On Convolution Dominated Operators

For a locally compact group $G$ we consider the algebra $CD(G)$ of convolution dominated operators on $L^{2}(G)$: An operator $A:L^2(G)\to L^2(G)$ is called convolution dominated if there exists $a\in L^1(G)$ such that for all $f \in L^2(G)$ $ |Af(x)| \leq a * |f| (x)$, for almost all $x \in G$. In the case of discrete groups those operators can be dealt with quite sufficiently if the group in question is rigidly symmetric. For non-discrete groups we investigate the subalgebra of regular convolution dominated operators $CD_{reg}(G)$. For amenable $G$ which is rigidly symmetric as a discrete group, we show that any element of $CD_{reg}(G)$ is invertible in $CD_{reg}(G)$ if it is invertible as a bounded operator on $L^2(G)$. We give an example of a symmetric group $E$ for which the convolution dominated operators are not inverse-closed in the bounded operators on $L^2(E)$.


Introduction
When one considers a convolution operator on the abelian group Z, then its matrix with respect to the canonical basis of l 2 (Z) is a Toeplitz matrix, i.e., it is constant along side diagonals. Conversely, a doubly infinite matrix, which is constant along its side diagonals and satisfies certain off-diagonal decay conditions, defines a convolution operator, when it is considered as acting with respect to the above basis. For the class of operators with decay in the sense of l 1 -summability N. Wiener [44] proved the Fourier transformed version of the following theorem Theorem 1.1 (Wiener's Lemma). If a two-sided infinite absolutely summable sequence a = (a(n)) n∈Z is invertible as a convolution operator on l 2 (Z), then the inverse is given by convolution with some b ∈ l 1 (Z).
Based on work of de Leeuw [9] and of Bochner and Phillips [8] on operator-valued Fourier transforms, several authors extended and applied Wiener's lemma to the case of matrices of operators, for which each side diagonal is uniformly bounded and these bounds are summable. The index set always had to be a discrete abelian group or a countable set [16], [42], [21], [17], [43], [1].
If G is a locally compact abelian group, for simplicity assumed to be compactly generated, then it admits a discrete co-compact subgroup H, and L p (G) = l p (H, L p (D)), where D is some fundamental domain. So vector valued Wiener Lemmata become applicable to classes of integral operators. Baskakov in [2,3,5,4] derives results for some of these. Shin and Sun in [41] give an account of those techniques.
If B is a Banach algebra and A a subalgebra then A is called spectral invariant in B when every element of A has the same spectrum in A as it has in B. In [26] and [27] Kurbatov shows among other results that for a locally compact abelian group the algebra of convolution dominated operators (see 2.1) is spectral in the bounded operators on L p (G). Farrell and Strohmer [12] extend this result to generalised Heisenberg groups (with compact center).
Plenty of work has been done with regularity assumptions on the kernel of an integral operator. Let us just mention interesting studies by Gröchenig and Klotz [18,19,20,24,25] on norm controlled inverse closedness of smoothness algebras in symmetric algebras. A comprehensive discussion of this important theme would go beyond the scope of this note.
In [13], together with Gröchenig we addressed the discrete nonabelian case using tools from abstract harmonic analysis to circumvent the restrictions of abelian Fourier transformation: Theorem 1.2. Let G be a discrete finitely generated group of polynomial growth. If a matrix A indexed by G satisfies the off-diagonal decay condition (1) |A(x, y)| ≤ a(xy −1 ), x, y ∈ G for some a ∈ ℓ 1 (G) and A is invertible on ℓ 2 (G), then there exists b ∈ ℓ 1 (G) such that |A −1 (x, y)| ≤ b(xy −1 ), x, y ∈ G, i. e. the algebra of matrices satisfying (1) is inverse-closed in B(L 2 (G)), the bounded operators on L 2 (G). Throughout this paper it is assumed that the involution of a Banach * -algebra is isometric. We recall the main line of the proof of this theorem, since this will be a guideline for us.
(i) The matrices satisfying (1) form a Banach * -algebra. We denote it by CD(G). (ii) Identify CD(G) with l 1 (G, l ∞ (G), T ), a twisted l 1 -algebra in the sense of Leptin [28], [29]. (iii) Based essentially on work of Leptin and Poguntke, in particular [35], we proved that CD(G) is symmetric Banach * -algebra. (iv) In a final step one relates the symmetry of a Banach * -algebra to the invertibility of certain of its elements. This is done with the help of Hulanicki's Lemma [22].
where σ A (a * a) denotes the spectrum of a * a in A.
