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ON THE EXTENSIONS OF INFINITE-DIMENSIONAL REPRESENTATIONS OF LIE SEMIGROUPS
ADOLF R. MIROTIN Received 22 January 2001
The necessary and sufficient conditions have been obtained for extendability of a Banach representation of a generating Lie semigroupSto a local representation of the Lie group Ggenerated bySwhen the tangent wedge ofSis a Lie semialgebra. The most convenient conditions we obtain correspond to the case of unitary representations. In this case, we give a criterion of global extendability ifGis exponential and solvable.
2000 Mathematics Subject Classification: 20M30, 22E99, 43A65.
1. Introduction. IfSis a subsemigroup of a topological groupGwith interior points andGis a left quotient group forS, it is easy to prove that every representation ofS by invertible operators on a Banach spaceᐂmay be extended, in a unique manner, to the representation ofGonᐂ(seeProposition 6.1below). It is easy to prove that every finite-dimensional representation of a generating Lie semigroupS can be extended to the local representation of the Lie groupGgenerated byS, ifGis connected and the tangent wedge ofS is a Lie semialgebra [14]. In this paper, using the infinitesimal method, we study the problem of extendability of a Banach representationsπ of a generating Lie semigroupS to the connected Lie groupG generated byS when the tangent wedge ofS is a Lie semialgebra. We show that the differentialdπ extends to the representation of the Lie algebra of Gif at least one operatorπ(s0),s0∈intS, is invertible. In this way, the necessary and sufficient conditions of a local extend- ability ofπhave been obtained in terms of the tangent objects. The most convenient conditions we obtain, correspond to the case of unitary representations. Thus, the problem of extendability of a representation ofStoGhas been reduced to the (group theoretical) problem of extendability of a local representation of Gto a global one (seeCorollary 2.2below). For exponential and solvableG, we give also necessary and sufficient conditions of global extendability of unitary representations.
The main results of this paper appeared earlier in [12]. One-dimensional and posi- tive representations ofS were studied in detail in [13].
Throughout, unless otherwise stated,Gdenotes a connected Lie group with unit e,L(G)its Lie algebra, and exp :L(G)→Gits exponential function. For a closed sub- semigroupSofGitstangent wedgeis defined by
L(S):=
X∈L(G): exp R+X
⊆S
. (1.1)
A closed subsemigroupS ⊆G is calledLie semigroup if S is the closure in G of expL(S), the semigroup, generated by
expL(S):=
expX:X∈L(S)
. (1.2)
After shrinkingGwe may assume thatSis ageneratingLie semigroup, that is,L(G) is the smallest Lie algebra containingL(S)(see [6]).
Definition1.1. A Lie subsemigroupS⊆Gisquasi-invariantif its tangent wedge L(S)is a Lie semialgebra, that is, for some Campbell-Hausdorff-neighborhoodBinL(G)
L(S)∩B
∗
L(S)∩B
⊆L(S), (1.3)
where∗denotes the Campbell-Hausdorff multiplication inL(G) X∗Y=(X+Y )+1
2[X,Y ]+···+Hn(X,Y )+···. (1.4) Note that everyinvariant (with respect to all inner automorphisms of the group G) generating Lie semigroupS is quasi-invariant [16, Proposition IV.7], [6, Theorem III.2.15], but the converse is false. As an example, take the subsemigroupS+⊂GL(2,R) consisting of all matrices with nonnegative entries [8, Example 6.3, page 182].
For a generating Lie semigroupSin a connected Lie groupGthe interior intSofS (with respect toG) is a dense ideal ofS, and intS⊆ expL(S)[16, Proposition IV.6], [6, Theorem V.1.10]. If, in addition,Sis quasi-invariant then, according to [6, Theorem II.2.13], we have
L(G)=L(S)−L(S). (1.5)
The basic reference in Lie semigroups is [6].
We will employ some preliminary results to prove our main theorems.
2. The G˙arding space. Letπbe a representation ofSon a Banach spaceᐂ, that is, a homomorphism ofSinto the multiplicative semigroup gl(ᐂ)of all bounded (linear) operators on ᐂ which is continuous with respect to the strong operator topology and satisfiesπ(e)=I, the identity operator onᐂ. For everyX∈L(S)the mapt π(exptX):R+→gl(ᐂ)is a one-parameterC0-semigroup of operators. We denote by A(X)ordπ(X)the generator of this semigroup.
Lemma 2.1. The representation π of a Lie semigroup S is uniquely determined bydπ.
Proof. Letπandπ1be two representations ofSin a Banach spaceᐂanddπ(X)= dπ1(X)for allX∈L(S). ThenC0-semigroupstπ(exptX)andtπ1(exptX)coincide (X∈L(S)). Therefore,π| expL(S) =π1| expL(S), andπ=π1by continuity.
