case 1 above, which shortened our original algebraic approach.More generally, note that no non-zero nilpotent operator on a finite-dimensional Hilbert space can be embedded into a C 0 -semigroup (see e.g.[11,Theorem 3.1]).j ℵ 0 " |N| denotes the first infinite cardinal, or simply: the cardinality of the naturals.
The embedding problem can also readily be extended to families of operators.In light of out generalisation of the Bhat-Skeide interpolation, we consider the following: Problem 2.8 Let H be a Hilbert space and I a non-empty index set.What is the smallest dimension that an auxiliary Hilbert space H 1 , can have, such that for each commuting family tS i u iPI Ď LpHq of bounded operators, the commuting family tI H 1 b S i u iPI Ď LpH 1 b Hq can be simultaneously embedded into a commuting family tT i u iPI of C 0 -semigroups on H 1 b H, in the sense that T i p1q " I H 1 b S i for i P I? Let κ denote this dimension.Lemma 1.4 yields that the κ is bounded above by κ max :" dimpL 2 p ś iPI Tqq.The arguments in the proof of Proposition 2.6 also yield κ ě ℵ 0 .If I is countable k then ℵ 0 ď κ ď κ max " ℵ 0 , whence the choice of auxiliary space in the Bhat-Skeide interpolation is optimal.If I is uncountable, then κ max ą ℵ 0 .l It would be nice to know if our generalisation of the Bhat-Skeide interpolation continues to provide the optimal solution for embedding in this case.
Finally, consider the space of d-tuples of commuting contractions for some d P N. In the above problem we determined that in the optimal worst-case scenario, the dimension of the auxiliary space needed for simultaneous embeddings is ℵ 0 .In the best-case scenario, this dimension is 1, which holds just in case a tuple can itself be simultaneously embedded (cf. the discussion at the start of this section).We can consider the subspace of such tuples and ask what topological properties it enjoys.Proposition 2.9 Let d P N and H be an infinite-dimensional Hilbert space.Let E d denote the set of all d-tuples of commuting contractions which can be simultaneously embedded into a commuting family of C 0 -semigroups on H. Then E d is dense in pCpHq d comm , pwq.Proof.Let tS i u d i"1 P CpHq d comm be arbitrary.By our generalisation of the Bhat-Skeide Interpolation, we know that the commuting d-tuple tI b S i u d i"1 P CpL 2 pT d q b Hq d can be simultaneously embedded.And for any unitary operator w : L 2 pT d q b H Ñ H one readily sees that tad w pI b S i qu d i"1 P CpHq d is a commuting d-tuple that can also be simultaneously embedded.Hence and it suffices to show that Ẽd is dense in pCpHq d comm , pwq.To this end consider an arbitrary tS i u d i"1 P CpHq d comm and finite F Ď H. Since dimpHq is infinite, there exists a unitary operator w : L 2 pT d q b H Ñ H satisfying w p1 1 b ξq " ξ for each ξ P F .m For n P N d 0 and ξ, η P F one obtains Since tS i u d i"1 was arbitrarily chosen, the density of Ẽd in pCpHq d comm , pwq follows.And by (2.5) the claim holds.k i.e. finite or countably infinite l Using elementary cardinal arithmetic one can readily obtain that dimpL 2 p ś iPI Tqq " maxt|I|, ℵ 0 u.So in particular, κmax ą ℵ 0 if I is an uncountable index set.

Numerical applications.
In this section we demonstrate how the Bhat-Skeide interpolation can be exploited for the purposes of time-discretisation of multi-parameter C 0 -semigroups on Hilbert spaces.

Interpolation under time-discretisation. Consider the following scenario:
Problem 2.10 Suppose a classical dynamical system is governed by an unknown commuting family t Ti u d i"1 of d contractive C 0 -semigroups on a Hilbert space H.We are provided with full information about the values tS i " Ti p1qu d i"1 .We seek a procedure to construct a commuting family tT i u d i"1 of C 0 -semigroups (on a finite-dimensional Hilbert space, if dimpHq ă 8), which recovers t Ti u d i"1 as much as possible based on the given information.The semigroups should be definable on arbitrarily fine discretisations of time points.

Since tS i u d
i"1 is clearly a commuting family of contractions, as a first attempt, one could apply Lemma 1.4 to obtain an interpolation tT i u d i"1 of tS i u d i"1 on the Hilbert space L 2 pT d q b H. Doing so, however, yields a semigroup on an infinite-dimensional space.Though, by discretising time points, one can control the growth of the dimension of the Hilbert space.For N P N let T N :" te ı2πt | t P as in Lemma 1.4.There exists a contractive homomorphism T pN q over the discrete monoid for all t P p 1 N Zq d , where r N P Lpℓ 2 pT d N q, L 2 pT d qq is a canonically defined isometry.In particular T pN q p0, 0, . . ., n i , . . ., 0q " I b Ti pnq for n P N 0 and i P t1, 2, . . ., du.
