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Holomorphic Realization of Unitary Representations of Banach--Lie Groups

In this paper we explore the method of holomorphic induction for unitary representations of Banach--Lie groups. First we show that the classification of complex bundle structures on homogeneous Banach bundles over complex homogeneous spaces of real Banach--Lie groups formally looks as in the finite dimensional case. We then turn to a suitable concept of holomorphic unitary induction and show that this process preserves commutants. In particular, holomorphic induction from irreducible representations leads to irreducible ones. Finally we develop criteria to identify representations as holomorphically induced ones and apply these to the class of so-called positive energy representations. All this is based on extensions of Arveson's concept of spectral subspaces to representations on Fr\'echet spaces, in particular on spaces of smooth vectors.


Introduction
This paper is part of a long term project concerned with a systematic approach to unitary representations of Banach-Lie groups in terms of conditions on spectra in the derived representation. A unitary representation π : G → U(H) is said to be smooth if the subspace H ∞ of smooth vectors is dense. This is automatic for continuous representations of finite dimensional groups but not in general (cf. [Ne10a]). For any smooth unitary representation, the derived representation dπ : g = L(G) → End(H ∞ ), dπ(x)v := d dt t=0 π(exp tx)v carries significant information in the sense that the closure of the operator dπ(x) coincides with the infinitesimal generator of the unitary one-parameter group π(exp tx).
We call (π, H) semibounded if the function s π : g → R∪{∞}, s π (x) := sup Spec(idπ(x)) = sup{ idπ( is bounded on the neighborhood of some point in g (cf. [Ne08, Lemma 5.7]). Then the set W π of all such points is an open invariant convex cone in the Lie algebra g.
We call π bounded if s π is bounded on some 0-neighborhood, which is equivalent to π being norm continuous (cf. Proposition 1.1). All finite dimensional unitary representations are bounded and many of the unitary representations appearing in physics are semibounded (cf. [Ne10c]). One of our goals is a classification of the irreducible semibounded representations and the development of tools to obtain direct integral decompositions of semibounded representations. To this end, realizations of unitary representations in spaces of holomorphic sections of vector bundles turn out to be extremely helpful. Clearly, the bundles in question should be allowed to have fibers which are infinite dimensional Hilbert spaces, and to treat infinite dimensional groups, we also have to admit infinite dimensional base manifolds. The main point of the present paper is to provide effective methods to treat unitary representations of Banach-Lie groups in spaces of holomorphic sections of homogeneous Hilbert bundles.
In Section 1 we explain how to parametrize the holomorphic structures on Banach vector bundles V = G × H V over a Banach homogeneous space M = G/H associated to a norm continuous representation (ρ, V ) of the isotropy group H. The main result of Section 1 is Theorem 1.6 which generalizes the corresponding results for the finite dimensional case by Tirao and Wolf ( [TW70]). As in finite dimensions, the complex bundle structures are specified by "extensions" β : q → gl(V ) of the differential dρ : h → gl(V ) to a representation of the complex subalgebra q ⊆ g C specifying the complex structure on M in the sense that T H (M ) ∼ = g C /q (cf. [Bel05]). The main point in [TW70] is that the homogeneous space G/H need not be realized as an open G-orbit in a complex homogeneous space of a complexification G C , which is impossible if the subgroup of G C generated by exp q is not closed. In the Banach context, two additional difficulties appear: The Lie algebra g C may not be integrable in the sense that it does not belong to any Banach-Lie group ( [GN03]) and, even if G C exists and the subgroup Q := exp q is closed, it need not be a Lie subgroup so that there is no natural construction of a manifold structure on the quotient space G/Q. As a consequence, the strategy of the proof in [TW70] can not be used for Banach Lie groups. Another difficulty of the infinite dimensional context is that there is no general existence theory for solutions of ∂-equations (see in particular [Le99]). The next step, carried out in Section 2, is to analyze Hilbert subspaces of the space Γ(V) of holomorphic sections of V on which G acts unitarily. In this context (ρ, V ) is a bounded unitary representation. The most regular Hilbert spaces with this property are those that we call holomorphically induced from (ρ, β). They contain (ρ, V ) as an H-subrepresentation satisfying a compatibility condition with respect to β. Here we show that if the subspace V ⊆ H is invariant under the commutant B G (H) = π(G) ′ , then restriction to V yields an isomorphism of the von Neumann algebra B G (H) with a suitably defined commutant B H,q (V ) of (ρ, β) (Theorem 2.12). This has remarkable consequences. One is that the representation of G on H is irreducible (multiplicity free, discrete, type I) if and only if the representation of (H, q) on V has this property. The second main result in Section 2 is a criterion for a unitary representation (π, H) of G to be holomorphically induced (Theorem 2.17).
Section 3 is devoted to a description of environments in which Theorem 2.17 applies naturally. Here we consider an element d ∈ g which is elliptic in the sense that the one-parameter group e R ad d of automorphisms of g is bounded. This is equivalent to the existence of an invariant compatible norm. Suppose that 0 is isolated in Spec(ad d).
Then the subgroup H := Z G (d) = {g ∈ G : Ad(g)d = d} is a Lie subgroup and the homogeneous space G/H carries a natural complex manifold structure. A smooth unitary representation (π, H) of G is said to be of positive energy if the selfadjoint operator −idπ(d) is bounded from below. Note that this is in particular the case if π is semibounded with d ∈ W π . Any positive energy representation is generated as a G-representation by the closed subspace V := (H ∞ ) p − , where q = p + ⋊ h C ⊆ g C is the complex subalgebra defining the complex structure on G/H and p − := p + . If the H-representation (ρ, V ) is bounded, then (π, H) is holomorphically induced by (ρ, β), where β is determined by β(p + ) = {0} (Theorem 3.7). In Theorem 3.14 we further show that, under these assumptions, π is semibounded with d ∈ W π . These results are rounded off by Theorem 3.12 which shows that, if π is semibounded with d ∈ W π , then π is a direct sum of holomorphically induced representations and if, in addition, π is irreducible, then V coincides with the minimal eigenspace of −idπ(d) and the representation of H on this space is automatically bounded. For all this we use refined analytic tools based on the fact that the space H ∞ of smooth vectors is a Fréchet space on which G acts smoothly ( [Ne10a]) and the R-action on H ∞ defined by π d (t) := π(exp td) is equicontinuous. These properties permit us to use a suitable generalization of Arveson's spectral theory, developed in Appendix A.