Corollary2.2. Let a Banach representationπ of a Lie semigroupS extend to a local representationT of the Lie groupGgenerated by S. If, in turn, T extends to a representationπˆof the wholeG, thenπˆis an extension ofπ toG.
Indeed, setπ1:=πˆ|S. Thenπ1and π coincide on the setU∩S, where U is an e-neighborhood inG, and sodπ1=dπ.
LetC0∞(S)be the space of all compactly supported functions inC∞(G)concentrated in intS. Forφ∈C0∞(S)andu∈ᐂ, set
u(φ):=
Gφ(x)π(x)udlx (2.1)
(dlxdenotes the left Haar measure inG), and denote byᏰSthe linear subspace ofᐂ generated by{u(φ):φ∈C0∞(S), u∈ᐂ}.For the caseS=Gwe get the G˙arding space ᏰG(cf. [1, Chapter 11, Section 1]). We denote also byᐂ∞the set of suchu∈ᐂthat the functionsπ(s)uis inC∞(intS).
Proposition 2.3. LetS be a quasi-invariant generating Lie semigroup in a con- nected Lie groupGand letπ be a representation ofSon a Banach spaceᐂ. Then the following assertions hold:
(i) ᏰS⊆ᐂ∞.
(ii) ᏰS⊆Ᏸ(A(X)), the domain ofA(X), andᏰS isA(X)-invariant andπ-invariant (X∈L(S)).
(iii) ᏰSis dense inᐂ.
(iv) The mapXA(X)|ᏰS extends uniquely to a linear mappingAˆbetweenL(G) and the linear space of operators onᏰS.
(v) Ifπ(s0)is invertible for somes0∈intS, thenᐂ∞⊆Ᏸ(A(X)) (X∈L(S)), and the mapXA(X)|ᐂ∞extends uniquely to a linear mappingAˆbetweenL(G)and the linear space of operatorsᐂ∞→ᐂ.
Proof. (i) Following the proof of Theorem 1 in [1, Chapter 11], lety(t)=exptY (Y∈L(G), t∈R)be a one-parameter subgroup inG,φ∈C0∞(S),s0∈intS. Then (cf.
[1, formula (14)]) t−1
π s0y(t)
−π s0
u(φ)=π s0
Gt−1 φ
y−1(t)x
−φ(x)
π(x)udlx. (2.2) Setting in (2.2)t→0, we get for the derivative of the functionsπ(s)u(φ) ats0
along the vectorY
Y π s0
u(φ)=π s0
uY φ˜
, (2.3)
where the function
Y φ˜
(x):=lim
t→0
φ
y(t)−1x
−φ(x)
t (2.4)
belongs toC0∞(S). Indeed, we know from the proof of Theorem 1 in [1, Chapter 11]
that ˜Y φ∈C0∞(G). Thus, it is sufficient to prove that the support supp ˜Y φ⊆intS if K:=suppφ⊂intS. Choose U ⊂G open such that K⊂U⊂U− ⊂intS (U− is the closure ofU). Then there is the neighborhoodN ofeinG withNK⊆U and δ >0 such thaty(t)∈Nfor|t|< δ. Since the support of the functionxφ(y(t)−1x)is y(t)K⊆U for|t|< δ,the support of the function xt−1(φ(y(t)−1x)−φ(x))is contained inU, too. It follows that supp ˜Y φ⊆U−⊂intS. Now from (2.3) we conclude thatu(φ)∈ᐂ∞.
(ii) Suppose thatY ∈L(S)andt >0. Then the preceding arguments are true for alls0∈S. In particular, setting in (2.2)s0=eand t→0+we conclude thatu(φ)∈ Ᏸ(A(X))and
A(Y )u(φ)=uY φ˜
. (2.5)
Equation (2.5) shows thatᏰS isA(Y )-invariant. The invariance ofᏰS with respect to π(s) (s∈S)follows from the equality
π(s)u(φ)=
Gφ(x)π(sx)udlx=
Gφ s−1x
π(x)udlx. (2.6) (iii) Sinceeis an adherent point for intS, there is a net of compact subsetsK⊂intS, which shrinks toe. By an argument similar to that given in the proof of Theorem 1 in [1, Chapter 11],ᏰS is dense inᐂ.
(iv) LetX1,X2∈L(S)andu∈ᏰS. Then A
X1+X2 u= d
dtπ expt
X1+X2 u|t=0
= d dtπ
exptX1exptX2
|t=0
=A X1
u+A X2
u.
(2.7)
The proof of the equalityA(cX1)u=cA(X1)u (c≥0)is similar to the preceding one.