Proof.We use the constructions of P i , U i , i P t1, 2, . . ., du as in the proof of Lemma 1.4.Set u h :" 1 ?
for f P ℓ 2 pT d N q, one can readily see that r N is an isometry.Moreover ranpr N q " linptp ś d i"1 U i pt i qq u 1{N | t P p 1 N Zq d uq is clearly closed under U i p k N q for k P Z, i P t1, 2, . . ., du.Clearly, P i p k N qu 1{N " u 1{N , and relying on this observation as well as the relations in (BSCR), one readily derives that ranpr N q is closed under P i p k N q for k P Z, i P t1, 2, . . ., du.By construction of the interpolants (see (2.3)), it follows that ranpr N q b H is closed under T i ptq for t P 1 N N 0 , i P t1, 2, . . ., du.Equivalently stated, it holds that ´śd i"1 T i pt i q ¯pr N b Iq " pr N b Iq T pN q ptq for all t P p 1 N N 0 q d , and for some function T pN q : p 1 N N 0 q d Ñ Lpℓ 2 pT d N q b Hq.Applying this expression repeatedly, one can verify that T pN q is a homomorphism.The interpolation properties of the T i further yields T pN q p0, 0, . . ., n m in particular, L 2 pT d q b H and H are unitarily equivalent, since by elementary arithmetic of transfinite cardinals, setting κ :" dimpHq one has dimpL 2 pT d q b Hq " dimpL 2 pT d qq ¨dimpHq " ℵ 0 ¨κ " maxtℵ 0 , κu " κ " dimpHq.
Proposition 2.12 The discretisation T pN q in Proposition 2.11 is concretely given by T pN q ptq pf b ξq " ÿ for t P p 1 N N 0 q d , f P ℓ 2 pT d N q, and ξ P H, where κpt, t 1 q :" ttu `1 1 t ttu u`t tt 1 u uě1 for t, t 1 P R.
Proof.Without loss of generality, assume f " δ pe ı2πt 1 i q d i"1 " r N ´śd i"1 U i pt 1 i q ¯u1{N for some t 1 P p 1 N t0, 1, . . ., N ´1uq d .For each i P t1, 2, . . ., du by (BSCR) one has P i pt i qU i pt 1 i q " U i pt 1 i qQ i pt i , t 1 i q, where Q i pt i , t 1 i q " I´pP i pt 1 i q´P pt 1 i `ti qq if t tt i u u`t tt 1 i u u ă 1, otherwise Q i pt i , t 1 i q " P i pt 1 i q´P pt 1 i `ti q.Observe that i qu 1{N " 0. Since U i , P j (and thus U i , Q j ) commute for i ‰ j induction yields pr N b Iq T pN q ptq pf b ξq " ´d ź i"1 Thus and since the isometry satisfies r N δ Note that if dimpHq is finite, then the dimension of the Hilbert space, on which the discretisation T pN q is defined, satisfies dimpℓ 2 pT d N q b Hq " N d dimpHq ă 8. Furthermore, the expression in (2.6) entails that the generalised Bhat-Skeide interpolation is a common contractionvalued dilation of its discretisations.
Finally, it is worth noting that the generalised Bhat-Skeide interpolation and its discretisations are contraction-valued dilations of intuitively defined piecewise multi-linear extensions of the d-parameter discrete semigroup S : N d 0 Ñ LpHq corresponding to tS i u d i"1 (cf.§1.1).The following result demonstrates this precisely: ´p1 ´t tt i u uqS for all t P R d ě0 , where v P LpH, L 2 pT d q b Hq is the isometric embedding defined by vξ " 1 1 b ξ for ξ P H, where 1 1 is the constant function equal to 1 everywhere on T d .
Similarly, for each N P N, the discretisation T pN q in Proposition 2.11 satisfies ´p1 ´t tt i u uqS for all t P p 1 N N 0 q d , where v N P LpH, ℓ 2 pT d N q b Hq is the isometric embedding defined by v N ξ " Proof.We first handle the discretised case.Let N P N, t P p 1 N N 0 q d , and ξ, η P H be arbitrary.Since 1, δ z " 1 for z P T d N , (2.7) yields T pN q ptq p1 b ξq " ¯ξ.For i P t1, 2, . . ., du the set of t 1 i P 1 N t0, 1, . . ., N ´1u for which t tt 1 i u u`t tt i u u ă 1 is given by 1 N t0, 1, . . ., p1 ´t tt i u uq ¨N ´1u, which has exactly p1 ´t tt i u uq ¨N elements.n Thus v N T pN q ptq v N ξ, η where the final expression holds by the above counting argument.Since this holds for all ξ, η P H, (2.9) holds.
Observe that pr N b Iq v N ξ " 1 " ´p1 ´t tt i u uqS for each t P p 1 N N 0 q d , N P N. Hence (2.8) holds for all t P pQ X r0, 8qq d .By right-sot-continuity of both sides of this expression (in t i for each i) the validity of (2.8) extends to all t P R d ě0 .
