On the complete metrisability of spaces of contractive semigroups

The space of unitary $C_{0}$-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology of uniform weak convergence on compact subsets of $\mathbb{R}_{+}$, is known to admit various interesting residual subspaces. Before treating the contractive case, the problem of the complete metrisability of this space was raised in [Eisner, 2010]. Utilising Borel complexity computations and automatic continuity results for semigroups, we obtain a general result, which in particular implies that the one-/multiparameter contractive $C_{0}$-semigroups constitute Polish spaces and thus positively addresses the open problem.


Introduction
In [2] the space of contractive C 0 -semigroups over a separable infinite-dimensional Hilbert space, and when viewed with the topology of uniform weak operator convergence on compact subsets of R`, was shown to constitute a Baire space. The main application of this is [2, Proposition 5.1], which relies on the approximation result in [17,Theorem 2.1] and shows that residual properties for the unitary case automatically transfer to the contractive case. In particular, this application renders meaningful the residuality results achieved in [6], [17,Corollary 3.2], and [4, §III.6 and §IV. 3.3]. Note also that residual properties of (contractive) operators on Banach spaces, as initiated in [3,5], have recently been studied in connection with hypercyclicity and the Invariant Subspace Problem in [9,10,11]. The continuous case remains to be investigated.
In this paper we improve upon the result in [2] and show that the space of contractive C 0semigroups is Polish (i.e. separable, completely metrisable). In fact, we prove this for spaces of more generally defined semigroups, including multiparameter semigroups. In particular, our result positively solves a problem raised in [4, §III.6.3] (cf. also [7,Remark 2.2]). There it was shown that, when viewed under the topology of uniform weak operator convergence on compact time intervals, the space of contractive C 0 -semigroups is not sequentially closed within the larger space of continuous contraction-valued functions. We shall reinforce this by studying the geometric properties of these spaces in a general setting, and providing a deeper reason for this failure (see Corollary 3.13). This renders the complete metrisability problem non-trivial.
The approach in [2,Theorem 1.20] involves studying and transferring properties from the subspace of unitary semigroups, which is a Baire space. This method crucially relies on the fact that contractive semigroups can be weakly approximated by unitary semigroups. These density results in turn arise from the theory of dilations (cf. [20,Theorem 1] and [17,Theorem 2.1]). By contrast, the approach here bypasses dependency upon dilation. Instead we directly classify the space of contractive C 0 -semigroups in terms of its Borel complexity within a larger, completely metrisable space. This complexity result in turn implies complete metrisability (see Theorem 4.2).
Our result encompasses a broad class of spaces on which the semigroups are defined. We provide basic examples in the main text and broaden this to a larger class in Appendix A. The generality of the main result may also be of interest to other fields. Multiparameter semigroups, for example, occur in structure theorems (see e.g. [18]), the study of diffusion equations in spacetime dynamics (see e.g. [25]), the approximation of periodic functions in multiple variables (see e.g. [24]), etc.

Definitions of spaces of semigroups
Throughout this paper, H shall denote a fixed separable infinite-dimensional Hilbert space. Furthermore, LpHq Ě CpHq Ě IpHq Ě U pHq denote (from left to right) the spaces of bounded linear operators, contractions, isometries, and unitaries over H. These can be endowed with the weak operator topology (wot) or the strong operator topology (sot).
Instead of working with semigroups defined on R`(continuous time) or over N 0 (discrete time), we shall more generally work with semigroups parameterised by a topological monoid.
Definition 2.1 Let pM,¨, 1q be a topological monoid. A semigroup over H on M shall mean any operator-valued function, T : M Ñ LpHq, satisfying T p1q " I and T pstq " T psqT ptq for s, t P M .
In other words, semigroups are just certain kinds of algebraic morphisms. Observe that the above definition applied to the topological monoid pR`,`, 0q yields the usual definition of an operator semigroup.
The continuous contractive semigroups defined on M may be viewed as subspaces of the function spaces C`M, CpHq˘, where CpHq may be endowed with either the wot-or sottopology. We summarise these spaces and their topologies as follows: Working with these definitions, one can readily classify some of these spaces as follows: Proposition 2.4 Let M be a locally compact Polish monoid. Then pF c w pM q, k wot q, pF c s pM q, k sot q, and pC s pM q, k sot q are Polish spaces. For a full proof see [2,Propositions 1.16 and 1.18]. For the reader's convenience, we sketch the arguments here.
