ON THE EXTENSIONS OF INFINITE-DIMENSIONAL REPRESENTATIONS OF LIE SEMIGROUPS

The necessary and sufficient conditions have been obtained for extendability of a Banach representation of a generating Lie semigroup S to a local representation of the Lie group G generated by S when the tangent wedge of S is 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 if G is exponential and solvable.


Introduction.
If S is a subsemigroup of a topological group G with interior points and G is a left quotient group for S, it is easy to prove that every representation of S by invertible operators on a Banach space ᐂ may be extended, in a unique manner, to the representation of G on ᐂ (see Proposition 6.1 below). It is easy to prove that every finite-dimensional representation of a generating Lie semigroup S can be extended to the local representation of the Lie group G generated by S, if G is connected and the tangent wedge of S is a Lie semialgebra [14]. In this paper, using the infinitesimal method, we study the problem of extendability of a Banach representations π of a generating Lie semigroup S to the connected Lie group G generated by S when the tangent wedge of S is a Lie semialgebra. We show that the differential dπ extends to the representation of the Lie algebra of G if at least one operator π(s 0 ), s 0 ∈ int S, is invertible. In this way, the necessary and sufficient conditions of a local extendability of π have 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 of S to G has been reduced to the (group theoretical) problem of extendability of a local representation of G to a global one (see Corollary 2.2 below). For exponential and solvable G, 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 positive representations of S were studied in detail in [13].
Throughout, unless otherwise stated, G denotes a connected Lie group with unit e, L(G) its Lie algebra, and exp : L(G) → G its exponential function. For a closed subsemigroup S of G its tangent wedge is defined by L(S) := X ∈ L(G) : exp R + X ⊆ S . (1.1) After shrinking G we may assume that S is a generating Lie semigroup, that is, L(G) is the smallest Lie algebra containing L(S) (see [6]). where * denotes the Campbell-Hausdorff multiplication in L(G) The basic reference in Lie semigroups is [6]. We will employ some preliminary results to prove our main theorems.

The Gȧrding space.
Let π be a representation of S on a Banach space ᐂ, that is, a homomorphism of S into the multiplicative semigroup gl(ᐂ) of all bounded (linear) operators on ᐂ which is continuous with respect to the strong operator topology and satisfies π(e) = I, the identity operator on ᐂ. For every X ∈ L(S) the map t π(exp tX) : R + → gl(ᐂ) is a one-parameter C 0 -semigroup of operators. We denote by A(X) or dπ (X) the generator of this semigroup.
Lemma 2.1. The representation π of a Lie semigroup S is uniquely determined by dπ .
Corollary 2.2. Let a Banach representation π of a Lie semigroup S extend to a local representation T of the Lie group G generated by S. If, in turn, T extends to a representationπ of the whole G, thenπ is an extension of π to G. Indeed, set π 1 :=π | S. Then π 1 and π coincide on the set U ∩ S, where U is an e-neighborhood in G, and so dπ 1 = dπ .
Let C ∞ 0 (S) be the space of all compactly supported functions in C ∞ (G) concentrated in int S. For φ ∈ C ∞ 0 (S) and u ∈ ᐂ, set (d l x denotes the left Haar measure in G), and denote by Ᏸ S the linear subspace of ᐂ generated by {u(φ) : φ ∈ C ∞ 0 (S), u ∈ ᐂ}. For the case S = G we get the Gȧrding space Ᏸ G (cf. [1, Chapter 11, Section 1]). We denote also by ᐂ ∞ the set of such u ∈ ᐂ that the function s π(s)u is in C ∞ (int S). Proposition 2.3. Let S be a quasi-invariant generating Lie semigroup in a connected Lie group G and let π be a representation of S on a Banach space ᐂ. Then the following assertions hold: The map X A(X) | Ᏸ S extends uniquely to a linear mappingÂ between L(G) and the linear space of operators on Ᏸ S .