Remark 2.14 Let H be a Hilbert space and d, N P N. By Proposition 2.11, the auxiliary space H 1 " ℓ 2 pT d N q, which has dimension N d , is sufficient to guarantee for every commuting family tS i u d i"1 Ď LpHq that tI b S i u d i"1 Ď LpH 1 b Hq can be simultaneously embedded into a commuting family tT i u d i"1 of discrete semigroups on H 1 b H over p 1 N N 0 , `, 0q.Analogous to §2.3, it would be useful to know whether the dimension of such a space is optimal, at least in the case of d " 1 and dimpHq ă ℵ 0 .

Remark 2.15
It would be interesting to know whether the discretised interpolation presented in this subsection can be applied in the numerical analysis of stochastic partial differential equations or in resampling methods in signal processing.
n Since t i P 1 N N 0 , one has t tt i u u P t0, 1 N , 2 N , . . ., 1 ´1 N u and thus p1 ´t tt i u uq ¨N P t1, 2, . . ., N u.

2.4.2
Approximations via scaled Bhat-Skeide interpolations.The methods in the previous section do not in general recover the original multi-parameter semigroup.However, by using ever finer interpolations, reconstruction is possible up to arbitrary approximation.Throughout this subsection we use the following notation: Given a projection p on a Hilbert space, we denote p K 0 :" p and p K 1 :" I ´p.Given d P N and a commuting d-tuple tS i u d i"1 of contractions on a Hilbert space H we let rtS i u d i"1 s B-Sk : R d ě0 Ñ CpL 2 pT d q b Hq denote the sot-continuous contractive homomorphism over R d ě0 on H corresponding to the generalised Bhat-Skeide interpolation of tS i u d i"1 as constructed in the proof of Lemma 1.4.Consider now a commuting family tT i u d i"1 of contractive C 0 -semigroups on an infinitedimensional Hilbert space H, or correspondingly an sot-continuous homomorphism T P S c s pR d ě0 , Hq (cf. the discussion in §1.1).For ε ą 0 and any unitary operator w : for t P R d ě0 , where t peq :" ppt t i ε u `ei qεq d i"1 for each e P t0, 1u d and where the unitaries U i pt i q and projections P i pt i q are the operators on L 2 pT d q defined as in the proof of Lemma 1.4.It is easy to see that rT s ε,w B-Sk defines an sot-continuous contractive homomorphism over R d ě0 on H, i.e. corresponds to a commuting family of d contractive C 0 -semigroups on H.Note that as per the discussion at the end of §2.1, these semigroups are furthermore unitary if (and only if) each of the T i are.
The construction in (2.10) can be understood as being unitarily similar to a Bhat-Skeide interpolation between the discretely spaced values t ś d i"1 T i pk i εq | k P N d 0 u of the family tT i u d i"1 .For ever smaller values of ε and appropriately chosen unitary operators w, one may ask whether these d-parameter semigroups approximate tT i u d i"1 .This is indeed the case.Proposition 2.16 Let d P N and H be an infinite-dimensional Hilbert space.For each T P S c s pR d ě0 , Hq there exists a net pT pαq q αPΛ Ď S c s pR d ě0 , Hq of homomorphisms unitarily similar to time-scaled Bhat-Skeide interpolations such that T pαq k-wot ÝÑ α T .
Proof.Let P Ď LpHq be the index set consisting of finite-rank projections on H directly ordered by p ě q :ô ranppq Ě ranpqq for p, q P P .For p P P , since H is infinite-dimensional, we may find a unitary operator w p : L 2 pT d q b H Ñ H satisfying w p p1 1 b ξq " ξ for each ξ P ranppq.m Let p0, 8q ˆP be the index set directly ordered by pε 1 , pq ě pε, qq :ô ranppq Ě ranpqq and ε 1 ď ε and consider the net ´T pε,pq :" rT s ε,wp B-Sk ¯pε,pqPp0, 8qˆP .
Fix arbitrary δ ą 0, F Ď H finite, and K Ď R d ě0 compact.Let τ ą 0 be sufficiently large such that r0, τ s d Ě K. Since T is uniformly sot-continuous on r0, 2τ s d , there exists ε 0 P p0, τ q such that sup s,tPr0, 2τ holds for each ξ P F .Let p 0 P P be the finite-rank projection onto linpF q.Consider an arbitrary pε, pq P p0, 8q ˆP with pε, pq ě pε 0 , p 0 q, i.e. ε ď ε 0 and ranppq Ě ranpp 0 q.For ξ, η P F and t P K one computes rT s ε,wp " per construction of w p and since ξ, η P F Ď ranpp 0 q Ď ranppq " ÿ ePt0,1u d ´d ź i"1 Now, by the commutativity of each U i pt i q with P j pt j q for i ‰ j as well as the commutativity of the projections tP i pt i qu d i"1 , and since tU i pt i qu d i"1 are Koopman operators (cf. the construction in the proof of Lemma 1.4), one has ν e ptq " ´d ź i"1 for each e P t0, 1u d .Furthermore i.e. tν e ptqu ePt0,1u d are non-negative reals summing to 1. Applying this to the above expression for the inner product thus yields pT pε,pq ptq ´T ptqqξ, η " prT s ε,wp ν e ptq ¨ pT pt peq q ´T ptqqξ, η .