In [Ne11] we shall use the techniques developed in the present paper to obtain a complete descriptions of semibounded representations for the class of hermitian Lie groups. One would certainly like to extend the tools developed here to Fréchet-Lie groups such as diffeomorphism groups and groups of smooth maps. Here a serious problem is the construction of holomorphic vector bundle structures on associated bundles, and we do not know how to extend this beyond the Banach context, especially because of the non-existing solution theory for ∂-equations (cf. [Le99]).
Several of our results have natural predecessors in more restricted contexts. In [BR07] Beltiţȃ and Ratiu study holomorphic Hilbert bundles over Banach manifolds M and consider the endomorphism bundle B(V) over M × M (using a different terminology). They also relate Hilbert subspaces H of Γ(V) to reproducing kernels, which in this context are sections of B(V) satisfying a certain holomorphy condition which under the assumption of local boundedness of the kernel is equivalent to holomorphy as a section of B(V) ([BR07, Thm. 4.2]; see also [BH98,Thm. 1.4] and [MPW97] for the case of finite dimensional bundles over finite dimensional manifolds and [Od92] for the case of line bundles). Beltiţȃ and Ratiu use this setup to realize certain representations of a C * -algebra A, which define bounded unitary representations of the unitary group G := U(A), in spaces of holomorphic sections of a bundle over a homogeneous space of the unit group A × on which U(A) acts transitively ([BR07, Thm. 5.4]). These homogeneous spaces are of the form G/H, where H is the centralizer of some hermitian element a ∈ A with finite spectrum, so that the realization of these representations by holomorphic sections could also be derived from our Corollary 3.9. This work has been continued by Beltiţȃ with Galé in a different direction, focusing on complexifications of real homogeneous spaces instead of invariant complex structures on the real spaces ( [BG08]).
In [Bo80] Boyer constructs irreducible unitary representations of the Hilbert-Lie group U 2 (H) = U(H) ∩ (1 + B 2 (H)) via holomorphic induction from characters of the diagonal subgroup. They live in spaces of holomorphic sections on homogeneous spaces of the complexified group GL 2 (H) = GL(H) ∩ (1 + B 2 (H)), which are restricted versions of flag manifolds carrying strong Kähler structures.
In the present paper we only deal with representations associated to homogeneous vector bundles. To understand branching laws for restrictions of representations to subgroups, one should also study situations where the group G does not act transitively on the base manifold. For finite dimensional holomorphic vector bundles, this has been done extensively by T. Kobayashi who obtained powerful criteria for representations in Hilbert spaces of holomorphic sections to be multiplicity free (cf. [KoT05], [KoT06]). It would be interesting to explore the extent to which Kobayashi's technique of visible actions on complex manifolds can be extended to Banach manifolds.
Notation: For a group G we write 1 for the neutral element and λ g (x) = gx, resp., ρ g (x) = xg for left multiplications, resp., right multiplications. We write g C for the complexification of a real Lie algebra g and x + iy := x − iy for the complex conjugation on g C , which is an antilinear Lie algebra automorphism. For two Hilbert spaces H 1 , H 2 , we write B(H 1 , H 2 ) for the space of continuous (=bounded) linear operators H 1 → H 2 and B p (H), 1 ≤ p < ∞, for the space of Schatten class operators A : H → H of order p, i.e., A is compact with tr((A * A) p/2 ) < ∞. Acknowledgment: We thank Daniel Beltiţȃ for a careful reading of earlier versions of this paper, for pointing out references and for various interesting discussions on its subject matter.

Holomorphic Banach bundles
To realize unitary representations of Banach-Lie groups in spaces of holomorphic sections of Hilbert bundles, we first need a parametrization of holomorphic bundle structures on given homogeneous vector bundles for real Banach-Lie groups. As we shall see in Theorem 1.6 below, formulated appropriately, the corresponding results from the finite dimensional case (cf. [TW70]) can be generalized to the Banach context.
The following observation provides some information on the assumptions required on the isotropy representation of a Hilbert bundle. It is a slight modification of results from [Ne09, Sect. 3].
Proposition 1.1. Let (π, H) be a unitary representation of the Banach-Lie group G for which all vectors are smooth. Then π : G → U(H) is a morphism of Lie groups, hence in particular norm continuous, i.e., a bounded representation.
Proof. Our assumption implies that, for each x ∈ g, the infinitesimal generator dπ(x) of the unitary one-parameter group π x (t) := π(exp G (tx)) is everywhere defined, hence a bounded operator because its graph is closed.
Therefore the derived representation leads to a morphism of Banach-Lie algebras dπ : g → B(H). Since the function s π is a sup of a set of continuous linear functionals, it is lower semi-continuous. Hence the function is a lower semi-continuous seminorm and therefore continuous because g is barreled (cf. [Bou07,§III.4.1]). We conclude that the set of all linear functional dπ(·)v, v , v ∈ H ∞ a unit vector, is equicontinuous in g ′ , so that the assertion follows from [Ne09, Thm. 3.1].
We now turn to the case where M is a Banach homogeneous space. Let G be a Banach-Lie group with Lie algebra g and H ⊆ G be a split Lie subgroup, i.e., the Lie algebra h of H has a closed complement in g, for which the coset space M := G/H carries the structure of a complex manifold such that the projection q M : G → G/H is a smooth H-principal bundle and G acts on M by holomorphic maps. Let m 0 = q M (1) ∈ M be the canonical base point and q ⊆ g C be the kernel of the complex linear extension of the map g → T m0 (G/H) to g C , so that q is a closed subalgebra of g C invariant under Ad(H) (cf. [Bel05,Thm. 15]). We call q the subalgebra defining the complex structure on M = G/H because specifying q means to identify T m0 (G/H) ∼ = g/h with the complex Banach space g C /q and thus specifying the complex structure on M .
This means that β is a continuous morphism of Banach-Lie algebras.
This observation leads us to the following structures. Definition 1.3. Let H ⊆ G be a Lie subgroup and q ⊆ g C be a closed subalgebra containing h C . If ρ : H → GL(V ) is a norm continuous representation on the Banach space V , then a morphism β : q → gl(V ) of complex Banach-Lie algebras is said to be an extension of ρ if (1) We write C(G, V ) ρ , resp., C ∞ (G, V ) ρ for the continuous, resp., smooth functions satisfying (2). (b) We associate to each x ∈ g C the left invariant differential operator on C ∞ (G, V ) defined by By complex linear extension, we define the operators L x+iy := L x + iL y for z = x + iy ∈ g C , x, y ∈ g.
(c) For any extension β of ρ, we write C ∞ (G, V ) ρ,β for the subspace of those elements f ∈ C ∞ (G, V ) ρ satisfying, in addition, (3) Remark 1.5. For x ∈ g, h ∈ H and any smooth function ϕ defined on an open right H-invariant subset of G, we have so that we obtain for each x ∈ g C and h ∈ H the relation The proof of the following theorem is very much inspired by [TW70].