Proof. (i) Following the proof of Theorem 1 in [1, Setting in (2.2) t → 0, we get for the derivative of the function s π (s)u(φ) at s 0 along the vector Y Y π s 0 u(φ) = π s 0 u Ỹ φ , (2.3) where the function belongs to C ∞ 0 (S). Indeed, we know from the proof of Theorem 1 in [1, Then there is the neighborhood N of e in G with NK ⊆ U and δ > 0 such that y(t) ∈ N for |t| < δ. Since the support of the function (ii) Suppose that Y ∈ L(S) and t > 0. Then the preceding arguments are true for all s 0 ∈ S. In particular, setting in (2.2) s 0 = e and t → 0+ we conclude that u(φ) ∈ Ᏸ(A(X)) and (2.5) The invariance of Ᏸ S with respect to π(s) (s ∈ S) follows from the equality Since e is an adherent point for int S, there is a net of compact subsets K ⊂ int S, which shrinks to e. By an argument similar to that given in the proof of Theorem 1 in [1,Chapter 11], Ᏸ S is dense in ᐂ.
(iv) Let X 1 ,X 2 ∈ L(S) and u ∈ Ᏸ S . Then (2.7) The proof of the equality A(cX 1 )u = cA(X 1 )u (c ≥ 0) is similar to the preceding one. Now let, by definition,Â The uniqueness of such an extention follows from (1.5), too.
(v) For X ∈ L(S) and s 0 ∈ int S the function t s 0 exp tX : R + → int S is differentiable. Therefore, for each u ∈ ᐂ ∞ and t ≥ 0 the function t π(s 0 exp tX)u = π(s 0 )π (exp tX)u is differentiable, as well. Since π(s 0 ) is invertible, it follows that the function t π(exp tX)u is differentiable at t = 0 and so u ∈ Ᏸ(A(X)). The last assertion follows as in (iv).

Analytical vectors.
The vector u ∈ ᐂ is called analytic for the representation π if the map s π(s)u is analytic on int S. Let ᐂ ω or ᐂ ω (π ) be the space of all analytic vectors for π . It is obvious that ᐂ ω ⊆ ᐂ ∞ . Note that ᐂ ω can be trivial even for Hilbert ᐂ and isometric π [3, Example 3.1.18].
Proposition 3.1. Let S be a quasi-invariant generating Lie semigroup in a connected Lie group G and let π(s 0 ) be invertible for some To prove the second statement, first note that Since s π(s exp tX)u is analytic on int S, it follows that s π(s)v is analytic on int S by Vitali theorem. Indeed, Thus, the family of analytic functions s h −1 (π (exp hX)u − π(s)u) (s ∈ int S) is uniformly bounded for s ∈ C and small h, and all the conditions of the Vitali theorem are satisfied.

The main lemma.
Recall thatÂ denotes the linear continuation from L(S) to L(G) of the map A : X (d/dt)π (exp tX)| t=0 (the right side is an operator on ᐂ ∞ whenever at least one operator π(s), s ∈ int S, is invertible).

Lemma 4.1. Let S be a quasi-invariant generating Lie semigroup in a connected Lie group G and let π be a representation of S on a Banach space
Applying the Taylor formula to the smooth function f (r ) = π(s 0 exp r X)u (|r | ≤ 2), we have Since for all k ∈ N and r ≥ 0 and π(s 0 ) is invertible, it follows from (4.2) that Now fix X 1 ,X 2 ∈ L(S) and choose δ>0 such that tX 1 , tX 2 , and tX 1 * tX 2 belong to B 0 ∩B for |t| < δ, where B is a Campbell-Hausdorff-neighborhood in L(G) and (1.3) holds.
Then π extends to a local unitary representation of G on Ᏼ. Moreover, if S algebraically generates G, for some e-neighborhood U ⊂ G the extension of π to a local representation of U is unique. Proof.
(1) First, prove that every vector u ∈ Ᏼ ω is analytic for the operator A(X) if X ∈ L(S) satisfies exp X ∈ int S. Indeed, Ᏼ ω ⊆ Ᏸ(A(X)) and since the function t exp tX : R → G is analytic, the function t π(exp tX)u has the same property in the same -neighborhood of the point t = 1. Therefore, u is an analytic vector of the one-parameter unitary group T 1 (t) which agrees with π(exp tX) for t ≥ 0 and has the generator A(X) (in the sense of semigroup theory). Standard Cauchy estimates for the disc {|t − 1| < } give for some M > 0, and so u is an analytic vector for A(X).