Now let, by definition, Aˆ
X1−X2 u:=A
X1 u−A
X2
u. (2.8)
This definition is correct, ˆAis linear onL(G)(see (1.5)) and ˆA|L(S)=AonᏰS. The uniqueness of such an extention follows from (1.5), too.
(v) ForX ∈L(S) and s0∈intS the functionts0exptX :R+ →intS is differ- entiable. Therefore, for each u∈ ᐂ∞ and t≥0 the function tπ(s0exptX)u= π(s0)π(exptX)uis differentiable, as well. Sinceπ(s0)is invertible, it follows that the functiontπ(exptX)uis differentiable att=0 and sou∈Ᏸ(A(X)). The last assertion follows as in (iv).
3. Analytical vectors. The vectoru∈ᐂis calledanalyticfor the representationπ if the mapsπ(s)uis analytic on intS. Letᐂωorᐂω(π)be the space of all analytic vectors forπ. It is obvious thatᐂω⊆ᐂ∞. Note thatᐂωcan be trivial even for Hilbert ᐂand isometricπ [3, Example 3.1.18].
Proposition 3.1. LetS be a quasi-invariant generating Lie semigroup in a con- nected Lie groupG and letπ(s0)be invertible for somes0∈intS. The spaceᐂω is π-invariant andA(X)-invariant forX∈L(S).
Proof. Letx∈S. Since the mappingssx: intS→intS is analytic, the same is sπ(sx)u=π(s)π(x)uforu∈ᐂω. Thereforeπ(x)u∈ᐂω, that is,ᐂωisπ(x)- invariant.
To prove the second statement, first note that ᐂω ⊆ Ᏸ(A(X)) (X ∈ L(S)) by Proposition 2.3(v). LetX∈L(S),u∈ᐂω, andv:=A(X)u. Ifs∈intS, then
π(s)v=π(s)d
dtπ(exptX)u|t=0= d
dtπ(sexptX)u|t=0. (3.1) Sincesπ(sexptX)uis analytic on intS, it follows thatsπ(s)v is analytic on intSby Vitali theorem. Indeed,
d
dtπ(sexptX)u|t=0=lim
h→0
π(sexphX)u−π(s)u
h . (3.2)
For sufficiently smallhand compactC⊂intS we have(s∈C) π(sexphX)u−π(s)u
h
≤sup
s∈C
π(s)
π(exphX)u−u h
≤const, (3.3)
because
h→0lim
π(exphX)u−u h
=A(X)u<∞. (3.4) Thus, the family of analytic functionss h−1(π(exphX)u−π(s)u) (s∈intS) is uniformly bounded fors∈Cand smallh, and all the conditions of the Vitali theorem are satisfied.
4. The main lemma. Recall that ˆA denotes the linear continuation fromL(S) to L(G) of the mapA:X(d/dt)π(exptX)|t=0 (the right side is an operator onᐂ∞ whenever at least one operatorπ(s),s∈intS, is invertible).
Lemma4.1. LetSbe a quasi-invariant generating Lie semigroup in a connected Lie groupGand letπbe a representation ofSon a Banach spaceᐂsuch that the operator π(s0)is invertible for somes0∈intS. Then Aˆis a representation of the Lie algebra L(G)by operators onᏰS orᐂω.
Proof. There exists a star-shaped neighborhoodB0⊂L(G)of 0 such thats0expr X
∈intSforX∈B0and|r| ≤2. Fix vectorsu∈ᏰS(orᐂω) andX∈B0∩L(S). Applying the Taylor formula
f (1)=f (0)+f(0)+ 1
2!f(0)+ 1 2!
1
0f(ξ)(1−ξ)2dξ (4.1) to the smooth functionf (r )=π(s0expr X)u (|r| ≤2), we have
π
s0expX u=π
s0
u+ d drπ
s0expr X
u|r=0+1 2
d2 dr2π
s0expr X u|r=0
+1 2
1 0
d3 dr3π
s0expr X
u|r=ξ(1−ξ)2dξ.
(4.2)
Since for allk∈Nandr≥0 dk drkπ
s0expr X u=π
s0
π(expr X) A(X)k
u (4.3)
andπ(s0)is invertible, it follows from (4.2) that π(expX)u=u+A(X)u+1
2
A(X)2
u+1 2
1
0π(expξX) A(X)3
u(1−ξ)2dξ. (4.4) Now fixX1,X2∈L(S)and chooseδ>0 such thattX1,tX2, andtX1∗tX2belong toB0∩B for|t|< δ, whereBis a Campbell-Hausdorff-neighborhood inL(G)and (1.3) holds.