Theorem 1.6. Let M = G/H, V be a complex Banach space and ρ : H → GL(V ) be a norm continuous representation. Then, for any extension β : q → gl(V ) of ρ, the associated bundle V := G × H V carries a unique structure of a holomorphic vector bundle over M , which is determined by the characterization of the holomorphic sections s : M → V as those for which s ∈ C ∞ (G, V ) ρ,β . Any such holomorphic bundle structure is G-invariant in the sense that G acts on V by holomorphic bundle automorphism. Conversely, every G-invariant holomorphic vector bundle structure on V is obtained from this construction. Proof.
Step 1: Let β be an extension of ρ and E ⊆ g be a closed subspace complementing h. Then E ∼ = g/h ∼ = T m0 (M ) implies the existence of a complex structure I E on E for which E → g/h is an isomorphism of complex Banach spaces. Therefore the I E -eigenspace decomposition is a direct decomposition into closed subspaces. The quotient map Pick open convex 0-neighborhoods U r ⊆ r and U q ⊆ q such that the BCH multiplication * defines a biholomorphic map µ : U r × U q → U, (x, y) → x * y, onto the open 0-neighborhood U ⊆ g C and that * defines an associative multiplication defined on all triples of elements of U .
Step 2: On the 0-neighborhood U ⊆ g C , we consider the holomorphic function For w ∈ g and z ∈ U g , the BCH product z * tw ∈ g C is defined if t is small enough, and we have where dλ * z (0) : g C → g C is the differential of the multiplication map λ * z (x) = z * x in 0. As (5) is complex linear in w ∈ g C , it follows that (L w f )(exp G z) = dF (z)dλ * z (0)w hold for every w ∈ g C , hence in particular for w ∈ q. For w ∈ q and z = µ(x, y), we thus obtain We conclude that In particular, we obtain f (gh) = ρ(h) −1 f (g) for g ∈ U G , h ∈ H 0 .
Step 3: Since H is a complemented Lie subgroup, there exists a connected submanifold Z ⊆ G containing 1 for which the multiplication map Z ×H → G, (x, h) → xh is a diffeomorphism onto an open subset of G. Shrinking U G , we may therefore assume that U G = U Z U H holds for a connected open 1-neighborhood U Z in Z and a connected open 1-neighborhood U H in H. Then defines a smooth function f : U Z H → GL(V ). That it extends f follows from the fact that u = zh ∈ U G with z ∈ U Z and h ∈ U H implies h ∈ H 0 , so that For w ∈ q, formula (4) leads to Step 4: For m ∈ M we choose an element g m ∈ G with g m .m 0 = m and put U m := gq M (U Z ), so that On this open subset of G, the function Next we note that the function F m defines a smooth trivialization The corresponding transition functions are given by ϕ m,n : U m,n := U m ∩ U n → GL(V ), gH → ϕ m,n (g) := F m (g) −1 F n (g).
To verify that these transition functions are holomorphic, we have to show that the functions ϕ m,n : G Um,n → B(V ) are annihilated by the differential operators L w , w ∈ q. This follows easily from the product rule and (b): We conclude that the transition functions (ϕ m,n ) m,n∈M define a holomorphic vector bundle atlas on V.
Step 5: To see that this holomorphic structure is determined uniquely by β, pick an open connected subset U ⊆ M containing m 0 on which the bundle is holomorphically trivial and the trivialization is specified by a smooth function F : Step 6: The above construction also shows that, for any holomorphic vector bundle structure on V, for which G acts by holomorphic bundle automorphisms, we may consider for a local trivialization given by F : Then β is a continuous linear map. To see that it does not depend on the choice of F , we note that, for any other trivialization F : G U → GL(V ), the function F −1 · F factors through a holomorphic function on U , so that We conclude that the right hand side of (7) does not depend on the choice of F . Applying this to functions of the form F g (g ′ ) := F (gg ′ ), defined on g −1 G U , we obtain in particular For w ∈ h, the relation β(w) = dρ(w) follows immediately from the H-equivariance of F . To see that β is an extension of ρ, it now remains to verify that it is a homomorphism of Lie algebras. For w 1 , w 2 ∈ q, we have which shows that β is a homomorphism of Lie algebras.
Example 1.7. We consider the special case where G is contained as a Lie subgroup in a complex Banach-Lie group G C with Lie algebra g C and where Q : induces an open embedding of M = G/H as an open G-orbit in the complex manifold G C /Q. In this case every holomorphic representation π : Q → GL(V ) defines an associated holomorphic Banach bundle V : Since the Lie algebra g C need not be integrable in the sense that it is the Lie algebra of a Banach-Lie group (cf. [GN03, Sect. 6]), the assumption G ⊆ G C is not general enough to cover every situation. One therefore needs the general Theorem 1.6.
In [GN03,Sect. 6] one finds examples of simply connected Banach-Lie groups G for which g C is not integrable. Here G is a quotient of a simply connected Lie group G by a central subgroup Z ∼ = R, G has a simply connected universal complexification η G : G → G C , and the subgroup exp G C (z C ) is not closed; its closure is a 2-dimensional torus.
Then the real Banach-Lie group G acts smoothly and faithfully by holomorphic maps on a complex Banach manifold M for which g C is not integrable. It suffices to pick a suitable tubular neighborhood

Hilbert spaces of holomorphic sections
In this section we take a closer look at Hilbert spaces of holomorphic sections in Γ(V) for holomorphic Hilbert bundles V constructed with the methods from Section 1. Here we are interested in Hilbert spaces with continuous point evaluations (to be defined below) on which G acts unitarily. This leads to the concept of a holomorphically induced representation, which for infinite dimensional fibers is a little more subtle than in the finite dimensional case. The first key result of this section is Theorem 2.12 relating the commutant of a holomorphically induced unitary G-representation (π, H) to the commutant of the representation (ρ, β) of (H, q) on the fiber V . This result is complemented by Theorem 2.17 which is a recognition devise for holomorphically induced representations. Remark 2.2. Let q : V → M be a holomorphic Hilbert bundle over M . Then we can represent holomorphic sections of this bundle by holomorphic functions on the dual bundle V * whose fiber (V * ) m is the dual space V * m of V m . We thus obtain an embedding Ψ : whose image consists of holomorphic functions on V * which are fiberwise linear. If G is a group acting on V by holomorphic bundle automorphisms, then G also acts naturally by holomorphic maps on implies that Ψ is equivariant with respect to the natural G-actions on Γ(V) and O(V * ).