Being the generator of T 1 (t), operator A(X) is antiselfadjoint and so, it is antisymmetric on Ᏼ ω (X ∈ L(S)). We claim that exp X j ∈ int S for some linear independent But the proof of the FS 3 -criterion in [1] shows that for connected G we get a unitary representation T of a local group W , where W is a neighborhood of unit in G, and a T -invariant dense linear subspace Ᏼ ∞ ⊆ Ᏼ such that for all X ∈ L(G) we have Ᏼ ω ⊆ Ᏼ ∞ ⊆ Ᏸ(T (X)) and dT (X) | Ᏼ ∞ =Â(X) | Ᏼ ∞ (the bar denotes the closure of an operator; see Lemmas 3, 4, and formula (11) therein). Since the one-parameter C 0 -semigroup T 1 (t) = π(exp tX), X ∈ L(S), leaves Ᏼ ω invariant, Ᏼ ω is an essential domain for A(X) by [3, Corollary 3.1.7]. Hence,Â(X) = A(X) | Ᏼ ω = A(X) for X ∈ L(S), and the definition of Ᏼ ∞ 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 quasiinvariant generating Lie semigroup S in a connected Lie group G.
(i) Let T be a local representation of G on ᐂ and let ᐂ 0 ⊆ ᐂ be a T -invariant dense linear subspace such that ᐂ 1 ⊆ ᐂ 0 ⊆ Ᏸ(T (X))∩Ᏸ(A(X)) for a some π -invariant dense linear subspace ᐂ 1 ⊆ ᐂ and dT (X) | ᐂ 0 = A(X) | ᐂ 0 for all X ∈ L(S). Then T | U ∩ S = π | U ∩ S for some e-neighborhood U ⊆ G, and π(s) is invertible for all s ∈ S.
(ii) If S algebraically generates G, then for some e-neighborhood U ⊆ G the extension of π to a local representation of U (if such exists) is unique.
Proof. (i) Let T be a representation of a local group W , where W is a neighborhood of e in G. Fix X ∈ L(S) and pick δ > 0 such that exp tX ∈ W for t ∈ R with |t| < δ. The map t T (exp tX) is a representation of the additive local group (−δ, δ) and for δ sufficiently small it extends to the one-parameter group T X on ᐂ with the generator dT (X) (cf. [2, Chapter 3, Section 6, Lemma 1]), and ᐂ 0 is T X -invariant. Consider also the C 0 -semigroup T 1 (t) = π(exp tX), t ∈ R + , with generator A(X). Note that generators of T X and T 1 coincide on ᐂ 0 and that ᐂ 0 and ᐂ 1 are essential domains for dT (X) and A(X), respectively, by [3, Corollary 3.1.7]. Since ᐂ 1 ⊆ ᐂ 0 ⊆ Ᏸ(A(X)), ᐂ 0 is an essential domain for A(X), too. Therefore, dT (X) = A(X) and hence T X (t) = T 1 (t) for all t ∈ R + . In particular, T (exp tX) = π(exp tX) for 0 ≤ t < δ. Let B 0 be a star-shaped neighborhood of zero in L(G) such that exp B 0 ⊂ W . Then the preceding equality implies that T (exp X) = π(exp X) for X ∈ B 0 ∩ L(S). But exp(B 0 ∩ L(S)) ⊃ U ∩ S for some e-neighborhood U ⊂ W , because L(S) is a semialgebra [15,Theorem III.9], and The equality T X (t) = T 1 (t), t ≥ 0, shows also that π(exp X) is invertible for X ∈ L(S). It follows that π(s 0 ) is invertible for s 0 ∈ exp L(S) ⊇ int S, and so is π(s) = π(ss 0 )π (s 0 ) −1 for all s ∈ S (s 0 ∈ int S).
(ii) As before, for an arbitrary small e-neighborhood U in G we have U ∩S = exp(B ∩ L(S)) (B ⊂ L(G) is a neighborhood of zero) by [15,Theorem III.9]. Then the set U ∩ σ X algebraically generates the full one-parameter semigroup σ X corresponding to X ∈ L(S). Therefore, the set U ∩ S algebraically generates the semigroup exp L(S) ⊇ int S. Being an ideal in S, intS algebraically generates G, too. Thus U ∩ S algebraically generates G and the local representation T is completely determined by its restriction T | U ∩ S. This completes the proof of Proposition 5.2.