ThentX1∗tX2∈L(S)fort∈[0,δ)and the Campbell-Hausdorff formula implies that tX1∗tX2=t
X1+X2 +1
2t2 X1,X2
+t3H3 X1,X2
+···. (4.5)
As exp(tX1∗tX2)=exptX1exptX2, we have fort∈[0,δ),u∈ᏰS
π exptX1
π exptX2
u=π exp
tX1∗tX2
u. (4.6)
If we substituteX=tX1∗tX2(t∈[0,δ))to (4.4), then π
exp
tX1∗tX2
u=u+A
tX1∗tX2 u+1
2 A
tX1∗tX22
u +1
2 1
0π expξ
tX1∗tX2
A
tX1∗tX2
3
u(1−ξ)2dξ.
(4.7)
In view of (4.5) and by the continuity of the linear mapXA(X)uˆ , A
tX1∗tX2
u=t A
X1
+A X2
u+1 2t2Aˆ
X1,X2
u+t3Aˆ H3
X1,X2
u+···
= t A
X1
+A X2
+1 2t2Aˆ
X1,X2
+t3R1
t,X1,X2
u.
(4.8)
Thus,
A
tX1∗tX2
2
u= t2
A X1
+A X2
2
+t3R2
t,X1,X2
u, A
tX1∗tX2
3
u=t3R3
t,X1,X2
u, (4.9)
where the functionstRn(t,X1,X2)u (n=1,2,3)are bounded fort→0. From (4.7) it follows, and since (4.8), (4.9), that fort∈[0,δ)
π exp
tX1∗tX2
u=u+t A
X1 +A
X2 u+1
2t2Aˆ X1,X2
+ A
X1 +A
X22 u +1
2t3 1
0π expξ
tX1∗tX2 R3
t,X1,X2
u(1−ξ)2dξ
=u+t A
X1 +A
X2 u+1
2t2Aˆ X1,X2 +
A X1
+A X2
2 u+o
t2 ,
(4.10) becauseπ(expξ(tX1∗tX2))is bounded fortsufficiently small and ξ∈[0,1]by the uniform boundedness principle.
By (4.6) andu∈ᏰS, the left-hand side in (4.10) is a smooth function oft. By the uniqueness of the Taylor polynomial, (4.10) implies
d2 dt2π
exp
tX1∗tX2
u|t=0=Aˆ X1,X2
u+
A X1
+A X2
2
u. (4.11)
On the other hand, let F(t)u=π
exptX1
π exptX2
u, t≥0, u∈ᏰS. (4.12) Then, for each∆t >0
∆F(t)u=π exptX1
π
exp∆tX1
π exptX2
π
exp∆tX2
u
−π
exptX1 π
exptX2 u
=π exptX1
π
exp∆tX1
π
exp∆tX2
v−v ,
(4.13)
wherev:=π(exptX2)u∈ᏰS (π(exptX)commutes withπ(expsX)for allX∈L(S), t,s≥0).
Therefore, d
dtF(t)u= lim
∆t→0π
exptX1 π
exp∆tX1π
exp∆tX2 v−v
∆t +π
exp∆tX1 v−v
∆t
=π
exptX1 A
X2 v+A
X1 v
=π
exptX1 A
X1 π
exptX2 u+π
exptX1 A
X2 π
exptX2 u.
(4.14) Sinceπ(exptXi)commutes withA(Xi),i=1,2,
d
dtF(t)u=A X1
F(t)u+F(t)A X2
u. (4.15)
BecauseA(X1)is close, it follows that d2
dt2F(t)u|t=0=A X1
A X1
u+A X2
u +A
X1
A X2
u+A X2
A X2
u
= A
X1
2
u+2A X1
A X2
u+ A
X2
2
u.
(4.16)
Now differentiating (4.6) twice att=0 we obtain forX1,X2∈L(S)by virtue of (4.11) A
X1
2
u+2A X1
A X2
u+
A X2
2
u=Aˆ X1,X2
u+
A X1
+A X2
2
u, (4.17) or
Aˆ X1,X2
u=Aˆ X1
,Aˆ X2
u. (4.18)
Since both sides of the last equality are bilinear, it is valid for allX1,X2∈L(G) by (1.5). This completes the proof.
Example4.2. LetS be as inLemma 4.1and Ᏺ a finite-dimensional subspace of Lp(S) (1≤p <∞)which is invariant under left translationsλ(s)f (x)=f (sx) (s,x∈ S). Thenλis a left regular representation ofS onᏲ. Sincetλ(texpX) (t≥0, X∈ L(S))is a one-parameterC0-semigroup onᏲ, the operatorλ(expX)is invertible. Thus λ(s)is invertible for alls∈ expL(S) ⊇intS anddλextends to a representation ˆA ofL(G)onᏲbyLemma 4.1. Then the local representationT ofGonᏲwithdT=Aˆ extendsλ. The representationT is global for simply connectedG.