(ii) Accordingly, a locally compact group G is called symmetric, if L 1 (G) is a symmetric Banach * -algebra. (iii) A locally compact (l.c.) group G is called rigidly symmetric if for any C * -algebra A the Banach * -algebra L 1 (G)⊗A is symmetric, where the projective tensor product L 1 (G)⊗A is endowed with its natural * -algebra structure.
In these notes we derive variants of Theorem 1.2 for integral operators on nondiscrete locally compact groups. This extension follows the ideas of the discrete case in [13], but requires non-trivial modifications. The special problem to be addressed is the measurability and integrability of certain kernels. We extend the results presented in the exposition [14] and provide proofs for them. Meanwhile some of our results have been reproved by I. Beltiţȃ and D. Beltiţȃ [7,6] and by Mȃntoiu [36] in the discrete case. After archiving (on arxiv) this paper A. R. Schep kindly informed us that our proposition 2.3 is a special case of a theorem in his dissertation [40].
It is not known yet if a symmetric group is already rigidly symmetric. An interesting example is the real ax + b group E, which is known to be a symmetric group but not symmetric as a discrete group [23]. We shall discuss a specific example of a non-symmetric algebra related to E showing non-symmetry of CD reg (E). So, in particular, CD reg (E) is not inverse-closed in B(L 2 (E)).

Convolution Dominated Operators
Let G be a locally compact group.
We define a norm on the space of convolution dominated operators by A CD = inf{ a 1 : (2) holds true } and denote by CD the normed linear space of convolution dominated operators.
It is clear that in this definition, necessarily, a ≥ 0 locally almost everywhere and that the CD-norm dominates the operator norm, in fact for 1 ≤ p ≤ ∞ and a ∈ L 1 (G) satisfying (2) It follows that A ∈ CD extends to a bounded operator on L p (G), 1 ≤ p < ∞ by continuity, and by duality to a bounded operator on L ∞ (G) too.
Proposition 2.2. With the involution of operators on L 2 (G) and composition of operators as product the space of convolution dominated operators is a Banach * -algebra.
Proof. Let A, B be convolution dominated operators and choose a, resp. b ∈ L 1 (G) according to (2). Then for f ∈ L 2 (G): From this and because L 1 (G) is an normed algebra under convolution it is clear that To see that the involution preserves the space CD take a ∈ L 1 (G) according to (2) and f, h ∈ L 2 (G): where a * (x) = a(x −1 )∆(x −1 ) is the involution on L 1 (G), and here, since a ≥ 0, a * (x) = a(x −1 )∆(x −1 ). Hence, |A * h(x)| ≤ a * ⋆ |h|(x) locally almost everywhere for all h ∈ L 2 (G), and it follows that A * ∈ CD with A * CD ≤ inf{ a 1 : To show that CD is a complete space we let (A i ) i∈N be a sequence in CD with i∈N A i CD convergent. Then we find a i ∈ L 1 (G) such that |A i f (x)| ≤ a i ⋆ |f |(x), for almost all x ∈ G and a i 1 ≤ A i CD + 2 −i . For A = i∈N A i this sum is convergent in the space of bounded operators, hence for f ∈ L 2 (G) and for a subsequence of the partial sums we have Af (x) = lim k

Proposition 2.3. For a given convolution dominated operator A there exists a locally integrable function F
Proof. let K, K ′ ⊂ G be compact sets and consider on C(K) × C(K ′ ) the form where f and h are extended as L 2 -functions vanishing outside K respectively K ′ . Take ϕ ∈ C cp (G), ϕ ≥ 0, then for f 1 , . . . , f n ∈ C(K) and h 1 , . . . , h n ∈ C(K ′ ) where ∆ denotes the modular function of G, ϕ * (z) = ϕ(z −1 )∆(z −1 ) and we write |K ′ | for the left Haar measure of the set K ′ . We take an approximate unit of L 1 (G) consisting of functions ϕ as above. Then we see that Hence f ⊗ h →< Af, h > extends to a linear form on the injective tensor product C(k) ⊗ ǫ C(K ′ ) = C(K × K ′ ), so there is a Borel measure µ such that for and by a computation much similar to the first part of the above one This last inequality extends to bounded measurable functions, and if H ⊂ K × K ′ is the characteristic function of a set of Haar measure 0 in G × G, then by Fubinis theorem, for almost all z ∈ G the function H(. , z) vanishes off a set of measure zero. The above estimate implies That is, µ is absolutely continuous with respect to the Haar measure on G × G, and by the Radon Nikodym theorem there exists a kernel F whenever f, h are continuous with support in K resp. K ′ . It is now a standard procedure to check the consistency of these kernels for different pairs of compact sets so that they define a locally integrable kernel F A on G × G, which represents A as claimed in the proposition.