Proof (of Proposition 2.4). First note that the spaces, pCpHq, wotq and pCpHq, sotq, are well-known to be Polish (see e.g. [14,Exercise 3.4 (5) and Exercise 4.9]). To prove the first claim, we need to show that pC`M, Y˘, k wot q is Polish, where Y :" pCpHq, wotq. Since M is a locally compact Polish space, and since for metrisable spaces, separability is equivalent to second countability, one can readily construct a countable collection of compact subsets,K Ď KpM q, such that tintpKq | K PKu covers M . Consider the spaces C`K, Y˘for K PK and endow these with the topology of uniform convergence, which makes them Polish spaces (see e.g. [14,Theorem 4.19] or [1,3.99]). The map Ψ : C`M, Y˘Ñ ś KPK C`K, Yf Þ Ñ pf | K q KPK is clearly well-defined. Since tintpKq | K PKu covers M and M is locally compact, the map is also clearly bicontinuous. The covering property also guarantees that every coherent sequence of continuous functions pf K q KPK P ś KPK C`K, Y˘corresponds to a unique continuous function, f :" Ť KPK f K P C`M, Y˘, satisfying Ψpf q " pf K q KPK . Thus Ψ is a homeomorphism between C`M, Y˘and the subspace of coherent sequences of continuous functions. Since the product of Polish spaces is Polish (see [1,Corollary 3.39]) and the subspace of coherent sequences is clearly closed under the product topology, it follows that pC`M, Y˘, k wot q is Polish. The second claim is obtained in the same manner, by replacing Y by pCpHq, sotq and k wot by k sot above. Finally, it is easy to verify that C s pM q is a closed subspace within pF c s pM q, k sot q, and thus that pC s pM q, k sot q too is Polish.
The aim of this paper is to show that pC s pM q, k wot q is completely metrisable for some topological monoids M , in particular for M " pR`,`, 0q. We can achieve this for a broad class of monoids, by appealing to the following condition: Definition 2.5 Call a topological monoid, M , 'good', if the contractive wot-continuous semigroups over H on M are automatically sot-continuous, i.e. if C w pM q " C s pM q holds.
All discrete monoids (including non-commutative ones) are trivially 'good'. By a classical result, pR`,`, 0q is 'good' (cf. [ We need to show that T is sot-continuous. For each k P t1, 2, . . . , du let π k : ś d i"1 M i Ñ M k denote the canonical projection, which is a (continuous) monoid homomorphism, and let r k : M k Ñ ś d i"1 M i denote the canonical (continuous) monoid homomorphism defined by r k ptq " p1, 1, . . . , t, . . . , 1q (the d-tuple with t in the k-th position and identity elements elsewhere) for all t P M k . For each k P t1, 2, . . . , du observe further that T k : M k Ñ CpHq define by T k :" T˝r k is a wot-continuous homomorphism. That is, each T k is a wot-continuous contractive semigroup over H on M k . Since each M k is 'good', these are sot-continuous. Observe now, that T ptq " T p ś d i"1 r k pπ k ptqqq " T pr 1 pπ 1 ptqqq¨T pr 2 pπ 2 ptqqq¨. . .¨T pr d pπ d ptqqq " T 1 pπ 1 ptqq¨T 2 pπ 2 ptqq¨. . .¨T d pπ d ptqq holds for all t P ś d i"1 M i . Since the algebraic projections are continuous and the T k are sotcontinuous and contractive, and since multiplication of contractions is sot-continuous, it follows that T is sot-continuous.
Thus we immediately obtain the following examples of 'good' monoids: Corollary 2.7 For each d P N the monoid R d , viewed under pointwise addition, is 'good'. If one more generally considers monoids which are closed subspaces of locally compact Hausdorff topological groups, a sufficient topological condition exists, which guarantees that a monoid is 'good' (see Definition A.2 and Theorem A.8). By Examples A.3-A.6 and Proposition A.7, the class of monoids satisfying this condition is closed under finite products and includes all discrete monoids, the non-negative reals under addition pR`,`, 0q, the p-adic integers under addition pZ p ,`, 0q for all p P P, and even non-discrete non-commutative monoids including naturally definable monoids contained within the Heisenberg group of order 2d´3 for each d ě 2.
3. The k wot -closure of the space of contractive semigroups The simplest approach to demonstrate the complete metrisability of pC s pM q, k wot q would be to show that this be a closed subspace within the function space pF c w pM q, k wot q, which we already know to be Polish (see Proposition 2.4). In [4, Example §III.6.10] and [7, Example 2.1] a construction is provided, which demonstrates that this fails in particular in the case of oneparameter contractive C 0 -semigroups. In this section we reveal that the deeper reason for this failure is that the closure of C s pM q within pF c w pM q, k wot q is always convex, whereas for a broad class of topological monoids, M , the subset C s pM q is not convex (see Corollary 3.13 below).