5. Extension of unitary representations. Our first statement deals with local extensions.
Theorem5.1. Letπ be a unitary representation on a Hilbert spaceᏴof a quasi- invariant generating Lie semigroupS in a connected Lie groupG and let one of the following conditions holds:
(1) Ᏼωis dense inᏴ;
(2) for some linear independent X1,...,Xd ∈ L(S) (d = dimL(G)) the Nelson operator
∆= d j=1
A Xj
2
(5.1) is essentially selfadjoint onᏰS.
Thenπextends to a local unitary representation ofGonᏴ. Moreover, ifSalgebraically generatesG, for somee-neighborhoodU⊂Gthe extension ofπto a local representation ofUis unique.
Proof. (1) First, prove that every vectoru∈Ᏼωis analytic for the operatorA(X) ifX∈L(S)satisfies expX∈intS. Indeed,Ᏼω⊆Ᏸ(A(X))and since the functiont exptX :R→G is analytic, the function tπ(exptX)u has the same property in the same-neighborhood of the pointt=1. Therefore,uis an analytic vector of the one-parameter unitary groupT1(t)which agrees withπ(exptX)fort≥0 and has the generatorA(X)(in the sense of semigroup theory). Standard Cauchy estimates for the disc{|t−1|< }give
A(X)n
u=T1(1) A(X)n
u= dn
dtnT1(t)u|t=1
≤n!M
n (5.2)
for someM >0, and souis an analytic vector forA(X).
Being the generator ofT1(t), operatorA(X)is antiselfadjoint and so, it is antisym- metric onᏴω (X∈L(S)). We claim that expXj∈intS for some linear independent Xj∈L(S),j=1,...,d. SinceL(S)is generating inL(G), zero is in the closure of intL(S) (with respect toL(G)). LetB1be a neighborhood of zero inL(G)such that exp|B1is a homomorphism on its image. Then exp(B1∩intL(S))is a nonvoid open subset ofS, and we may choose linear independentXj∈B1∩intL(S),j=1,...,d.
Now if (1) holds, then for the representation ˆA of the Lie algebra L(G)by oper- ators on Ᏼω (see Lemma 4.1) all the conditions of the FS3-criterion (cf. [1, Chap- ter 11, Section 6, Theorem 5]), except the simply connectedness ofG, are satisfied.
But the proof of theFS3-criterion in [1] shows that for connectedG we get a uni- tary representationT of a local groupW, whereW is a neighborhood of unit inG, and aT-invariant dense linear subspaceᏴ∞⊆Ᏼsuch that for allX∈L(G)we have Ᏼω⊆Ᏼ∞⊆Ᏸ(T (X)) anddT (X)|Ᏼ∞=A(X)ˆ |Ᏼ∞ (the bar denotes the closure of an operator; see Lemmas 3, 4, and formula (11) therein). Since the one-parameter C0-semigroup T1(t)= π(exptX), X ∈L(S), leavesᏴω invariant, Ᏼω is an essen- tial domain for A(X) by [3, Corollary 3.1.7]. Hence, ˆA(X)=A(X)|Ᏼω =A(X) for X∈L(S), and the definition ofᏴ∞in [1, Chapter 11, Section 6, Lemma 3] entails that Ᏼ∞⊆Ᏸ(A(X)). The first assertion follows now from the next proposition.
Proposition 5.2. Let π be a representation on a Banach space ᐂ of a quasi- invariant generating Lie semigroupSin a connected Lie groupG.
(i) LetT be a local representation ofGonᐂand letᐂ0⊆ᐂbe aT-invariant dense linear subspace such thatᐂ1⊆ᐂ0⊆Ᏸ(T (X))∩Ᏸ(A(X))for a someπ-invariant dense linear subspaceᐂ1⊆ᐂanddT (X)|ᐂ0=A(X)|ᐂ0for allX∈L(S). ThenT|U∩S= π|U∩S for somee-neighborhoodU⊆G, andπ(s)is invertible for alls∈S.
(ii) IfSalgebraically generatesG, then for somee-neighborhoodU⊆Gthe extension ofπto a local representation ofU(if such exists) is unique.