Since t P K Ď r0, τ s d , by the choice of ε 0 one has t peq " ppt t i ε u `ei qεq d i"1 P r0, τ `εs d Ď r0, τ `ε0 s d Ď r0, 2τ s d for each e P t0, 1u d .Noting further that |t i ´pt t i ε u `ei qε| " |t t t i ε u u ´ei | ¨ε ď maxt1 ´t t t i ε u u, t t t i ε u uu ¨ε ď ε ď ε 0 , one may apply the uniform continuity condition in (2.11) to the above computation, yielding for ξ, η P F .Hence pT pε,pq q pε,pqPp0, 8qˆP By construction, the multi-parameter semigroups in (2.10) are unitarily similar to timescaled Bhat-Skeide interpolations.As such, each such semigroup is completely determined by a few pieces of information: a time-scale ε ą 0, a d-tuple tS i u d i"1 of commuting contractions on the Hilbert space H, and a unitary operator w : L 2 pT d q b H Ñ H.The class of such semigroups thus admits a simple parameterisation and simple connections to discrete-time processes.This provides good reason to find these approximants interesting in and of themselves.
The following result is likely well-known, but demonstrates a simple application of these approximants.
Proposition 2.17 Let d P N and tT i u d i"1 be a commuting family of contractive C 0 -semigroups on an infinite-dimensional Hilbert space H. Then tT i u d i"1 has a simultaneous unitary dilation if and only if tT i pt i qu d i"1 has a simultaneous power-dilation for all t P R d ě0 .Proof.The 'only if'-direction is a straightforward observation.Towards the 'if'-direction, first let T : R d ě0 Ñ CpHq be the corresponding sot-continuous homomorphism corresponding to tT i u d i"1 .By [8, Theorem 1.18] (see also Proposition 2.19 below) it suffices to find k wot -approximants for T which are unitarily dilatable.To this end it suffices to consider the approximants prT s ε,wp B-Sk q pε,pqPp0, 8qˆP constructed in Proposition 2.16.Let ε ą 0 and w : L 2 pT d q b H Ñ H be arbitrary.By assumption, tS i :" T i pεqu d i"1 admits some power-dilation pH 1 , tV i u d i"1 , rq.Consider now the Bhat-Skeide interpolations rtS i u d i"1 s B-Sk and rtV i u d i"1 s B-Sk .By the discussion at the end of §2.1, we know that the latter corresponds to a commuting family of d unitary C 0 -semigroups.By working with the generalised Bhat-Skeide construction (see (2.3)) it is a straightforward exercise to see that pL 2 pT d qbH 1 , rtV i u d i"1 s B-Sk , idb rq constitutes a unitary dilation of rtS i u d i"1 s B-Sk .That is, rtT i pεqu d i"1 s B-Sk and hence also rT s ε,w B-Sk " ad w ˝rtT i u d i"1 s B-Sk pε ´1¨q are unitarily dilatable.Since this holds for all ε, w, it follows that T is unitarily dilatable.Proposition 2.18 (Parrott, 1970).Let H 0 be a Hilbert space.Every pair of non-commuting unitaries R 1 , R 2 P LpH 0 q yields a tuple of commuting contractions which admits no power-dilation.o o Recall that E ij P M 2 pCq denotes the elementary matrix with 1 in the pi, jq-th entry, and 0's elsewhere.
A proof of this can be found in [33, §3] and [44, §I.6.3].Note that in Parrott's presentation, it suffices to require R 1 to be unitary and R 2 to be a contraction.We also make use of the following relation established in [8]: Proposition 2.19 (Equiv. of dilations and approximations).Let H be an infinitedimensional Hilbert space and pG, M q P tpR d , R d ě0 q, pZ d , N d 0 q, | d P Nu.Let A, A ˚, A, A ˚, D be as defined above (see §1.3).Then A " A ˚and A " A ˚" D.
Proof.By the 1:1-correspondence between S u s pM, Hq and ReprpG : Hq mentioned in §1.1, we have A " A ˚.By [8,Theorem 1.18], which is applicable to pG, M q (see [8,Example 1.10] and [5, Appendix A, Example 4.5, p. 522]) and to our infinite-dimensional Hilbert space H, unitary dilatability of an element T P X is equivalent to T being k wot -approximable via elements of A (cf. [8,Definition 1.15]).Thus A " A ˚" D.
Choose now two non-commuting unitaries R 1 , R 2 P LpH 0 q (this is possible, since dimpH 0 q ě 2).Set S 1 :" R 1 b E 21 , S 2 :" R 2 b E 21 , and S 3 :" I H 0 b E 21 .Since tS i u 3 i"1 Ď LpHq is a commuting family of contractions (cf.Proposition 2.18), by the generalised Bhat-Skeide interpolation (Lemma 1.4), there exists a commuting family, tT i u 3 i"1 of contractive C 0 -semigroups on L 2 pZq b H " H, such that ś 3 i"1 T i pn i q " ś 3 i"1 pI L 2 pZq b S i q n i for all n " pn i q 3 i"1 P N 3 0 .Set T i :" pI H q tPR ě0 for i P I z t1, 2, 3u.