Before we proceed, we require a few definitions. In the following M shall denote an arbitrary topological monoid. We continue to use H to denote a separable infinite-dimensional Hilbert space and I H (or simply I) for the identity operator.
Definition 3.1 For n P N and u 1 , u 2 , . . . , u n P LpHq we shall call pu 1 , u 2 , . . . , u n q an isometric partition of the identity, just in case the following axioms hold: (P1) uj u i " δ ij¨I for all i, j P t1, 2, . . . , nu.
(P2) ř n i"1 u i ui " I. Note that by axiom (P1) the operators in an isometric partition are necessarily isometries.
Remark 3.2 Let n P N. If pu 1 , u 2 , . . . , u n q is an isometric partition of I, then, letting H i :" ranpu i q for i P t1, 2, . . . , nu, it is easy to see that (P1) and (P2), together with the fact that each u i is necessarily an isometry, imply that the H i are mutually orthogonal closed subspaces of H and that H " for each i, then letting u i P LpHq be any isometries with ranpu i q " H i for each i P t1, 2, . . . , nu, one can readily see that (P1) and (P2) are satisfied. Thus, isometric partitions of I can be constructed from orthogonal decompositions into infinite-dimensional closed subspaces of H and vice versa.
Of course, these observations only apply for infinite-dimensional Hilbert spaces.
Definition 3.3 Let n P N, pu 1 , u 2 , . . . , u n q be an isometric partition of I, and T 1 , T 2 , . . . , T n P F c w pM q. Denote via p Say that A is closed under finite joins just in case for all n P N, all isometric partitions u :" pu 1 , u 2 , . . . , u n q of I, and all T 1 , T 2 , . . . , T n P A, it holds that The property of being closed under finite joins is a key ingredient in proving the convexity of the closure of subsets (see Lemma 3.7 below). We first provide some basic observations about which subsets are closed under finite joins.
Proposition 3.5 Let A P tF c w pM q, F c s pM q, C w pM q, C s pM qu. Then A is closed under finite joins. Proof. First consider the case A " F c w pM q. Let n P N, u :" pu 1 , u 2 , . . . , u n q be an isometric partition of I, and T 1 , T 2 , . . . , T n P A. We need to show that T :" p À n i"1 T i q u is in A. Applying the properties of the partition yields for all ξ P H and all t P M . And since by construction, T p¨q " ř n i"1 u i T i p¨qui , it clearly holds that T is wot-continuous. Thus T is a wot-continuous contraction-valued function, i.e. T P A. Hence A is closed under finite joins. The case of A " F c s pM q is analogous. Next we consider the case A " C w pM q. Let n P N, u :" pu 1 , u 2 , . . . , u n q be an isometric partition of I, and T 1 , T 2 , . . . , T n P A. We need to show that T :" p À n i"1 T i q u is in A. Since A Ď F c w pM q and F c w pM q is closed under finite joins, we already know that T P F c w pM q, i.e. that T is contraction-valued and wot-continuous. To show that T P A, it remains to show that T is a semigroup. Since each of the T i are semigroups, applying the properties of the partition yields T pstq for all s, t P M . Thus T is a wot-continuous contractive semigroup, i.e. T P A. Hence A is closed under finite joins. The case of A " C s pM q is analogous. Proposition 3.6 Let A Ď F c w pM q and let A be the closure of A within pF c w pM q, k wot q. If A is closed under finite joins, then so too is A.
Proof. Let n P N, u :" pu 1 , u 2 , . . . , u n q be an isometric partition of I, and T 1 , T 2 , . . . , T n P A. We need to show that T :" p À n j"1 T j q u is in A. To see this, we may simply fix a net ppT closed under finite joins, we have T piq :" p À n j"1 T piq j q u P A for all i. We also clearly have Since H is a separable Hilbert space, it admits a countable orthonormal basis (ONB) B Ď H, which we shall fix. It suffices to show for S, T P A and α, β P r0, 1s with α`β " 1, that R :" αS`βT P A. To do this, we need to show that R can be approximated within the k wot -topology by elements in A. In order to achieve this, it suffices to fix arbitrary K P KpM q, F Ď B finite, and ε ą 0, and show that someR P A exists satisfying sup tPK | pRptq´Rptqqe, e 1 | ă ε, for all e, e 1 P F .