Existence of analytic vectors
Proof. The manifold V is a G-equivariant quotient of the product manifold G × V on which G acts analytically by left multiplications in the left factor. Since the quotient map q : G × V → V is a real analytic submersion, the action of G on V is also analytic.
Then H is called a reproducing kernel Hilbert space and since it is determined uniquely by the kernel Q, it is denoted H Q (cf.
. In this case we call H Q a G-invariant reproducing kernel Hilbert space.
Lemma 2.5. Let G be a Banach-Lie group acting analytically via by holomorphic maps on the complex manifold M .
(a) Let H ⊆ O(M ) be a reproducing kernel Hilbert space whose kernel Q is a Ginvariant holomorphic function on M × M . Then the elements (Q m ) m∈M representing the evaluation functionals in H are analytic vectors for the representation of G, defined by π(g)f : and since Q and the G-action are real analytic, this function is analytic on M . Now [Ne10b, Thm. 5.2] implies that Q m is an analytic vector.
(b) As in Remark 2.2, we realize Γ(V) by holomorphic functions on the dual bundle V * . We thus obtain a reproducing kernel Hilbert space For the corresponding G-invariant kernel Q on V * this means that Q αm = ev * m v, so that the assertion follows from (a).

The endomorphism bundle and commutants
The goal of this section is Theorem 2.12 which connects the commutant of the Grepresentation on a Hilbert space H V of a holomorphically induced representation with the corresponding representation (ρ, β) of (H, q) on V . The remarkable point is that, under natural assumptions, these commutants are isomorphic, so that both representations have the same decomposition theory. This generalizes an important result of S. Kobayashi concerning irreducibility criteria for the G-representation on and the the corresponding extension β : q ⊕ q → gl(B(V )) by We write L := (G × G) × H×H B(V ) for the corresponding holomorphic Banach bundle over M × M (Theorem 1.6).
Remark 2.6. For every g ∈ G, we have isomorphisms . Accordingly, we have for every pair (g 1 , g 2 ) ∈ G × G an isomorphism This defines a map For h 1 , h 2 ∈ H, we then have so that γ factors through a bijection γ : L → B(V). This provides an interpretation of the bundle L as the endomorphism bundle of V (cf. [BR07]).
Since the group G acts (diagonally) on the bundle L, it makes sense to consider G-invariant holomorphic sections.
Lemma 2.7. The space Γ(L) G of G-invariant holomorphic sections of L has the property that the evaluation map Proof. If s(1, 1) = 0, then the corresponding holomorphic section s ∈ Γ(L) vanishes on the totally real submanifold and hence on all of M × M . For the function s : showing that s(1, 1) lies in B H (V ).
Remark 2.8. In general, the holomorphic bundle L does not have any non-zero holomorphic section, although its restriction to the diagonal ∆ M has the trivial section R given by R (m,m) = id Vm for every m ∈ M .
If R : M × M → L is a holomorphic section, then we obtain for every v ∈ V and n ∈ M a holomorphic section R n,v : Therefore V has nonzero holomorphic sections if L does, and this is not always the case.
defines a holomorphic section of B(V). Since this assertion is local, it follows from the corresponding assertion for trivial bundles treated in [Ne00, Lemma A.III.9(iii)]. The corresponding smooth function We conclude that R ∈ Γ(L) G , so that it is completely determined by . In particular, Ψ is completely determined by ev 1 •Ψ. This follows also from its equivariance, which leads to Ψ is also contained in C ∞ (G × G, B(V )) ρ, β , hence defines a holomorphic section of the bundle L.
Definition 2.10. (a) For (ρ, β) as in Theorem 1.6 and the corresponding associated bundle V = G × H V , a unitary representation (π, H) of G is said to be holomorphically induced from (ρ, β) if there exists a realization Ψ : H → Γ(V) as an invariant Hilbert space with continuous point evaluations whose kernel R ∈ Γ(L) G satisfies R(1, 1) = id V .
(b) Since this condition determines R by Lemma 2.7, it also determines the reproducing kernel space Ψ(H) and its norm. We conclude that, for every pair (ρ, β), there is at most one holomorphically induced unitary representation of G up to unitary equivalence. Accordingly, we call (ρ, β, V ) inducible if there exists a corresponding holomorphically induced unitary representation of G.
If this is the case, then we use the isometric embedding ev * 1 : V → H to identify V with a subspace of H and note that the evaluation map ev 1 : H → V corresponds to the orthogonal projection p V : H → V .
is holomorphically induced. Since ev 1 is H-equivariant, the closed subspace V ⊆ H is H-invariant and the H-representation on this space is equivalent to (ρ, V ), hence in particular bounded.
Moreover, V ⊆ H ω follows from Lemma 2.5(b). Since the evaluation maps ev g : H → V separate the points, the analyticity of the elements ev * g v even implies that H ω is dense in H.
If V is of the form (H ∞ ) n for a subalgebra n ⊆ g C , then it is invariant under the commutant B G (V ), but we do not know if this is always true for holomorphically induced representation. To make the following proposition as flexible as possible, we assume this naturality condition of V (cf. Remark 2.18 below).
Theorem 2.12. Suppose that (π, H V ) is holomorphically induced from the representation (ρ, β) of (H, q) on V and that is an isomorphism of von Neumann algebras.
Proof. By assumption, every A ∈ B G (H V ) preserves V , so that A| V can be identified with the operator p V Ap V ∈ B(V ), where p V : H → V is the orthogonal projection.
According to Example 2.9(b), it is contained in C ∞ (G × G, B(V )) ρ, β , hence defines an element R ∈ Γ(L) G . In view of this section vanishes in the base point, hence on all of M × M (Lemma 2.7). We conclude that 0 = R(g 1 , g 2 ) = P p V π(g −1 1 g 2 )p V (1 − P ) = P π(g −1 1 g 2 )(1 − P ), so that, for every g ∈ G, we have P π(g)(1 − P ) = 0. This leads to P H V2 = {0}, and hence to H V1 ⊥H V2 . We derive that H V = H V1 ⊕ H V2 is an orthogonal direct sum. Therefore the orthogonal projection P ∈ B G (H V ) onto H V1 leaves V invariant and satisfies P | V = P . This proves that R is surjective.
Remark 2.13. The preceding proof shows that, under the assumptions of Theorem 2.12, the range of the injective map ev : The preceding theorem has quite remarkable consequences because it implies that the representations (π, H V ) and (ρ, V ) decompose in the same way.
Corollary 2.14. Suppose that (π, H V ) is holomorphically induced from (ρ, β), that V is B G (V )-invariant, and that β(q) = β(h C ), so that B H (V ) = B H,q (V ). Then the G-representation (π, H V ) has any of the following properties if and only if the H-representation (ρ, V ) does.