(2) Let condition (2) of Theorem 5.1 be satisfied. As it was mentioned above, A(X) is an antiselfadjoint operator and Ᏸ S is an essential domain for A(X), X ∈ L(S), by [3, Corollary 3.1.7]. Therefore, operator iA(X) | Ᏸ S has a selfadjoint closure. Thus for the representationÂ of the Lie algebra L(G) by operators on Ᏸ S all the conditions of the Nelson criterion (cf. [1, Chapter 11, Section 5, Theorem 2]), excepting the simply connectedness of G, are satisfied. The proof of this criterion (see especially Lemma 1 therein) shows that for connected G we get a local unitary representation T of some neighborhood N of unit in G such that for X ∈ L(G) with exp X ∈ N we have T (exp X) = eÂ (X) := T X (1), where T X is a one-parameter unitary group on Ᏼ with the generatorÂ(X) (in the sense of semigroup theory). Choose, as in the proof of Proposition 5.2, a neighborhood of zero B 0 ⊂ L(G) such that exp B 0 ⊂ N and exp(B 0 ∩ L(S)) contains U ∩S for some e-neighborhood U ⊂ G. SinceÂ(X) = A(X) for X ∈ L(S), T X (t) = T 1 (t) := π(exp tX) for X ∈ L(S), t ≥ 0. Thus for X ∈ B 0 ∩ L(S) we have T (exp X) = π(exp X) and T | U ∩ S = π | U ∩ S. In other words, T is an extension of π to U . The uniqueness of such an extention follows from Proposition 5.2 immediately.
We call a Lie group G exponential if exp L(G) = G.
(the strong operator convergence). Moreover, such an extension is unique.
Next we have (5.4) and the same formula for exp(Y 1 − Y 2 ). The necessity of (3) follows. Now let (2) and (3) hold. By Theorem 5.1, there is a local unitary representation T which extends π to some e-neighborhood N ⊆ G. For X ∈ L(G) denote by T X the unitary one-parameter group in Ᏼ such that T X (t) = T (exp tX) for t ∈ R with exp tX ∈ N. The proof of Theorem 5.1 yields that the antiselfadjoint operatorÂ(X) is the generator of T X . We extend T to the full G settingπ(exp X) := T X (1). To prove the correctness of this definition one should show that exp and that the Trotter multiplicative formula can be applied to the unitary group T X (cf. [17,Theorem VIII.31]). Indeed, by definitionÂ(X) = (A(X 1 ) − A(X 2 )) | Ᏸ S . Since the operator A(X 1 ) − A(X 2 ) ⊇Â(X) is antisymmetric, it has an antisymmetric closure A(X 1 ) − A(X 2 ) ⊇Â(X), and the hypermaximality property of a selfadjoint operator implies A(X 1 ) − A(X 2 ) =Â(X), an antiselfadjoint operator. Now by the Trotter formula mentioned above so the mappingπ is well defined by virtue of (3). Let G 1 be the simply connected Lie group with L(G 1 ) = L(G). Then there exist eneighborhoods U ⊆ N and U 1 ⊆ G 1 and a local Lie group morphism ϕ : U → U 1 such that ϕ and its reciprocal ϕ −1 both are analytic. Write π 1 (x) := T (ϕ −1 (x)), x ∈ U 1 , and extend π 1 to a (unique) unitary representation π 1 of G 1 on Ᏼ. Then for each X ∈ L(G) and sufficiently large n ∈ N we havê π(exp X) = T X (1) = T X 1 n n = T exp 1 n X n = π 1 ϕ exp 1 n X n = π 1 ϕ exp 1 n X n . (5.6) Hence, the mapping f (X) :=π(exp X)v is analytic on L(G) for every v in Ᏼ ω (π 1 ), which is a π 1 -invariant dense linear subspace of Ᏼ, and thus the map (X, Y ) π(exp X)π(exp Y )v is separately analytic on L(G) × L(G) (formula (5.6) shows that Ᏼ ω (π 1 ) isπ(exp X)-invariant).