To constructR, we first construct an isometric partition, pw 0 , w 1 q, of I, such that w 0 fixes the vectors in F . This can be achieved as follows: Since the ONB B is infinite, a partition tB 0 , B 1 u of B exists satisfying F Ď B 0 and |B 0 | " |B 1 | " |B| " dimpHq. There thus exist bijections f 0 : B Ñ B 0 and f 1 : B Ñ B 1 and since B 0 Ě F , we may assume without loss of generality that f 0 | F " id F . Using these bijections we obtain (unique) isometries w 0 , w 1 P LpHq satisfying w 0 e " f 0 peq and w 1 e " f 1 peq for all e P B. In particular, ranpw 0 q " linpB 0 q and ranpw 1 q " linpB 1 q. Now since tB 0 , B 1 u partitions B, we have H " linpBq " linpB 0 q ' linpB 1 q. As per Remark 3.2 it follows that pw 0 , w 1 q satisfies the axioms of an isometric partition of I. Now set u 0 :" ? αw 0`? βw 1 and u 1 :" ? βw 0´? αw 1 . One can easily derive from the fact that pw 0 , w 1 q is an isometric partition of I, that u :" pu 0 , u 1 q also satisfies the axioms of an isometric partition of I. Moreover, since w 0 was chosen to fix the vectors in F , applying the properties of the partition pw 0 , w 1 q yields  for all e P F . Finally setR :" pS À T q u . Since A is closed under finite joins, by Proposition 3.6 A is also closed under finite joins, and hence the constructed operator-valued function,R, lies in A. For all e, e 1 P F we obtain pRptq´Rptqqe, e 1 " u 0 Sptqu0 e, e 1 ` u 1 T ptqu1e, e 1 ´ Rptqe, e 1 " Sptqu0 e, u0 e 1 ` T ptqu1 e, u1e 1 ´ pαSptq`βT ptqqe, e 1
This establishes that the convex hull of A is contained in A. Thus A is convex. By Proposition 3.5 and Lemma 3.7 we thus immediately obtain the general result: Corollary 3.8 For all topological monoids, M , the closure of C s pM q within pF c w pM q, k wot q is convex.
We now provide a large class of topological monoids, M , for which C s pM q is not convex. Definition 3.10 Let pG,¨,´1, 1q be a topological group. Say that M Ď G is a topological submonoid, just in case M is endowed with the subspace topology, contains the neutral element 1, and is closed under¨.
In particular, if G is a topological group and M Ď G is a topological submonoid, then M is itself a topological monoid. satisfying U 0 pt 0 q ‰ I H 0 . By irreducibility and since G is separable, it necessarily holds that dimpH 0 q ď ℵ 0 " dimpHq. If dimpH 0 q " dimpHq, then we may assume without loss of generality that H 0 " H, and thus that U 0 is a representation of G over H. If dimpH 0 q is finite, we may assume that H 0 Ă H and view the orthogonal complement H 1 :" H K 0 within H. Replacing U 0 by t P G Þ Ñ U 0 ptq ' I H 1 yields an sot-continuous unitary representation of G over H which satisfies U 0 pt 0 q ‰ I H . In both cases, restricting U 0 to M yields a non-trivial sot-continuous unitary semigroup over H on M . Hence M has non-trivial unitary semigroups over H. Lemma 3.12 Suppose that M has non-trivial unitary semigroups over H. Then C s pM q is not a convex subset of F c w pM q. Proof. Let S :" Ip¨q be the trivial sot-continuous unitary semigroup over H on M . And by non-triviality we may fix some sot-continuous unitary semigroup T P C s pM qztIp¨qu. In particular, T pt 0 q ‰ I for some t 0 P M , which we shall fix. Choose any α, β P p0, 1q with α`β " 1. It suffices to show that R :" αS`βT R C s pM q. Suppose per contra that R P C s pM q. Then by the semigroup law we have 0 " Rpstq´RpsqRptq "`αSpstq`βT pstq˘´`αSpsq`βT psq˘`αSptq`βT ptq"`α SpsqSptq`βT psqT ptq˘´`αSpsq`βT psq˘`αSptq`βT ptq" αp1´αq SpsqSptq`βp1´βq T psqT ptq´αβ SpsqT ptq´βα T psqSptq " αβ SpsqSptq`βα T psqT ptq´αβ SpsqT ptq´βα T psqSptq " αβ pSpsq´T psqqpSptq´T ptqq. for all s, t P M . Since α, β ‰ 0 setting s :" t :" t 0 in the above yields pI´uq 2 " 0, where u :" T pt 0 q P U pHq. Since u is unitary, a basic application of the spectral mapping theorem yields that the spectrum of u is t1u. By the Gelfand theorem (see [19, Theorem 2.1.10]), it follows that T pt 0 q " u " I, which is a contradiction. Applying Corollary 3.8, Proposition 3.11, and Lemma 3.12 yields: Corollary 3.13 Let G be a locally compact Polish group and suppose that M Ď G is a topological submonoid with M ‰ t1u. Then the closure of C s pM q within pF c w pM q, k wot q is convex, whilst C s pM q itself is not convex. In particular, C s pM q is not a closed subspace within pF c w pM q, k wot q.