We claim that tT i u 3 i"1 and hence tT i u iPI do not have simultaneous unitary dilations.Suppose, towards a contradiction, that pH 1 , tU i u 3 i"1 , rq is a simultaneous unitary dilation for tT i u 3 i"1 .Set V i :" U i p1q for each i P I. Then for n " pn i q 3 i"1 P has a power-dilation, where R 1 1 :" As this is a contradiction, tT i u 3 i"1 admits no simultaneous unitary dilation.Construction with bounded generators: For each i P I fix a net pT pαq i q αPΛ i of Yosidaapproximants, which converges to T i in the topology of uniform sot-convergence on compact subsets of R ě0 (written: k sot -convergence).p Set Λ :" ś 3 i"1 Λ i .The net ptT pα i q i u 3 i"1 q αPΛ consists of commuting families of contractive C 0 -semigroups and converges to tT i u 3 i"1 wrt. the topology of uniform sot-convergence on compact subsets of R 3 ě0 (cf.[8, Lemma 2.4 and Proposition 2.6-7]).Let T, T pαq P S c s pR 3 ě0 , Hq be the 3-parameter contractive C 0 -semigroups corresponding to tT i u 3 i"1 and tT pα i q i u 3 i"1 respectively for α P Λ (cf. the discussion in §1.1).
We now claim that for some index α P Λ, the commuting family tT pα i q i u 3 i"1 of contractive C 0 -semigroups (with bounded generators) has no simultaneous unitary dilation.If this were not the case, then each T pαq has a unitary dilation.By Proposition 2.19, it follows that tT pαq | α P Λu Ď D " A ˚where D Ď S c s pR 3 ě0 , Hq ": X is the set of unitarily dilatable elements and A ˚" tU | R 3 ě0 | U P ReprpR 3 : Hqu and the closure is computed inside pX, k wot q.From the above k sot -convergence, it follows that pT pαq q αPΛ k-wot ÝÑ T and thus T P A ˚" D. Thus T must be unitarily dilatable, i.e. tT i u 3 i"1 has a simultaneous unitary dilation.This contradicts the above construction.Thus setting Ti :" T pα i q i for i P t1, 2, 3u and some suitable index α P Λ, and Ti :" pI H q tPR ě0 for i P I z t1, 2, 3u, one has that t Ti u iPI is a commuting family of contractive C 0 -semigroups with bounded generators and which admits no simultaneous unitary dilation.

Residuality results
As discussed in the introduction, there is a 1:1-correspondence between commuting families tT i u d i"1 of d contractive C 0 -semigroups on a Hilbert space H and sot-continuous contractive homomorphisms T : R d ě0 Ñ LpHq.Similarly, there is a 1:1-correspondence between commuting families tS i u d i"1 of d contractions on H and contractive homomorphism S : N d 0 Ñ LpHq.Thus an appropriate general setting is to consider topological submonoids M of a topological group G, and the space S c s pM, Hq of all sot-continuous contractive homomorphisms T : M Ñ LpHq.We endow this with the k wot -topology (the topology of uniform wot-convergence on compact subsets of M , see §1.1) and consider in particular the subspace S u s pM, Hq of all sot-continuous unitary homomorphisms U : M Ñ LpHq.Recall that in the case of pG, M q P tpR d , R d ě0 q, pZ d , N d 0 q | d P Nu the map ReprpG : s pM, Hq is a bijection.In this section we first demonstrate the genericity of elements of S u s pM, Hq with dense orbits under the unitary action.This is then used to develop a 0-1-dichotomy for the residuality of S u s pM, Hq within pS c s pM, Hq, k wot q under modest assumptions.Since we now have a more complete picture on the existence of non-dilatable commuting families of contractive C 0 -semigroups as well as non-dilatable commuting families of contractions, we are able to decide this dichotomy for the concrete cases of M P tR d ě0 , N d 0 | d P Nu.
3.1 Universal elements.Let M be a topological monoid and H be a Hilbert space.We define an action of the group U pHq (the unitary operators on H) on S c s pM, Hq via pad w T qpxq :" ad w T pxq :" w T pxq w for w P U pHq, T P S c s pM, Hq, and x P M .One can readily see that this is a well-defined action.This gives rise to the equivalence relation T " u T 1 :ô Dw P U pHq : ad w T " T 1 for T, T 1 P S c s pM, Hq.We denote the equivalence classes, or orbits, by rT s u :" tT 1 P S c s pM, Hq | T " u T 1 u " tad w T | w P U pHqu for T P S c s pM, Hq.Definition 3.1 Let Ω u pM, Hq Ď S c s pM, Hq be the set elements T P S c s pM, Hq for which rT s u is dense in pS c s pM, Hq, k wot q.That is, Ω u pM, Hq is the set of elements with dense orbits.Refer to the elements in Ω u pM, Hq as universal elements.
p Let A i be the generator of T i .For each α P p0, 8q, the α-th Yosida-approximant is given by T pαq i " pe tA pαq i q tPR ě0 where A pαq i " αA i pαI ´Ai q ´1.For the well-definedness of these approximants and proof of the convergence T

Proposition 3.2 (Residuality of universal elements).