(iii) Type I, II or III.
Proof. (i) follows from the fact that, according to Schur's Lemma, irreducibility means that the commutant equals C1.
(ii) is clear because multiplicity freeness means that the commutant is commutative.
(iii) is clear because the type of a representation is defined as the type of its commutant as a von Neumann algebra.
(iv) That a unitary representation decomposes discretely means that its commutant is an ℓ ∞ -direct sum of factors of type I. Hence the G representation on H V has this property if only if the H-representation on V does. Corollary 2.14(i) is a version of S. Kobayashi's Theorem in the Banach context (cf. [Ko68]).
Problem 2.15. Theorem 2.12 should also be useful to derive direct integral decompositions of the G-representation on H V from direct integral decompositions of the H-representation on V .
Suppose that the bounded unitary representation (ρ, V ) of H is of type I and holomorphically inducible. Then it is a direct integral of irreducible representations (ρ j , V j ). Are these irreducible representations of H also inducible?
Proof. Since γ is (H, q)-equivariant, we have a well-defined G-equivariant bijection This implies that γ(H V1 ) = H V2 and that the map γ : H V1 → H V2 is unitary because both spaces have the same reproducing kernels (cf. Definition 2.10).

Realizing unitary representations by holomorphic sections
We conclude this section with a result that helps to realize certain subrepresentations of unitary representations in spaces of holomorphic sections. We continue in the setting of Section 1, where M = G/H is a Banach homogeneous space with a complex structure defined by the subalgebra q ⊆ g C . From the discussion in Remark 2.11, we know that every holomorphically induced representation satisfies the assumptions (A1/2) in the theorem below, which is our main tool to prove that a given unitary representation is holomorphically induced.
Then the following assertions hold: (i) β is an extension of ρ defining on V := G × H V the structure of a complex Hilbert bundle.
(ii) If p V : H → H denotes the orthogonal projection to V , then maps H into C ∞ (G, V ) ρ,β ∼ = Γ(V), and we thus obtain a G-equivariant unitary isomorphism of the closed subspace H V := span π(G)V with the representation holomorphically induced from (ρ, β).
, and it is also easy to see that β(Ad(h)x) = π(h)β(x)π(h) −1 for h ∈ H and x ∈ q. Therefore β is an extension of ρ, and we can use Theorem 1.6 to see that β defines the structure of a holomorphic Hilbert bundle on V.
For v ∈ D V , w ∈ H ∞ and x ∈ q, we further derive from (A2) that For v ∈ H ∞ and x ∈ q, we now obtain Clearly Φ(v) = 0 is equivalent to v⊥π(G)V , so that (ker Φ) ⊥ = H V . Since (A2) implies that V ⊆ H ∞ , the same holds for span(π(G)V ). This shows that The corresponding kernel Q : G × G → B(V ) is given by so that we have in particular Q(1, 1) = id V .
(iii) To see that V consists of analytic vectors, we may w.l.o.g. assume that H = H V and hence that H ⊆ Γ(V) is holomorphically induced and that Φ = id H . Let ev 1H : H → V 1H ∼ = V be the evaluation map. The corresponding map is simply given by evaluation in 1 ∈ G. Now ev 1 | V : V → V is the identity, so that ev * 1 : V → H is simply the isometric inclusion. Hence the analyticity of v = ev * 1 v = ev * 1H v follows from Lemma 2.5. Remark 2.18. Suppose that there exist subalgebras p ± ⊆ g C with q = h C ⋊ p + and Ad(H)p ± = p ± . Then, for every unitary representation (π, H) of G, the closed subspace V := (H ∞ ) p − is invariant under H and B G (H). If the representation (ρ, H) on V is bounded, then the dense subspace D V := (H ∞ ) p − satisfies (A2) if we put β(p + ) = {0}. Theorem 2.17 now implies that V ⊆ H ω , so that we see in particular that V = (H ∞ ) p − is a closed subspace of H. Moreover, Corollary 2.14 applies.
The following remark sheds some extra light on condition (A2). The action of G on V by holomorphic bundle automorphisms leads to a homomorphismσ V : g C → V(V) and each x ∈ q m thus leads to a linear vector field −β m (x) on the fiber V m . Passing to derivatives in the formula (g.s)(m) := g.s(g −1 m), we obtain for x ∈ q m (x.s)(m) = −β m (x) · s(m).
In particular, V ⊥ ∩ H ∞ is invariant under the derived action of q m , so that one can expect that the adjoint operators coming from q m act on V m As we have seen in Lemma 2.5(b), the subspace im(ev * m ) of H consists of smooth vectors, so that V ∩ H ∞ is dense in V . Remark 2.21. Let (π, H) be a smooth unitary representation of the Lie group G and V ⊆ H be a closed H-invariant subspace. We then obtain a natural G-equivariant map η : G/H → Gr V (H), gH → π(g)V . If this map is holomorphic, then we can pull back the natural bundle V → Gr V (H) from Example 2.20 and obtain a realization of H in Γ(η * V). This works very well if the representation (π, H) is bounded because in this case π : G → U(H) is a morphism of Banach-Lie groups, but if π is unbounded, then it seems difficult to verify that η is smooth, resp., holomorphic.
If (A1/2) in Theorem 2.17 are satisfied, then V ⊆ H ω ⊆ H ∞ implies that, the operators dπ(x), x ∈ g, are defined on V . Since they are closable, the graph of these restrictions is closed, which implies that the restrictions dπ(x)| V : V → H are continuous linear operators. We thus obtain a natural candidate for a tangent map

Realizing positive energy representations
In this section we fix an element d ∈ g = L(G) for which the one-parameter group e R ad d ⊆ Aut(g) is bounded, i.e., preserves an equivalent norm. We call such elements elliptic. Then H := Z G (d) is a Lie subgroup and if 0 is isolated in Spec(D), then G/H carries a natural complex structure. The class of representations which one may expect to be realized by holomorphic sections of Hilbert bundles V over G/H is the class of positive energy representations, which is defined by the condition that the selfadjoint operator −idπ(d) is bounded below.

The splitting condition
Let d ∈ g be an elliptic element. Then is a closed subgroup of G, not necessarily connected, with Lie algebra h = z g (d) = ker(ad d). Since g contains arbitrarily small e R ad d -invariant 0-neighborhoods U , there exists such an open 0-neighborhood with exp G (U ) ∩ H = exp G (U ∩ L(H)). Therefore H is a Lie subgroup of G, i.e., a Banach-Lie group for which the inclusion H ֒→ G is a topological embedding.