Proof. (i) LetTbe a representation of a local groupW, whereWis a neighborhood ofeinG. FixX∈L(S)and pickδ >0 such that exptX∈Wfort∈Rwith|t|< δ. The
maptT (exptX)is a representation of the additive local group(−δ,δ)and forδ sufficiently small it extends to the one-parameter groupTXonᐂwith the generator dT (X)(cf. [2, Chapter 3, Section 6, Lemma 1]), andᐂ0isTX-invariant. Consider also the C0-semigroupT1(t)=π(exptX), t∈R+, with generatorA(X). Note that generators ofTX and T1 coincide onᐂ0and that ᐂ0 and ᐂ1 are essential domains fordT (X) and A(X), respectively, by [3, Corollary 3.1.7]. Sinceᐂ1⊆ᐂ0⊆Ᏸ(A(X)), ᐂ0 is an essential domain forA(X), too. Therefore,dT (X)=A(X)and henceTX(t)=T1(t)for allt∈R+. In particular,T (exptX)=π(exptX)for 0≤t < δ. LetB0be a star-shaped neighborhood of zero in L(G) such that expB0⊂ W. Then the preceding equality implies thatT (expX)=π(expX)forX∈B0∩L(S). But exp(B0∩L(S))⊃U∩S for somee-neighborhoodU⊂W, becauseL(S)is a semialgebra [15, Theorem III.9], and henceT|U∩S=π|U∩S.
The equalityTX(t)=T1(t), t≥0, shows also thatπ(expX)is invertible forX∈ L(S). It follows that π(s0)is invertible fors0∈ expL(S) ⊇intS, and so isπ(s)= π(ss0)π(s0)−1for alls∈S (s0∈intS).
(ii) As before, for an arbitrary smalle-neighborhoodUinGwe haveU∩S=exp(B∩
L(S))(B⊂L(G)is a neighborhood of zero) by [15, Theorem III.9]. Then the setU∩ σX algebraically generates the full one-parameter semigroup σX corresponding to X∈L(S). Therefore, the setU∩Salgebraically generates the semigroupexpL(S) ⊇ intS. Being an ideal inS, intS algebraically generatesG, too. ThusU∩Salgebraically generatesGand the local representationTis completely determined by its restriction T|U∩S. This completes the proof ofProposition 5.2.
(2) Let condition (2) ofTheorem 5.1be satisfied. As it was mentioned above,A(X) is an antiselfadjoint operator andᏰS is an essential domain forA(X), X∈L(S), by [3, Corollary 3.1.7]. Therefore, operatoriA(X)|ᏰS has a selfadjoint closure. Thus for the representation ˆAof the Lie algebra L(G)by operators on ᏰS all the condi- tions of the Nelson criterion (cf. [1, Chapter 11, Section 5, Theorem 2]), excepting the simply connectedness ofG, are satisfied. The proof of this criterion (see especially Lemma 1 therein) shows that for connectedGwe get a local unitary representation T of some neighborhoodNof unit in Gsuch that forX∈L(G)with expX∈Nwe haveT (expX)=eA(X)ˆ :=TX(1), whereTXis a one-parameter unitary group onᏴwith the generator ˆA(X) (in the sense of semigroup theory). Choose, as in the proof of Proposition 5.2, a neighborhood of zeroB0⊂L(G)such that expB0⊂Nand exp(B0∩ L(S))containsU∩Sfor somee-neighborhoodU⊂G. Since ˆA(X)=A(X)forX∈L(S), TX(t)=T1(t):= π(exptX) for X ∈L(S), t ≥0. Thus for X ∈B0∩L(S) we have T (expX)=π(expX)andT|U∩S=π|U∩S. In other words,Tis an extension ofπ toU. The uniqueness of such an extention follows fromProposition 5.2immediately.
We call a Lie groupGexponentialif expL(G)=G.
Theorem5.3. LetGbe a connected exponential and solvable Lie group,S a quasi- invariant generating Lie subsemigroup of G. A unitary representation π of S on a Hilbert spaceᏴextends to the wholeGif and only if condition (2) ofTheorem 5.1and the following condition (3) hold:
(3)ifX1,2,Y1,2∈L(S)satisfyexp(X1−X2)=exp(Y1−Y2), then
n→∞lim π exp1
nX1 π exp1 nX2
−1n
= lim
n→∞ π exp1
nY1 π exp1 nY2
−1n
(5.3) (the strong operator convergence). Moreover, such an extension is unique.
Proof. The existence of a basisX1,...,Xd∈L(S)has been proved above (Theorem 5.1), and the necessity of (2) follows from the Nelson-Stinespring theorem (cf. [1, Chap- ter 11, Section 2, Theorem 2]).
Next we have
exp X1−X2
=lim
n→∞ exp1
nX1 exp1 nX2
−1n
, (5.4)
and the same formula for exp(Y1−Y2). The necessity of (3) follows.