Considering G " pR d ,`, 0q with d ě 1 and M :" R d Ď G, the conditions of Corollary 3.13 are satisfied. In particular, the subspace of one-/multiparameter C 0 -semigroups is not closed within the larger space of wot-continuous contraction-valued functions.

Complete metrisability results
As indicated in the introduction, we shall demonstrate the complete metrisability of C s pM q by directly classifying its Borel complexity within the larger Polish space, pF c w pM q, k wot q. By the previous section we know that in a very general setting, C s pM q is not closed within pF c w pM q, k wot q. Hence we require weaker conditions, which determine when subspaces are completely metrisable. For this we rely on the following classical result from descriptive set theory (see [14,Theorem 3.11] for a proof): Lemma (Alexandroff 's lemma). Let X be a completely metrisable space. Then A Ď X viewed with the relative topology is completely metrisable if and only if it is a G δ -subset of X.
We now present the main result. Proof. Since t1u is a compact subset of M , it is easy to see that X :" tT P F c w pM q | T p1q " Iu is a closed subset in pF c w pM q, k wot q and thus pX, k wot q is Polish (cf. Proposition 2.4). By Alexandroff's lemma it thus suffices to prove that C s pM q is a G δ -subset of X.
To proceed, observe that since M is a locally compact Polish space, it is σ-compact, i.e. there exists a countable collection of compact subsets,K Ď KpM q, such that Ť KPK K " M . W. l. o. g. one may assume thatK is closed under finite unions. Since H is a separable Hilbert space, it admits a countable ONB B Ď H. For each finite F Ď B define π F :" Proj linpF q P LpHq, i.e., the projection onto the closed subspace generated by F . Using these, we construct d K,F,e,e 1 pT q :" sup s,tPK | pT psqπ F T ptq´T pstqqe, e 1 | for each K P KpM q, F Ď B finite, e, e 1 P H, and T P X, and V ε;K,F pT q :" č e,e 1 PF tT P X | sup tPK | pT ptq´T ptqqe, e 1 | ă εu, W ε;K,F,e,e 1 :" tT P X | d K,F,e,e 1 pT q ă εu for each ε ą 0, K P KpM q, F Ď B finite, e, e 1 P H, andT P X. We can now present our strategy for the rest of the proof: To show that C s pM q is a G δ -subset of X, it suffices to show (I) that the W -sets defined in (4.1) are open and (II) that If these two statements hold, then by assumption of M being 'good', (I) + (II) will yield that C s pM q " C w pM q are equal to a G δ -subset of X, which will complete the proof.
Towards (I), fix arbitrary ε ą 0, K P KpM q, F Ď B finite, and e, e 1 P B, and consider an arbitrary element,T P W ε;K,F,e,e 1. We need to show thatT is in the interior of W ε;K,F,e,e 1. By continuity of multiplication in the topological monoid, M , the set K¨K " tst | s, t P Ku is compact. Setting K 1 :" K Y pK¨Kq and F 1 :" F Y te, e 1 u, it suffices to show that V ε 1 ;K 1 ,F 1 pT q Ď W ε;K,F,e,e 1 (4.3) holds for some ε 1 ą 0, since clearlyT is an element of the left hand side and by definition of the k wot -topology, the V -sets are clearly open. We determine ε 1 as follows. First note that by virtue ofT being in W ε;K,F,e,e 1 r :" ε´d K,F,e,e 1 pT q ą 0 (4.4) holds. Since the unit disc, D 1 " tz P C | |z| ď 1u, is compact, the map pa, bq P D 2 1 Þ Ñ ab P C is uniformly continuous, and hence some ε 1 ą 0 exists, such that for all a, b, a 1 , b 1 P D 1 with |a´a 1 | ă ε 1 and |b´b 1 | ă ε 1 . We may also assume without loss of generality that ε 1 ă r 4 . With this ε 1 -value, the left hand side of (4.3) is now determined. It remains to show that the inclusion holds.
Since the elements in X are all contraction-valued functions and the ONB, B, consists of unit vectors, it holds that T ptqξ, η P D 1 for all T P X, t P M , and ξ, η P B. Now consider an arbitrary T in the left hand side of (4.3). Let s, t P K be arbitrary. Then s, t, st P K 1 , so that by the choice of F 1 and by virtue of T being inside V ε 1 ;K 1 ,F 1 pT q, we have for all e 2 P F . Since F is an orthonormal collection, the choice of ε 1 and (4.5) yield for all s, t P K. Thus " ε´r`r 2 ă ε, whence T P W ε;K,F,e,e 1 . Hence the inclusion in (4.3) holds, as desired.