Remark 2.4. In the above proof we have seen that in terms of the kernel F A of a convolution dominated operator A the inequality (2) may be rewritten as , locally almost everywhere (l.a.e.) Remark 2.5. Conversely, if F : G × G → C is a locally integrable function, such that for some a ∈ L 1 (G): a bounded operator on L 2 (G) can be defined, which clearly is dominated by λ(a). For this it suffices to check the Schur conditions ess sup x |F (x, y)| dy < ∞ and ess sup y |F (x, y)| dx < ∞.
Remark 2.6. The kernel of a convolution dominated operator satisfies the Schur conditions and hence represents the operator in the sense that for all f ∈ L 2 (G) the following integral converges l.a.e. and Now let a i , i ∈ N be a sequence such that (2.1) holds true for each i and such that a i 1 ≤ A cd + 2 −i . Then, for n ∈ N, let b n = a 1 ∧ . . . ∧ a n , where ∧ denotes the operation of taking the pointwise minimum of integrable functions. The functions b n are bounded below, and form a decreasing sequence in L 1 (G). Thus they converge to b, say. It is easily seen from (3) that this limit satisfies (2), further b 1 = A CD . We conclude: If G is a discrete group then an element A ∈ CD may be represented uniquely by its matrix with respect to the basis given by the unit masses placed at the group elements A(x, y) = (Aδ y |δ x ). Denote m z as the z-th side-diagonal of the matrix. Then the matrix is the direct sum of its side diagonals and therefore where D m ∈ B(l 2 (G)) is the multiplication operator with m ∈ l ∞ (G). In [13] this was used to show that is surjective from a certain twisted L 1 -algebra onto CD. In fact this map is an isometric * -algebra isomorphism. We shall next define the twisted L 1 -algebra and the map R, which unfortunately is no longer surjective. Let Ruc(G) denote the space of bounded right uniformly continuous functions, i. e. those F ∈ L ∞ (G) such that ess sup x∈G |f (y −1 x) − f (x)| → 0 as y → 0. (This definition of right uniform continuity follows [39,Ch. 3,1.8(vi)]) This space is just the subspace of L ∞ of those elements on which left translation acts norm continuously. It is a closed subspace containing only continuous functions.
For y ∈ G denote T y left translation on Ruc(G), that is T y n(z) = n(y −1 z) n ∈ Ruc(G). We consider the map T : y → T y as a homomorphism of G into the group of isometric automorphisms of the C * -algebra Ruc(G), which is continuous when the latter group is endowed with the strong operator topology. With this homomorphism we form the twisted L 1 -algebra L = L 1 (G, Ruc(G), T ) in the sense of Leptin [28,29,30]. The underlying Banach space of L is the space of Ruc(G)valued Bochner integrable functions on G, but we will often interpret it as the projective tensor product Thus for an element f ∈ L 1 (G, Ruc(G)) we denote its value in Ruc(G) by f (x), x ∈ G, and we write f (x)(z) or f (x, z) for the value of this Ruc-function at z ∈ G.
The twisted convolution of h, f ∈ L is defined by The properties of the projective tensor product ensure that R : a ⊗ m → λ(a) • D m extends to a norm-nonincreasing linear map from L 1 (G, Ruc(G)) to B(L 2 (G)).
This shows that R maps into CD and does not increase the norm as a map from L to CD.
As a consequence: R : L 1 (G, Ruc(G), T ) → CD is isometric. Definition 2.9. Elements in the image of R we call regular convolution dominated operators, and denote the whole image by CD reg .
The continuous functions vanishing at infinity C 0 (G) are a closed two-sided ideal in Ruc(G) and as is easily seen this implies that L 1 (G, C 0 (G), T ) is a closed twosided ideal in L 1 (G, Ruc(G), T ).
Remark 2.10. It follows from [35,Theorem 4] that L 1 (G, C 0 (G), T ) is simple and symmetric. The representation ρ in the beginning of their proof, we denoted it by R, maps L 1 (G, C 0 (G), T ) into an ideal (in CD reg ) of compact operators. Moreover the operator norm closure of R(L 1 (G, C 0 (G), T )) equals the compact operators [34].
3. Symmetry of the twisted L 1 -algebra In this section we shall show that the twisted L 1 -algebra L 1 (G, Ruc(G), T ) is a symmetric Banach * -algebra. To this end we shall first recall a criterion for the symmetry of a Banach * -algebra. Remark 3.2. If ρ is a contractive representation of A on a Banach space E then we are given a Banach A module in the sense of Leptin. The representation is preunitary in the above sense if the Banach A module is preunitary in the sense of Leptin [33].