Let M be a locally compact Polish topological monoid and H be a separable infinite-dimensional Hilbert space.Then Ω u pM, Hq is a dense G δ -subset in pX, k wot q where X :" S c s pM, Hq.Proof.Borel complexity: As per the discussion in §1.1 after Question 1.2, given the conditions on M and H we know that pX, k wot q is a second countable space.There thus exists a countable basis O for pX, k wot q.Observe that for w P U pHq the map X which is a G δ -set.
Density: For T P Ω u pM, Hq and T 1 P rT s u , it holds that rT 1 s u " rT s u is dense and thus T 1 P Ω u pM, Hq.Thus if Ω u pM, Hq is non-empty, then it contains all the elements in the (dense) orbit of an element, and is thus itself dense.Thus it suffices to show that Ω u pM, Hq is non-empty.To construct a universal element, since pX, k wot q is second countable and thus separable, we can fix a family pT n q nPN Ď X such that tT n | n P Nu is dense in pX, k wot q.Since H is infinite-dimensional, it holds that H 1 :" À nPN H is isomorphic to H, i.e. there exists a unitary operator v P LpH 1 , Hq.We now construct T : M Ñ LpHq via T pxq :" v ´à nPN T n pxq ¯vf or x P M .One can readily check that T is pointwise well-defined and contraction-valued.It is routine to check that T is also sot-continuous and a homomorphism.Thus T P S c s pM, Hq " X.We now show that rT s u is dense.To this end it suffices to prove that T n P rT s u for each n P N.
Let pr n q nPN Ď LpH, H 1 q be isometries associated with the direct sum, i.e. r mr n " δ mn I H for m, n P N and ř nPN r n r n " I H 1 (computed strongly).In particular we can express the above construction more concretely as T pxq " ÿ nPN vr n T n pxq r nv ˚(3.12) for x P M , where the sum is computed strongly.
Let n 0 P N be arbitrary.Let P Ď LpHq be the index set consisting of finite-rank projections on H, directly ordered by p ě q :ô ranppq Ě ranpqq for p, q P P .Since vr n 0 P LpHq is an isometry and dimpHq is infinite, for each p P P there exists a unitary operator w p P LpHq such that w p p " vr n 0 p and thus w p vr n 0 p " p. (3.13) We now show that rT s u Ě pad wp T q pPP k-wot ÝÑ T n 0 .To this end, let ξ, η P H be arbitrary.Let p 0 P P be the finite-rank projection onto lintξ, ηu.For p P P with p ě p 0 one has w p vr n 0 ξ " w p vr n 0 pξ and thus ppad wp T qpxq ´Tn 0 pxqqξ, η 3.3 Application to the multi-parameter setting.We now consider the concrete cases of pG, M q P tpZ d , N d 0 q, pR d , R d ě0 q | d P Nu.For the discrete setting, we shall make use of the following reformulation of the 0-1-dichotomy for the pw-topology: Proposition 3.5 Let H be a separable infinite-dimensional Hilbert space and d P N. Let E Ď CpHq d comm and assume that E is unitarily invariant in the sense that tad w S i u d i"1 " twS i w ˚ud i"1 P E for all tS i u d i"1 P E and unitaries w P U pHq (e.g.E " U pHq d comm ).Then exactly one of the following holds: either E is dense in pCpHq d comm , pwq or E is meagre in this space.In the case of E " U pHq d comm , the first option in this dichotomy can be furthermore strengthened to: E is a dense G δ -subset.
Proof.Set M :" N d 0 and X :" S c s pM, Hq.By Remark 1.1, pX, k wot q -pCpHq d comm , pwq topologically via ϕ : S Þ Ñ tSp0, 0, . . ., 1 i , . . ., 0qu d i"1 .By the assumptions on E it is straightforward to see that A Ď ϕ ´1pEq Ď X and that ϕ ´1pEq is " u -invariant.Via the homeomorphism one also has ϕ ´1pEq " ϕ ´1pEq, where E denotes the closure of E in pCpHq d comm , pwq.By Lemma 3.4 exactly one of the following holds: either ϕ ´1pEq is dense in pX, k wot q (resp.a dense G δ -set if ϕ ´1pEq " S u s pM, Hq) or ϕ ´1pEq " ϕ ´1pEq is meagre in this space.Applying the homeomorphism ϕ yields the claimed dichotomy.
We can now decide the 0-1-dichotomy in both the discrete and continuous settings.