To prove (II), consider the first inclusion of (4.2). Let T P C s pM q be arbitrary. To show that T is in the G δ -set in the middle of (4.2), consider arbitrary fixed ε ą 0, K P KpM q, F 0 Ď B finite, and e, e 1 P B. Our goal is to find some finite F Ď B with F Ě F 0 , such that T P W ε;K,F,e,e 1.
To this end, we rely on the fact that T is a contractive semigroup and observe that for all finite F Ď B the functions f F : KˆK Ñ Rp s, tq Þ Ñ | pT psqπ F T ptq´T pstqqe, e 1 | satisfy f F ps, tq " | pT psqπ F T ptq´T psqT ptqqe, e 1 | " | pI´π F qT ptqe, T psq˚e 1 | ď }T psq˚e 1 }f F ptq ďf F ptq for all s, t P K. Furthermore, the sot-continuity of T guarantees thatf F is continuous. Now consider the net pf F q F , where the indices run over all finite F Ď B, ordered by inclusion.
Note that the correspondingly indexed net of projections, pπ F q F , is monotone, and, since Ť F ĎB finite F " B and B is a basis for H, it holds that π F ÝÑ F I weakly (in fact strongly).
Clearly thenf pointwise and monotone. Since K is compact and thef F are continuous for all F , by Dini's Theorem (cf. [1, Theorem 2.66]) the monotone pointwise convergence in (4.6) is in fact uniform convergence. Hence, by the definition of the net, for some finite F Ď B with F Ě F 0 d K,F,e,e 1 pT q " sup s,tPK f F ps, tq ď sup tPKfF ptq ă ε, and thereby T P W ε;K,F,e,e 1.
To complete the proof of (II), we treat the second inclusion in (4.2). So let T P X be an arbitrary element in the G δ -set in the middle of (4.2). To show that T P C w pM q, it is necessary and sufficient to show that T pstq " T psqT ptq for all s, t P M . So fix arbitrary s, t P M . It suffices to show that pT psqT ptq´T pstqqe, e 1 " 0 for all basis vectors, e, e 1 P B. So fix arbitrary e, e 1 P B.
Note that sinceK covers M and is closed under finite unions, there exists some K PK, such that s, t P K. Fix this compact set. Now consider the net pd K,F,e,e 1 pT qq F , whose indices run over all finite F Ď B, ordered by inclusion. Since T is in the set in the middle of (4.2), working through the definitions yields @ε P Q`: @F 0 Ď B finite : DF Ď B finite, s.t. F Ě F 0 : d K,F,e,e 1 pT q ă ε, which is clearly equivalent to lim inf Now, since s, t P K, it holds that | pT psqT ptq´T pstqqe, e 1 | ď | T psqpI´π F qT ptqe, e 1 | looooooooooooooomooooooooooooooon pT psqπ F T ptq´T pstqqe, e 1 | loooooooooooooooooomoooooooooooooooooon ďd K,F,e,e 1 pT q for all finite F Ď B. Since π F ÝÑ F I weakly (see above), we have d 1 F ÝÑ F 0. Noting (4.7), taking the limit inferior of the right hand side of the above expression thus yields pT psqT ptq´T pstqqe, e 1 " 0. Since e, e 1 P B were arbitrarily chosen, and B is a basis for H, it follows that T pstq " T psqT ptq. This completes the proof.

Remark 4.3
The proof of Theorem 4.2 reveals that in fact claims (I) and (II) hold, provided the topological monoid M is at least σ-compact. And if M is furthermore 'good', then these again imply that that C s pM q is a G δ -subset in pF c w pM q, k wot q. The stronger assumption of M being locally compact is only relied upon to obtain that pF c w pM q, k wot q is itself completely metrisable (cf. the proof of Proposition 2.4), and thus via Alexandroff's lemma that G δ -subsets of this space are completely metrisable.
Finally, by Corollary 2.7 we obtain: Definition A.1 A topological monoid, M , shall be called extendible, if there exists a locally compact Hausdorff topological group, G, such that M is topologically and algebraically isomorphic to a closed subset of G.
If M is extendible to G via the above definition, then one can assume w. l. o. g. that M Ď G.
Definition A.2 Let G be a locally compact Hausdorff group. Call a subset A Ď G positive in the identity, if for all neighbourhoods, U Ď G, of the group identity, U X A has non-empty interior within G.
Example A.3 (The non-negative reals). Consider M :" R`viewed under addition. Since M Ď R is closed, we have that M is an extendible locally compact Hausdorff monoid. For any open neighbourhood, U Ď R, of the identity, there exists an ε ą 0, such that p´ε, εq Ď U and thus U X M Ě p0, εq ‰ ∅. Hence M is positive in the identity.