Our assumption implies that α t := e t ad d defines an equicontinuous one-paramter group of automorphisms of the complex Banach-Lie algebra g C . For δ > 0, we consider the Arveson spectral subspace p + := g C ([δ, ∞[) (cf. Definition A.5). Applying Proposition A.14 to the Lie bracket g C × g C → g C , we see that p + is a closed complex subalgebra. For f ∈ L 1 (R), α(f ) := R f (t)α t dt and x ∈ g C , the relations α(f )x = α(f )x and f (ξ) = f (−ξ) imply that p − := p + = g C (] − ∞, −δ]). To make the following constructions work, we assume the splitting condition: In view of Lemma A.16, it is satisfied for some δ > 0 if and only if 0 is isolated in Spec(ad d).
Since Ad(H) commutes with e R ad d , the closed subalgebras p ± ⊆ g C are invariant under Ad(H) and e R ad d . Now g ∩ (p + ⊕ p − ) is a closed complement for h in g, so that M := G/H carries the structure of a Banach homogeneous space and q := h C + p + ∼ = p + ⋊ h C defines a G-invariant complex manifold structure on M (cf. Section 1).
Remark 3.1. (a) For bounded derivations of compact L * -algebras similar splitting conditions have been used by Beltiţȃ in [Bel03] to obtain Kähler polarizations of coadjoint orbits. In [Bel04] this is extended to bounded normal derivations of a complex Banach Lie algebra.
(b) If g is a real Hilbert-Lie algebra, then one can use spectral measures to obtain natural complex structures on G/H even if the splitting condition is not satisfied, i.e., 0 need not be isolated in the spectrum of ad d ([BRT07, Prop. 5.4]).
Example 3.2. If α factors through an action of the circle group T = R/2πZ, then the Peter-Weyl Theorem implies that the sum n∈Z g n C of the corresponding eigenspaces g n C := ker(ad d − in1) is dense in g C with h C = g 0 C . Since the operator ad d is bounded, only finitely many g n C are non-zero, so that we actually have g C = n∈Z g n C . From [g n C , g m C ] ⊆ g n+m C it follows that p ± := ±n>0 g n C are closed subalgebras for which g C = p + ⊕ h C ⊕ p − is direct. In this case the splitting condition is always satisfied and Spec(D) ⊆ iZ.
If, conversely, d ∈ g is an element for which the complex linear extension of ad d to g C is diagonalizable with finitely many eigenvalues in iZ, then e R ad d ⊆ Aut(g C ) is compact, hence preserves a compatible norm. An important special situation, where we have all this structure are hermitian Lie groups (cf. [Ne11]). In this case we simply have p ± = g ±1 C .

Positive energy representations
Proposition 3.4. Let (π, H) be a smooth unitary representation of the Banach-Lie group G, d ∈ g be elliptic, and P : B(R) → L(H) be the spectral measure of the unitary one-parameter group π d (t) := π(exp G td). Then the following assertions hold: (i) H ∞ carries a Fréchet structure for which π d (t) t∈R defines a continuous equicontinuous action of R on H ∞ . In particular, H ∞ is invariant under π d (f ) for every f ∈ L 1 (R).
(ii) For every closed subset E ⊆ R, we have H ∞ (E) = (P (E)H) ∩ H ∞ for the corresponding spectral subspace.
(iv) For closed subsets E, F ⊆ R, the Arveson spectral subspaces g C (F ) satisfies Proof. (i) We may w.l.o.g. assume that the norm on g is invariant under e R ad d . On H ∞ we consider the Fréchet topology defined by the seminorms with respect to which the action of G on H ∞ is smooth (cf. [Ne10a,Thm. 4.4]). In particular, the bilinear map is continuous because it can be obtained as a restriction of the tangent map of the G-action.
In view of the relation π d (t)dπ(x)π d (t) −1 = dπ(e t ad d x) for t ∈ R, x ∈ g, the isometry of e t ad d on g implies that the seminorms p n on H ∞ are invariant under π d (R). Since the R-action on H ∞ defined by the operators π d (t) is smooth, hence in particular continuous, we obtain with Definition A.5 an algebra homomorphism (cf. Definition A.5 below), and this implies (i).
(ii) Since H ∞ (E) = H(E) ∩ H ∞ follows immediately from the definition of spectral subspaces (Remark A.6), this assertion is a consequence of Lemma 3.3(ii).
(iii) We write the open set E as an increasing union of compact subsets E n and observe that n P (E n )H is dense in P (E)H. For every n, there exists a compactly supported function h n ∈ C ∞ c (R, R) such that supp(h n ) ⊆ E, 0 ≤ h n ≤ 1, and h n En = 1.
Remark 3.5. Combining Lemma 3.3(i) with Proposition 3.4(i), we derive that the subspace H ∞ of H is invariant under all operators P ( f ) = f (−idπ(d)) for f ∈ L 1 (R). This implies in particular to the operators P (h), h ∈ S(R), but not to the spectral projections P (E). If E ⊆ R is open, Proposition 3.4(iii) provides a suitable approximate invariance.
The following proposition is of key importance for the following. It contains the main consequences of Arveson's spectral theory for the actions on g C and H ∞ .
Proposition 3.6. If d ∈ g is elliptic with 0 isolated in Spec(ad d), then for any smooth positive energy representation (π, H) of G, the H-invariant subspace V := (H ∞ ) p − satisfies H = span(π(G)V ).

Proof. First we show that
For some ε ∈]0, δ[, we consider the closed subspace where Proof. Since π(G)V spans a dense subspace of H, i.e., H = H V (Proposition 3.6), the assertion follows from Remark 2.18. Corollary 3.9. If d ∈ g is elliptic with 0 isolated in Spec(ad d), then every bounded representation of G is holomorphically induced from the representation (ρ, β) of (H, q) From Corollary 2.14 we obtain in particular: Corollary 3.10. A positive energy representation (π, H) of G for which the representation (ρ, V ) of H is bounded is a direct sum of irreducible ones if and only if (ρ, V ) has this property.
Example 3.11. The complex Banach-Lie algebra g is called weakly root graded if there exists a finite reduced root system ∆ such that g contains the corresponding finite dimensional semisimple Lie algebra g ∆ and for some Cartan subalgebra h ⊆ g ∆ , the Lie algebra g is a direct sum of finitely many ad h-eigenspaces. Now suppose that g is a real Banach-Lie algebra for which g C is weakly root graded, that h is invariant under conjugation and, for every x ∈ h ∩ ig, the derivation ad x has real spectrum. Then the realization results for bounded unitary representations of G which follows from [MNS09, Thm. 5.1], applied to their holomorphic extensions G C → GL(H), can be derived easily from Corollary 3.9.