Now let (2) and (3) hold. ByTheorem 5.1, there is a local unitary representationT which extendsπto somee-neighborhoodN⊆G. ForX∈L(G)denote byTXthe uni- tary one-parameter group inᏴsuch thatTX(t)=T (exptX)fort∈Rwith exptX∈N. The proof ofTheorem 5.1yields that the antiselfadjoint operator ˆA(X)is the gener- ator ofTX. We extendTto the fullGsetting ˆπ(expX):=TX(1).To prove the correct- ness of this definition one should show that expX=expY impliesTX(1)=TY(1)for X,Y∈L(G). LetX=X1−X2whereX1,X2∈L(S). We claim thatA(X1)−A(X2)=A(X)ˆ and that the Trotter multiplicative formula can be applied to the unitary groupTX
(cf. [17, Theorem VIII.31]). Indeed, by definition ˆA(X)=(A(X1)−A(X2))|ᏰS. Since the operatorA(X1)−A(X2)⊇A(X)ˆ is antisymmetric, it has an antisymmetric closure A(X1)−A(X2)⊇A(X)ˆ , and the hypermaximality property of a selfadjoint operator im- pliesA(X1)−A(X2)=A(X)ˆ , an antiselfadjoint operator. Now by the Trotter formula mentioned above
TX(1)=eA(X1)−A(X2)=lim
n→∞
e(1/n)A(X1)e−(1/n)A(X2)n
=lim
n→∞ π exp1
nX1 π exp1 nX2
−1n
,
(5.5)
so the mapping ˆπ is well defined by virtue of (3).
LetG1be the simply connected Lie group withL(G1)=L(G). Then there existe- neighborhoodsU⊆NandU1⊆G1and a local Lie group morphismϕ:U→U1such thatϕand its reciprocalϕ−1both are analytic. Writeπ1(x):=T (ϕ−1(x)),x∈U1, and extendπ1to a (unique) unitary representationπ1ofG1onᏴ. Then for eachX∈L(G) and sufficiently largen∈Nwe have
π(ˆ expX)=TX(1)= TX
1 n
n
= T exp1 nX
n
= π1 ϕ exp1 nX
n
=π1
ϕ exp1 nX
n .
(5.6)
Hence, the mappingf (X):=π(ˆ expX)v is analytic onL(G)for every v inᏴω(π1), which is a π1-invariant dense linear subspace of Ᏼ, and thus the map (X,Y )
π(ˆ expX)π(ˆ expY )v is separately analytic onL(G)×L(G)(formula (5.6) shows that Ᏼω(π1)is ˆπ(expX)-invariant).
SinceGis exponential and solvable, for eachx0∈Gthere is anX0∈L(G)such that x0=expX0and exp is regular atX0[18, Theorem IV.2.44] (see also [4]). So the mapping exp−1=log is analytic on the neighborhoodU:=expBofx0for some neighborhoodB ofX0inL(G). Therefore the functionxπ(x)vˆ =f (logx) (x∈U)is analytic forv∈ Ᏼω(π1)too, and hence the map(X,Y )π(ˆ expXexpY )vis analytic onL(G)×L(G).
BecauseT is a local representation, the equality
π(ˆ expXexpY )v=π(ˆ expX)π(ˆ expY )v (5.7) holds for allX,Y∈L(G)with sufficiently small norms, andv∈Ᏼω(π1). The separate analyticity of both sides of (5.7) implies that this formula is valid for allX,Y∈L(G), v∈Ᏼω(π1). By continuity, (5.7) holds for allv∈Ᏼso ˆπis a unitary representation of G(the continuity of ˆπ onGis a consequence of its continuity onN). The application ofCorollary 2.2shows that ˆπ is an extension ofπtoG.
Finally, letπbe a unitary extension ofπtoG. Note thatᏴω(π)⊆Ᏼω(π). Since for allX1,X2∈L(S),u∈Ᏼω(π),
dπ X1−X2
u=dπ X1
u−dπ X2
u=dπ X1
u−dπ X2
u (5.8)
and (1.5) hold,πis completely determined byπ, which completes the proof.
6. The case of Banach representations. We begin with a very simple case when G is the left quotient group forS [10, Chapter 1, Section 1.10], that is, S⊂G and G=S−1S.
Proposition6.1. LetG be a topological group and a group of left quotients for S,intS= ∅. The representationπ ofS on a Banach spaceᐂis extendable to a repre- sentation ofGonᐂ if and only if all operatorsπ(s),s∈S, are invertible. Moreover, such an extension is unique.
Proof. To prove the sufficiency, for everyx∈G,x=a−1b (a,b∈S), set ˆπ(x):= π(a)−1π(b). Let, in addition,x=c−1d (c,d∈S). Thenac−1=p−1qfor somep,q∈ S. Sincepa=qc,π(p)π(a)=π(q)π(c). On the other hand,a−1b=c−1d implies pb=qd. Therefore,π(p)π(b)=π(q)π(d)or
π(p)π(a)πˆ a−1b
=π(q)π(c)πˆ c−1d
. (6.1)
Thus, we proved that ˆπ(a−1b)=π(cˆ −1d)and the definition of ˆπis consistent.