Since G is exponential and solvable, for each x 0 ∈ G there is an X 0 ∈ L(G) such that x 0 = exp X 0 and exp is regular at X 0 [18,Theorem IV.2.44] (see also [4]). So the mapping exp −1 = log is analytic on the neighborhood U := exp B of x 0 for some neighborhood B of X 0 in L(G). Therefore the function Because T is a local representation, the equalitŷ holds for all X, Y ∈ L(G) with sufficiently small norms, and v ∈ Ᏼ ω (π 1 ). The separate analyticity of both sides of (5.7) implies that this formula is valid for all X, Y ∈ L(G), v ∈ Ᏼ ω (π 1 ). By continuity, (5.7) holds for all v ∈ Ᏼ soπ is a unitary representation of G (the continuity ofπ on G is a consequence of its continuity on N). The application of Corollary 2.2 shows thatπ is an extension of π to G. Finally, let π be a unitary extension of π to G. Note that Ᏼ ω (π ) ⊆ Ᏼ ω (π ). Since for all X 1 ,X 2 ∈ L(S), u ∈ Ᏼ ω (π ), and (1.5) hold, π is completely determined by π , which completes the proof.

The case of Banach representations.
We begin with a very simple case when G is the left quotient group for S [10, Chapter 1, Section 1.10], that is, S ⊂ G and G = S −1 S. Proposition 6.1. Let G be a topological group and a group of left quotients for S, int S = ∅. The representation π of S on a Banach space ᐂ is extendable to a representation of G on ᐂ if and only if all operators π(s), s ∈ S, are invertible. Moreover, such an extension is unique.
Finally,π is continuous on G, because it is continuous on int S. The uniqueness of π is obvious. Corollary 6.2. Let S be a generating Lie subsemigroup in a connected Lie group G and let one of the following conditions be satisfied: (1) G is nilpotent.
Then every representation of S by invertible operators on a Banach space ᐂ extends uniquely to the representation of G on ᐂ.
Indeed, by [7,Theorem 3.46], G is a left quotient group for S. The general case is more complicated. The following theorem describes the tangent map of a Banach representation of S which extends to a local representation of G. We assume below that Ᏸ(Â(X)) = Ᏸ S . (c) Ᏸ S is invariant with respect to R(λ,Â(X)), X ∈ L(G). In this case, all operators π(s), s ∈ S, are invertible. Moreover, if S algebraically generates G, for some e-neighborhood U in G the extension of π to a local representation of U is unique.
Proof. Let T be a representation of a local Lie group U ⊂ G (U is a neighborhood of unit) which extends π and leaves Ᏸ S invariant. The fact that T | U ∩ S = π | U ∩ S entails that dT (X)u =Â(X)u (= A(X)u) for all X ∈ L(S), u ∈ Ᏸ S . In view of formula (1.5), the last equality is valid for all X ∈ 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], for ᐂ 0 = Ᏸ S . Now let π satisfies (a), (b), and (c). We claim that π(s) is invertible for each s ∈ S. Indeed, since Ᏸ S is an essential domain for A(X), X ∈ L(S), condition (b) implies that A(X) is a generator of a C 0 -group by Gelfand theorem [5]. Thus operators π(exp tX), t ≥ 0, are invertible and the invertibility of π(s) follows as in Proposition 5.2. According to Lemma 4.1,Â is a representation of L(G) by operators on Ᏸ S . Again, by the Krein-Shihvatov theorem in the Kirillov formulation for ᐂ 0 = Ᏸ S , T 0 =Â, there exists a representation T of some local Lie group W with L(W ) = L(G) such that dT (X) | Ᏸ S =Â(X), X ∈ L(G), and Ᏸ S is T -invariant. After shrinking W we may assume that W is an e-neighborhood in G and then apply Proposition 5.2 with ᐂ 0 = ᐂ 1 = Ᏸ S . Proof. To demonstrate the sufficiency, note that for simply connected G the local representation T constructed in Theorem 6.3 extends uniquely to the representation π of the whole G. Since every symmetric e-neighborhood generates G as a semigroup, the extensionπ leaves Ᏸ S invariant, as well. Finally,π | S = π by Lemma 2.1.
The uniqueness of the extension can be proved as in Theorem 5.3. This completes the proof.