The following theorem shows that, assuming that (π, H) is semibounded with d ∈ W π permits us to get rid of the quite implicit assumption that the H-representation on V is bounded. It is an important generalization of Corollary 3.9 to semibounded representations.
Theorem 3.12. Let (π, H) be a semibounded unitary representation of the Banach-Lie group G and d ∈ W π be an elliptic element for which 0 is isolated in Spec(ad d). We write P : B(R) → B(H) for the spectral measure of the unitary one-parameter group π d (t) := π(exp(td). Then the following assertions hold: (i) The representation π| H of H is semibounded and, for each bounded measurable subset B ⊆ R, the H-representation on P (B)H is bounded.
(ii) The representation (π, H) is a direct sum of representations (π j , H j ) for which there exist H-invariant subspaces D j ⊆ (H ∞ j ) p − for which the H-representation ρ j on V j := D j is bounded and span π j (G)V j is dense in H j . Then the representations (π j , H j ) are holomorphically induced from (ρ j , β j , V ), where β j (p + ) = {0}.  (ii) We apply Zorn's Lemma to the ordered set of all pairwise orthogonal systems of closed G-invariant subspaces satisfying the required conditions. Therefore it suffices to show that if H = {0}, then there exists a non-zero H-invariant subspaces D ⊆ (H ∞ ) p − for which the H-representation ρ on D is bounded.
For s := inf Spec(−idπ(d)) and 0 < ε < δ, we may take the space D := H ∞ ∩ P ([s, s + ε[)H from the proof of Proposition 3.6. As we have seen there, it is annihilated by dπ(p − ) and the boundedness of the H-representation on its closure follows from (i).
That the representations (π j , H j ) are holomorphically induced from the bounded H-representations on V j follows from Theorem 2.17.
From (i) we know that the H-representation on U is bounded and on the dense subspace D U we have dπ(p − )D U = {0}. Therefore (A1/2) in Theorem 2.17 are satisfied and this proves Claim 2.
Claim 3: U = P ({s})H for every t > s. In view of Claim 2, Corollary 2.14 implies that the H-representation on U is irreducible. Since π d commutes with H, it follows in particular that ρ(exp Rd) ⊆ T1. The definition of s now shows that U ⊆ P ({s})H.
For 0 < ε < δ and t < δ + ε, the proof of Proposition 3.6 implies that P ([s, To see that we actually have equality, we note that Claim 2 shows that P ([s, This completes the proof of (iii).
Remark 3.13. For finite dimensional Lie groups the classification of irreducible semibounded unitary representations easily boils down to a situation where one can apply Theorem 3.12. Here d ∈ g is a regular element whose centralizer h = t is a compactly embedded Cartan subalgebra and the corresponding group T = H is abelian and V is one-dimensional (cf. [Ne00]). In this case 0 is trivially isolated in the finite set Spec(ad d).
The following theorem provides a bridge between the seemingly weak positive energy condition and the much stronger semiboundedness condition.
Theorem 3.14. Let d ∈ g be elliptic with 0 isolated in Spec(ad d). Then a smooth unitary representation (π, H) of G for which the representation ρ of H on (H ∞ ) p − is bounded satisfies the positive energy condition if and only if π is semibounded with d ∈ W π .
Proof. If π semibounded with d ∈ W π , then we have in particular (15). It remains to show the converse if all the assumptions of the theorem are satisfied. Recall that the splitting condition (SC) is satisfied because 0 is isolated in Spec(ad d). We note that the representation ad p + of h on p + is bounded with Spec(ad p + (−id)) ⊆]0, ∞[. Therefore the invariant cone is continuous bilinear and H-equivariant, we see with Proposition A.14 that the sub- For every v ∈ V ⊆ H ω , the Poincaré-Birkhoff-Witt Theorem shows that it contains the subspace To see that C ⊆ W π , it now suffices to show that Ad(G)C has interior points. Let p := (p + + p − ) ∩ g and note that this is a closed H-invariant complement of h in g. The map F : h × p → g, F (x, y) := e ad y x is smooth and Since the operators ad x, x ∈ h, preserve p, the operator dF (x, 0) is invertible if and only if ad x : p → p is invertible, and this is the case for any x ∈ C because ad x| p ± are invertible operators. This proves that C is contained in the interior of F (C, p), and hence that C ⊆ W π . Therefore π is semibounded with d ∈ W π .

A Equicontinuous representations
In this appendix we first explain how a continuous representation π : G → GL(V ), i.e., a representation defining a continuous action of G on V , of the locally compact group G on the complete locally convex space V can be integrated to a representation of the group algebra L 1 (G), provided π(G) is equicontinuous. If G is abelian, we use this to extend Arveson's concept of spectral subspaces to representations on complete locally convex spaces.

A.1 Equicontinuous groups
If (π, V ) is a continuous representation of a compact group on a locally convex space, then π(G) is an equicontinuous subgroup of GL(V ) (cf. Lemma A.3) and if V is finite dimensional, then each equicontinuous subgroup of GL(V ) has compact closure. Therefore, in the context of locally convex spaces, equicontinuous groups are natural analogs of compact groups.
Proposition A.1. Let V be a locally convex space. For a subgroup G ⊆ GL(V ), the following are equivalent: (1) G is equicontinuous.
(2) For each 0-neighborhood U ⊆ V , the set g∈G gU is a 0-neighborhood.
(3) There exists a basis of G-invariant absolutely convex 0-neighborhoods in V .
(4) The topology on V is defined by the set of G-invariant continuous seminorms.
(5) Each equicontinuous subset of the dual space V ′ is contained in a G-invariant equicontinuous subset. Proof.
(1) ⇔ (2): The equicontinuity of G means that for each 0-neighborhood U there exists a 0-neighborhood W in V with gW ⊆ U for each g ∈ G, i.e., W ⊆ g∈G gU .
(2) ⇔ (3) follows from the local convexity of V . (3) ⇔ (5): First we note that a subset K ⊆ V ′ is equicontinuous if and only if its polar is a 0-neighborhood and, conversely, for each 0-neighborhood U ⊆ V , its polar set is equicontinuous. If (3) holds and K ⊆ V ′ is equicontinuous, then K is a 0-neighborhood and (3) implies the existence of a G-invariant absolutely convex 0-neighborhood U , contained in K. Then U ⊇ K ⊇ K and the G-invariance of U imply (5). If, conversely, (5) holds and W ⊆ V is a 0-neighborhood, then W is equicontinuous and (5) implies the existence of a G-invariant equicontinuous subset K ⊇ W . Then K ⊆ W is an absolutely convex G-invariant 0-neighborhood in V . In view of the preceding proposition, this is equivalent to the existence of a G-invariant norm defining the topology.