Example A.4 (The p-adic integers). Consider M :" Z p with p P P, viewed under addition and with the topology generated by the p-adic norm. Since M Ď Q p is clopen, it is an extendible locally compact Hausdorff monoid. Since M is clopen, it is clearly positive in the identity.
Example A.5 (Discrete cases). Let G be a discrete group, and let M Ď G contain the identity and be closed under group multiplication. Clearly, M is a locally compact Hausdorff monoid, extendible to G and positive in the identity. For example one can take the free-group F 2 with generators ta, bu, and M to be the algebraic closure of t1, a, bu under multiplication.
Example A.6 (Non-discrete, non-commutative cases). Let d P N with d ą 1 and consider the space, X, of R-valued dˆd matrices. Topologised with any matrix norm (equivalently the strong or the weak operator topologies), this space is homeomorphic to R d 2 and thus locally compact Hausdorff. Since the determinant map X Q T Þ Ñ detpT q P R is continuous, the subspace of invertible matrices tT P X | detpT q ‰ 0u is open and thus a locally compact Hausdorff topological group. The subspace, G, of upper triangular matrices with positive diagonal entries, is a closed subgroup and thus locally compact Hausdorff. Letting G 0 :" tT P G | detpT q " 1u, G`:" tT P G | detpT q ą 1u, and G´:" tT P G | detpT q ă 1u, it is easy to see that M :" G 0 YG`is a topologically closed subspace containing the identity and is closed under multiplication. Moreover M is a proper monoid, since the inverses of the elements in G`are clearly in GzM . Consider now an open neighbourhood, U Ď G, of the identity. Since inversion is continuous, U´1 is also an open neighbourhood of the identity. Since, as a locally compact Hausdorff space, G satisfies the Baire category theorem, and since G`Y G´is clearly dense (and open) in G, and thus comeagre, we clearly have pU XU´1qXpG`YG´q ‰ ∅. So either U X G`‰ ∅ or else U´1 X G´‰ ∅, from which it follows that U X G`" pU´1 X G´q´1 ‰ ∅. Hence in each case U X M contains a non-empty open subset, viz. U X G`. So M is extendible to G and positive in the identity.
Next, consider the subgroup, G h Ď G, consisting of matrices of the form T " I`T , wherẽ T is a strictly upper triangular matrix with at most non-zero entries on the top row and right hand column. That is, G h is the continuous Heisenberg group, H 2d´3 pRq, of order 2d´3. The elements of the Heisenberg group occur in the study of Kirillov's orbit method (see [15]) and have important applications in physics (see e.g. [16]). Clearly, G h is topologically closed within G and thus locally compact Hausdorff. Now consider the subspace, M h :" tT P G h | @i, j P t1, 2, . . . , du : T ij ě 0u, of matrices with only non-negative entries. This is clearly a topologically closed subspace of G h containing the identity and closed under multiplication. Moreover, for each S, T P M h ztIu we have ST " I``pS´Iq`pT´Iq`pS´IqpT´Iq˘P M h ztIu, which implies that no non-trivial element in M h has its inverse in M h , making M h a proper monoid. Consider now an open neighbourhood, U Ď G h , of the identity. Since G h is homeomorphic to R 2d´3 , there exists some ε ą 0, such that U Ě tT P G h | @pi, jq P I : T ij P p´ε, εqu, where I :" tp1, 2q, p1, 3q, . . . , p1, dq, p2, dq, . . . , pd´1, dqu. Hence U X M h Ě tT P G h | @pi, jq P I : T ij P p0, εqu -V, where V is clearly a non-empty open subset of G h , since the 2d´3 entries in the matrices can be freely and independently chosen. Thus M h is extendible to G h and positive in the identity.
Finally, we may consider the subgroup, G u :" UT 1 pdq, of upper triangular matrices over R with unit diagonal. The elements of UT 1 pdq have important applications in image analysis (see e.g. [16] and [21, §5.5.2]) and representations of the group have been studied in [23,Chapter 6]. Setting M u :" tT P G u | @i, j P t1, 2, . . . , du : T ij ě 0u, one may argue similarly to the case of the continuous Heisenberg group and obtain that G u is locally compact and that M u is a proper topological monoid which is furthermore extendible to G u and positive in the identity.
The following result allows us to generate infinitely many examples from basic ones: Proposition A.7 Let n P N and let M i be locally compact Hausdorff monoids for 1 ď i ď n. Assume for each i ă n that M i is extendible to a locally compact Hausdorff group G i , and that M i is positive in the identity of G i . Then M :" ś n i"1 M i is a locally compact Hausdorff monoid which is extendible to G :" ś n i"1 G i and positive in the identity. Proof. The extendibility of M to G is clear. Now consider an arbitrary open neighbourhood, U , of the identity in G. For each 1 ď i ď n, one can find open neighbourhoods, U i , of the identity in G i , so that U 1 :" Thus M X U has non-empty interior. Hence M is positive in the identity.