Now letx=a−1bandy=c−1d (a,b,c,d∈S)are arbitrary elements ofG, and let bc−1=r−1s, wherer ,s∈S. Thenxy=(r a)−1sdand we have
π(xy)ˆ =
π(r a)−1
π(sd)=π(a)−1π(r )−1π(s)π(d). (6.2) Sincer b=sc,π(r )−1π(s)=π(b)π(c)−1. Therefore
π(xy)ˆ =π(a)−1π(b)π(c)−1π(d)=π(x)ˆ π(y).ˆ (6.3) Finally, ˆπis continuous onG, because it is continuous on intS. The uniqueness of πˆis obvious.
Corollary6.2. LetS be a generating Lie subsemigroup in a connected Lie group Gand let one of the following conditions be satisfied:
(1)Gis nilpotent.
(2)L(G)is a compact Lie algebra.
(3)Sis invariant inG.
Then every representation ofSby invertible operators on a Banach spaceᐂextends uniquely to the representation ofGonᐂ.
Indeed, by [7, Theorem 3.46],Gis a left quotient group forS.
The general case is more complicated. The following theorem describes the tangent map of a Banach representation ofSwhich extends to a local representation ofG. We assume below thatᏰ(A(X))ˆ =ᏰS.
Theorem 6.3. Let S be a quasi-invariant generating Lie subsemigroup in a con- nected Lie groupG. The representationπ ofS on a Banach spaceᐂis extendable to a local representation ofGonᐂwhich leavesᏰS invariant if and only if the following conditions hold:
(a) the operatorsA(X)ˆ are closable for allX∈L(G);
(b) there exist constants C and >0such that for allX∈L(G)with the norms
|X|<1and|Reλ|> the resolventsR(λ,A(X))ˆ are defined and Rn
λ,A(X)ˆ ≤ C
|Reλ|−n, n∈N; (6.4)
(c) ᏰSis invariant with respect toR(λ,A(X))ˆ ,X∈L(G).
In this case, all operatorsπ(s),s∈S, are invertible. Moreover, ifSalgebraically gen- eratesG, for somee-neighborhoodUinGthe extension ofπ to a local representation ofUis unique.
Proof. LetT be a representation of a local Lie groupU⊂G(Uis a neighborhood of unit) which extendsπ and leavesᏰS invariant. The fact thatT|U∩S=π|U∩S entails thatdT (X)u=A(X)u (=ˆ A(X)u)for allX∈L(S),u∈ᏰS. In view of formula (1.5), the last equality is valid for allX∈L(G)by linearity, and conditions (a), (b), (c) follow from the Krein-Shihvatov theorem [11] in the Kirillov formulation [9, Section 10.5, Theorem 2], forᐂ0=ᏰS.
Now letπ satisfies (a), (b), and (c). We claim thatπ(s)is invertible for eachs∈S. Indeed, sinceᏰSis an essential domain forA(X),X∈L(S), condition (b) implies that A(X)is a generator of aC0-group by Gelfand theorem [5]. Thus operatorsπ(exptX), t≥0, are invertible and the invertibility ofπ(s)follows as inProposition 5.2. Accord- ing toLemma 4.1, ˆAis a representation ofL(G) by operators onᏰS. Again, by the Krein-Shihvatov theorem in the Kirillov formulation forᐂ0=ᏰS,T0=Aˆ, there exists a representationT of some local Lie groupW withL(W )=L(G) such thatdT (X)| ᏰS=A(X)ˆ ,X∈L(G), andᏰS isT-invariant. After shrinkingW we may assume that Wis ane-neighborhood inGand then applyProposition 5.2withᐂ0=ᐂ1=ᏰS.
Corollary6.4. LetS be a quasi-invariant generating Lie semigroup in a simply connected Lie groupG. The representationπ ofS on a Banach spaceᐂextends to a representation of the fullGonᐂwhich leavesᏰSinvariant if and only if conditions (a), (b), and (c) ofTheorem 6.3hold. Moreover, such an extension is unique.
Proof. To demonstrate the sufficiency, note that for simply connectedGthe local representationT constructed inTheorem 6.3extends uniquely to the representation πˆof the wholeG. Since every symmetrice-neighborhood generatesGas a semigroup, the extension ˆπ leavesᏰSinvariant, as well. Finally, ˆπ|S=πbyLemma 2.1.
The uniqueness of the extension can be proved as inTheorem 5.3. This completes the proof.
Remark6.5. The results in Sections4, 5, and6are true with anyA(X)- andπ- invariant dense linear subspaceᏰ⊆ᐂ∞∩Ᏸ(A(X)),X∈L(S),in place ofᏰS.
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Adolf R. Mirotin: Department of Mathematics, Gomel State University, Sovietskaya, 104,246699Gomel, Belarus
E-mail address:[email protected]