(b) If G ⊆ GL(V ) is equicontinuous, then each G-invariant continuous seminorm p on V defines a homomorphism G → Isom(V p ), where V p is the completion of the quotient space V /p −1 (0) with respect to the norm induced by p. We thus obtain an embedding G ֒→ i∈I Isom(V pi ), where (p i ) i∈I is a fundamental system of G-invariant continuous seminorms on V .
(c) If G ⊆ GL(V ) is equicontinuous, then Proposition A.1(5) implies that each G-orbit in V ′ is equicontinuous. A.2 Arveson spectral theory on locally convex spaces Definition A.4. A representation (π, V ) of the group G on the locally convex space V is called equicontinuous if π(G) ⊆ B(V ) (the space of continuous linear operators on V ) is an equicontinuous group of operators, which is equivalent to the existence of a family of G-invariant continuous seminorms defining the topology (cf. Proposition A.1).
Definition A.5. (a) Let (π, G) be an equicontinuous strongly continuous action of the locally compact group G on the complete complex locally convex space V .
We write P G for the set of π(G)-invariant continuous seminorms on V . This set carries a natural order defined by p ≤ q if p(v) ≤ q(v) for every v ∈ V . Let V p be the completion of V /p −1 (0) with respect to p. For p ≤ q we then obtain a contractive map ϕ pq : V q → V p of Banach spaces. Since V is complete, the natural map is a topological isomorphism, where the right hand side carries the product topology.
We now obtain on each V p a continuous isometric representation (π p , V p ) of G. Therefore each f ∈ L 1 (G) defines a bounded operator where dg stands for a left Haar measure on G ([HR70, (40.26)]). Since these operators satisfy for p ≤ q the compatibility relation ϕ pq • π q (f ) = π p (f ), the product operator (π p (f )) p∈P G on p∈P G V p preserves the closed subspace lim ←− V p ∼ = V , so that we obtain a continuous linear operator π(f ) ∈ B(V ). This leads to a representation π : (L 1 (G), * ) → B(V ).
(18) (b) If G is abelian and G := Hom(G, T) is its character group, then we have the Fourier transform We define the spectrum of (π, V ) by i.e., the hull or cospectrum of the ideal ker π L 1 (G). Accordingly, we define the spectrum of an element v ∈ V by Remark A.6. In [Ar74,p. 225] it is shown that which implies in particular that V (E) is a closed subspace which is clearly π(G)invariant. Note that the condition supp( f ) ∩ E = ∅ means that f vanishes on a neighborhood of E.
Remark A.7. In [Jo82, Thm. 3.3] it is shown that for one-parameter groups of isometries of a Banach space X with infinitesimal generator A, the Arveson spectrum of an element v ∈ X coincides with the complement of the maximal open subset G ⊆ R for which the map z → (z − A) −1 v extends holomorphically to (C \ R) ∪ G (see also [Ta03, Ch. XI]). Definition A.10. Let g be a complete locally convex Lie algebra and x ∈ g be such that ad x generates a continuous equicontinuous one-parameter group π : R → Aut(g) of automorphisms, i.e., π is strongly differentiable with π ′ (0) = ad x. Then we obtain for each closed subset E ⊆ R a spectral subspace g C (E).
As we shall see below, the following lemma is an important technical tool.
Lemma A.13. For every closed subset E ⊆ G, the space V (E) is the intersection of the subspaces R(E + N ), where N is an identity neighborhood in G.
Proof. Since every identity neighborhood N in G contains a compact one, it suffices to consider the spaces R(E + N ) for identity neighborhoods N which are compact. Hence the assertion follows from [Ar74, Prop. 2.2] if V is a Banach space.
We first show that V (E) ⊆ R(E + N ). We know already that this is the case if V is Banach. To verify it in the general case, let v ∈ V (E) and U be a neighborhood of v. We have to show that U intersects R(E + N ). Let p ∈ P G be a G-invariant seminorm for which the ε-ball U p ε (v) = {w : p(v − w) < ε} is contained in U . Since the result holds for the G-representation (π p , V p ) on the Banach space V p and the image v p of v in this space belongs to V p (E), the open ε-ball U ε (v p ) in V p intersects R(E + N, π p ), hence contains a finite sum n j=1 π p (f j )w j with supp( f j ) ⊆ E + N compact. Since the image of V in V p is dense and U ε (v p ) is open, we may even assume that w j = v j,p for some v j ∈ V . Then π p (f j )v j,p < ε implies that n j=1 π(f j )v j ∈ U ∩ R(E + N ). We conclude that v ∈ R(E + N ) = R(E + N ).
We now arrive with Remarks A.8 and A.12 at because all sets E + N are closed.
Proposition A.14. Assume that (π j , V j ), j = 1, 2, 3 are continuous equicontinuous representations of the locally compact abelian group G on the complete locally convex spaces V j and that β : V 1 × V 2 → V 3 is a continuous equivariant bilinear map. Then we have for closed subsets E 1 , E 2 ⊆ G the relation Proof. Pick an identity neighborhood U 0 ⊆ G and choose a symmetric 1-neighborhood U with U U ⊆ U 0 . In view of Lemma A.13, the assertion follows, if we can show that To verify this relation, pick v 1 , v 2 ∈ V and f 1 , f 2 ∈ L 1 (R) such that the functions f j are compactly supported with supp( f j ) ⊆ E j + U . We have to show that β(π 1 (f 1 )v 1 , π 2 (f 2 )v 2 ) ∈ V 3 (E 1 + E 2 + U 0 ), i.e., that for any f 3 ∈ L 1 (R) for which f 3 vanishes on a neighborhood of E 1 + E 2 + U 0 , we have π 3 (f 3 )β(π 1 (f 1 )v 1 , π 2 (f 2 )v 2 ) = 0.
Remark A.15. Now we consider the case G = R and a strongly continuous representation (α, V ) of R by isometries. We know already from Lemma A.9 that V 0 := V ({0}) is the set of fixed points. We would like to have a decomposition where V ± are closed invariant subspaces satisfying for some δ > 0. We claim that this implies that the latter inclusions are equalities. In fact, for any three pairwise disjoint closed subsets E 1 , E 2 , E 3 , it follows from the relation and Remark A.8 that the sum V (E 1 ) + V (E 2 ) + V (E 3 ) is direct. Our assumption (19) now implies that V satisfies Suppose, conversely, that D is not invertibble and that the splitting condition is satisfied. Then im(D) = V + ⊕ V − is a closed subspace, V 0 = ker D, and V = im(D) ⊕ ker(D). Hence [Si70] implies that 0 is isolated in Spec(D).