Our argumentation for the generalised continuity result builds on [8, Theorem 5.8].
Theorem A.8 Let M be a topological monoid such that M is extendible to a locally compact Hausdorff group G and such that M is positive in the identity. Then for any Banach space, E, every wot-continuous semigroup, T : M Ñ LpEq, is sot-continuous. In particular, M is 'good'.
(Note that a semigroup over a Banach space E on a topological monoid is defined analogously to Definition 2.1.) Proof. First note that the principle of uniform boundedness applied twice to the wotcontinuous function, T , ensures that T is norm-bounded on all compact subsets of M . Fix now a left-invariant Haar measure, λ, on G and set S :" tF Ď G | F a compact neighbourhood of the identityu. Consider arbitrary F P S and x P E. By the closure of M in G as well as positivity in the identity, M X F is compact and contains a non-empty open subset of G. It follows that 0 ă λpM X F q ă 8. The wot-continuity of T , the compactness (and thus measurability) of M X F , and the norm-boundedness of T on compact subsets ensure that describes a well-defined element x F P E 2 . Exactly as in [8,Theorem 5.8], one may now argue by the wot-continuity of T and compactness of M X F that in fact x F P E for each x P E and F P S. Moreover, since M is locally compact, and T is wot-continuous with T p1q " I, one readily obtains that each x P E can be weakly approximated by the net, px F q F PS , ordered by inverse inclusion. So D :" tx F | x P E, F P Su is weakly dense in E. Since the weak and strong closures of any convex subset in a Banach space coincide (cf. [1,Theorem 5.98]), it follows that the convex hull, copDq, is strongly dense in E. Now, to prove the sot-continuity of T , we need to show that is strongly continuous for all x P E. Since M is locally compact and T is norm-bounded on compact subsets of M , the set of x P E such that (A.2) is strongly continuous, is itself a strongly closed convex subset of E. So, since copDq is strongly dense in E, it suffices to prove the strong continuity of (A.2) for each x P D.
To this end, fix arbitrary x P E, F P S and t P M . We need to show that T pt 1 qx F ÝÑ T ptqx F strongly for t 1 P M as t 1 ÝÑ t.
First recall, that by basic harmonic analysis, the canonical left-shift, defined via pL t f qpsq " f pt´1sq for s, t P G and f P L 1 pGq, is an sot-continuous morphism (cf. [22, Proposition 3.5.6 (λ 1 -λ 4 )]). Now, by compactness, f :" 1 M XF P L 1 pGq and it is easy to see that }L t 1 f´L t f } 1 " λpt 1 pM X F q ∆ tpM X F qq for t 1 P M . The sot-continuity of L thus yields for t 1 P M as t 1 ÝÑ t. Fix now a compact neighbourhood, K Ď G, of t. For t 1 P M X K and ϕ P E 1 one obtains | T pt 1 qx F´T ptqx F , ϕ | " | x F , T pt 1 q˚ϕ ´ x F , T ptq˚ϕ | }T psq}¨}x}¨}ϕ}¨λpt 1 pM X F q ∆ tpM X F qq since t, t 1 P M X K.
Since K 1 :" pM X KqpM X F q is compact, and T is uniformly bounded on compact sets (see above), it holds that C :" sup sPK 1 }T psq} ă 8. The above calculation thus yields }T pt 1 qx F´T ptqx F } " supt| T psqx F´T ptqx F , ϕ | | ϕ P E 1 , }ϕ} ď 1u ď 1 λpM XF q¨C¨} x}¨λpt 1 pM X F q ∆ tpM X F qq for all t 1 P M sufficiently close to t. By (A.3), the right-hand side of (A.4) converges to 0 and hence T pt 1 qx F ÝÑ T ptqx F strongly as t 1 ÝÑ t. This completes the proof.
Remark A.9 In the proof of Theorem A.8, weak continuity only played a role in obtaining the boundedness of T on compact sets, as well as the well-definedness of the elements in D. In [12, Theorem 9.3.1 and Theorem 10.2.1-3] a proof of the classical continuity result exists under weaker conditions, viz. weak measurability, provided the semigroups are almost separably valued. It would be interesting to know whether the above approach can be adapted to these weaker assumptions.

ÒÓÛÐ Ñ ÒØº
The author is grateful to Tanja Eisner for her feedback, to Konrad Zimmermann for his helpful comments on the results in the appendix, and to the referee for their constructive feedback.