Proof (of Corollary 1.7).Discrete setting: Let d P N and pG, M q :" pZ d , N d 0 q.Let X " CpHq d comm , A " U pHq d comm , and D Ď X be the set of d-tuples admitting a power-dilation.The claim A " D follows from Proposition 2.19 under consideration of the correspondence between commuting families of contractions and homomorphisms mentioned in §1.1 and the topological isomorphism mentioned in Remark 1.1.By the strict 0-1-dichotomy in Proposition 3.5 either A is a dense G δ -subset in pX, pwq or A is meagre (in particular not dense) in this space.If d ď 2, then by Andô's power-dilation theorem for pairs of contractions (see [2]) one has A " D " X, whence the first option in the dichotomy holds.Continuous setting: Let d P N and pG, M q :" pR d , R d ě0 q.Let X " S c s pM, Hq, A " S u s pM, Hq, and D Ď X be the set of d-parameter semigroups admitting a unitary dilation.The claim A " D was established in Proposition 2. 19.By the strict 0-1-dichotomy in Lemma 3.4, either A is a dense G δ -subset in pX, k wot q or A is meagre in this space.If d ď 2, then by S lociński's theorem on simultaneous unitary dilations of pairs of commuting contractive C 0semigroups (see [41], [42,Theorem 2], and [38,Theorem 2.3]) one has A " D " X, whence the first of option in the dichotomy holds.And if d ě 3, by our counter-examples in Theorem 1.5 one has X z A " X z D ‰ ∅, whence the second option in the dichotomy holds.i"1 purely in terms of operator-theoretic properties of the generators tA i u d i"1 at least in the case of bounded generators.This has been established for example in [7] for simultaneous regular unitary dilatability via the concept of complete dissipativity (see Theorem 1.1 and Definition 2.8 in this reference).This conjecture, however, turned out to be false.For d " 3, Varopoulos and Kaijser (see [45], [46, Theorem 1 and Addendum]), Crabb and Davie [4], and various others (cf.[40,Remark 3.6]) provided counterexamples to the von Neumann inequality for finite-dimensional Hilbert spaces.From these counterexamples, it is easy to derive that for any infinite-dimensional Hilbert space, H, and d ě 3, there is a commuting family of d contractions for which the von Neumann inequality fails, r i.e.Remark 3.9 (Rigidity).A C 0 -semigroup T on H is said to be rigid if a sequence r0, 8q Q pt n q nPN Ñ 8 exists such that pT pt n qq nPN sot ÝÑ I.In [13,Theorem IV.3.20],rigidity for unitary C 0 -semigroups on separable infinite-dimensional Hilbert spaces was shown to be residual i.e. within pS u s pR ě0 , Hq, k wot q.This result was later extended in [6, Theorem 1.3] to the contractive case i.e. within pS c s pR ě0 , Hq, k wot q.It would be nice to know if there is an appropriate notion of rigidity for d-parameter C 0 -semigroups, d P N. In light of Corollary 1.7, it would be of interest to determine whether for d ě 2 either the rigidity or non-rigidity of d-parameter contractive C 0 -semigroups is residual.of C 0 -semigroups on H. t For d " 1 it is shown in [12,Theorem 3.2], that E 1 is residual in pCpHq, wotq.This in fact also holds wrt. the pw-topology, since the result simply builds on the embeddability of all unitary operators and since U pHq is residual in pCpHq, pwq (see [15,Theorem 4.1]).

Remark 3.8 (von Neumann inequality
For d ě 1, this approach can be generalised as follows: For tS i u d i"1 P U pHq d comm , similar to the proof of [12,Lemma 3.1], the spectral mapping theorem for commutative C ˚-algebras can be applied to simultaneously diagonalise the S i to multiplication operators over a semifinite measure space (see e.g.[32,Theorem 1.3.6],[35, §3.3.1 and §3.4.1], and [24]).That is, one can find a semi-finite measure space pX, µq, measurable R-valued functions θ 1 , θ 2 , . . ., θ d P L 8 pX, µq, and a unitary operator u P LpH, L 2 pX, µqq, such that S i " u ˚Me ıθ i p¨q u for each i P t1, 2, . . ., du.Clearly, tT i :" pu ˚Me ıtθ i p¨q uq tPR ě0 u d i"1 is a commuting family of (unitary) C 0semigroups satisfying T i p1q " S i for each i. q The von Neumann inequality is stated this way e.g. in [20], [26, §5].In [33,44], the right-hand bound in (3.15) is given as sup λPtzPC||z|ď1u d |ppλ 1 , λ 2 , . . ., λ d q|.The two bounds are, however, the same due to the maximum modulus principle for holomorphic functions.
r Letting H 0 be a sufficiently large finite-dimensional Hilbert space such that there are commuting contractions tR i u 3 i"1 for which the von Neumann inequality fails.Set S i :" R i for i P t1, 2, 3u and S i :" I H 0 for i P t4, 5, . . ., du.Given an infinite-dimensional H, we may fix an isometry r : H 0 Ñ H. Then tr S i r ˚ud i"1 is a d-tuple of commuting contractions for which the von Neumann inequality continues to fail.
s in the sense that tw S i w ˚ud i"1 P E d for all tS i u d i"1 P E d and w P U pHq.
dimpHq ě ℵ 0 .It is however unclear whether E d (or Ẽd ) is a G δ -set, from which the residuality of embeddable d-tuples of commuting contractions would follow.

Remark 1 . 1
Let d P N and consider the space pCpHq d comm Ď CpHq d , pwq of d-tuples of commuting contractions on H under the weak polynomial topology.This is given by the convergence con-

T
psqT ptq " pU psq b IqpP psq b S tsu `pI ´P psqq b S tsu`1 q ¨pU ptq b IqpP ptq b S ttu `pI ´P ptqq b S ttu`1 q " pU psq b IqpU ptq b Iq ¨pQps, tq b S tsu `pI ´Qps, tqq b S tsu`1 q ¨pP ptq b S ttu `pI ´P ptqq b S ttu`1 q.

T 1 "P
ptq ˚T ptq " ´P ptq b pS ˚qttu `pI ´P ptqq b pS ˚qttu`1 ¯pU ptq ˚b Iq ¨pU ptq b Iq ´P ptq b S ttu `pI ´P ptqq b S ttu`ptq b pS ˚qttu S ttu `pI ´P ptqq b pS ˚qttu`1 S ttu`1 " P ptq b I `pI ´P ptqq b I " I, piq t : Z Ñ Z and p piq t : Z Ñ R via ϕ piq t pzq :" i , . . ., 0q " pr N b Iq ˚Ti pnqpr N b Iq " pr N b Iq ˚pI b S n i qpr N b Iq " I b S n i " I b Ti pnq for n P N 0 , i P t1, 2, . . ., du.

2. 5
Non-dilatable families of semigroups.Building on the following counter-examples, the generalised Bhat-Skeide interpolation provides sufficient means to prove Theorem 1.5.

Remark 3 . 6 Remark 3 . 7
In [7, Proposition 5.3], examples of d-parameter contractive C 0 -semigroups are constructed which admit no regular unitary dilation (a stronger notion of dilation, see [7, Definition 2.2]) for d ě 2. It would be interesting to know whether for d ě 3 these examples admit unitary dilations.It would be nice to find a simple characterisation for the simultaneous unitary dilatability of arbitrary d-parameter C 0 -semigroups tT i u d

Remark 3 . 10 (
Genericity of embeddability).Let H be a separable infinite-dimensional Hilbert space and d P N. Let E d Ď CpHq comm be the subset of d-tuples of commuting contractions tS i u d i"1 P CpHq d comm that can be simultaneously embedded into commuting families tT i u d i"1 [30,mm Ď CpHq d of d-tuples of commuting contractions on H under the pw-topology as well as the subspaces A :" U pHq d For the construction of the product of arbitrarily many probability spaces, see e.g.[30, Example 14.37].gFor Hilbert spaces H 1 , H 2 the Hilbert space tensor product H 1 bH 2 can be viewed as being isomorphic to ℓ 2 pB 1 ˆB2 q, where B i Ď H i is an orthonormal basis (ONB) for each i.For ξ 1 , η 1 P H 1 and ξ 2 , η 2 P H 2 we have ξ 1 bξ 2 , η 1 bη 2 H 1 bH 2 " ξ 1 , η 1 H 1 ξ 2 , η 2 H 2 , where e.g.ξ 1 b ξ 2 " p ξ 1 , e 1 H 1 ξ 2 , e 2 H 2 q pe 1 ,e 2 qPB 1 ˆB2 .For the algebraic and geometric definitions as well as tensor products of bounded operators, we refer the reader to [28, §2.6], [32, §6.3] [34, E3.2.19-21]. f 1the sets LpHq and LpH, H 1 q denote the bounded linear operators on H resp. the bounded linear operators between H and H 1 .Further, CpHq and U pHq denote the set of contractions and unitaries on H respectively.And for d P N, CpHq d For Hilbert spaces H 1 , H 2 , ..., H n , n P N, the tensor productH 1 b H 2 b . . . . ..bH nshall always denote the Hilbert space tensor product, i.e. the completion of the algebraic tensor product under the induced norm. •
d ‰ CpHq d comm .Now, clearly, E d is unitarily invariant, s contains U pHq d comm , and is closed within pCpHq d comm , pwq.Supposing further that dimpHq " ℵ 0 , by Proposition 3.5 it follows that either E d " CpHq d comm or E d is meagre in pCpHq d comm , pwq.Since the first option is ruled out, it follows for each d ě 3 that E d is meagre, or in other words the collection of d-tuples of commuting contractions not satisfying the von Neumann inequality is residual in pCpHq d comm , pwq.
Thus E d Ě U pHq d comm and clearly E d is unitarily invariant.By Proposition 3.5 it follows that exactly one of the following holds: E d is dense in pCpHq d comm , pwq or E d is meagre in this space.Applying Corollary 1.7, if d P t1, 2u then U pHq d comm and thus also E d are residual in pCpHq d comm , pwq.For d ě 3, one can no longer argue in this way, since U pHq d comm is no longer residual in pCpHq d comm , pwq.We nonetheless ascertained in Proposition 2.9 that E d (as well as a particular subset Ẽd Ď E d defined in (2.5)) is at least dense in pCpHq d comm , pwq, provided