It is clear that the image, under U, of a ρ(A)-invariant subspace is invariant under π(A). We may and do assume that U(E) is dense in H.
The following question appears naturally. If ρ is an algebraically irreducible, preunitary representation, can the representation π in the definition be chosen topologically irreducible? The answer is positive: Proposition 3.3. Let A be a Banach * -algebra with approximate identity. If ρ : A → B(E) is an algebraically irreducible, preunitary representation of A then there is a topologically irreducible * representation π extending ρ.
Proof. The representation ρ is preunitary, hence its kernel is a * -ideal, and possibly replacing A by A/kern(ρ) we may assume that ρ is faithful. Since an algebraically irreducible representation of a Banach * -algebra is equivalent to a contractive one, we may use from [32]: Following the concept used in the discrete case [13] one would like to define a map But this does not work since Q(f ) is not Bochner measurable because λ : G → B(L 2 (G)), is strongly continuous, but not norm continuous.
The problem can be worked around by showing, completely analogously to [37], that algebraically irreducible representations of certain twisted L 1 -algebras of G remain irreducible when "restricted" to the discretised group.
So, as before let G be a locally compact group, A a Banach * -algebra, with isometric involution and a left approximate identity. Further we assume that T : G → Aut(A) is a continuous homomorphism from G into the group of *automorphisms of A, where Aut(A) is endowed with the strong operator topology, i.e. y → T y a is continuous from G to A, for all a ∈ A. With these data we form the twisted L 1 -algebra (as above) L = L 1 (G, A, T ).
Let E be a nontrivial linear space and let ρ : L → B(E) be a non-trivial algebraically irreducible representation on it. Given ξ 0 ∈ E, ξ 0 = 0, one has a norm on E: with respect to which E is a complete space. In fact it is the quotient of L with respect to the maximal modular left ideal {f ∈ L : ρ(f )ξ 0 = 0}. Different ξ ∈ E define different but equivalent norms.
As in the proof of [30, Satz 3], ρ : G → B(E) and ρ : A → B(E) are representations of the group respectively of the Banach algebra A. The operators ρ(x), x ∈ G, are isometries and ρ does not increase norms. Here the operators ρ(x), x ∈ G and ρ(a), a ∈ A do not necessarily commute, but we have the relation ρ(y −1 )ρ(a)ρ(y) = ρ(T y − 1 a), and furthermore for all Unfortunately the proof in [30] is done with the hypothesis of dealing with a *representation. Apart from some algebraic identities the main ingredient is [30, Satz 2] in its consequence (1.2) loc.cit. . Now we take the group G with the discrete topology, denote it G d . We form the twisted L 1 -algebra L d := l 1 (G, A, T ), and define a representation of it on E by Proof. We follow the proof of [37,Theorem 2]. Assume that E ′ ⊂ E is a non-trivial ρ d (L d ) invariant subspace; we have to show that E ′ = E. To this end we take a fixed nonzero ξ 0 ∈ E ′ , and the corresponding norm, see (6), on E. Claim 1. We claim that for ξ ∈ E and ε > 0 there is h ∈ l 1 (G, A, T ) such that The claim implies the assertion of the lemma. For, if η ∈ E is given we have to find h ∈ L d such that ρ d (h)ξ 0 = η, and this is done inductively as follows: First we find h 1 ∈ L d such that If h 1 , . . . , h n are already defined with and Since It remains to establish the claim. So let δ > 0 be a positive real number to be determined later. By definition of the norm on E we find f ∈ L 1 (G, A, T ), with Since the space of continuous, compactly supported, Denote S the support of f 1 and |S| its Haar measure. Since ρ is a strongly continuous representation of G and since f is uniformly continuous, there is a neighbourhood U of the identity such that As S is compact it can be covered by finitely many translates x 1 U, . . . , x m U of U. We make this covering disjoint V 1 := x 1 U ∩ S and inductively V k := (x k U ∩ S) \ ∪ j<k V j . The V j are measurable pairwise disjoint subsets of S, hence m 1 |V j | ≤ |S|. Now let a j = f 1 (x j ) ∈ A, j = 1, . . . , m, and f 2 ∈ L 1 (G, A, T ) be given by Then Then Moreover, The second term can be estimated by For the first term we use that ρ : A → B(E) is bounded by one and that s ∈ V j can be written s = x j u with u ∈ U: Altogether we found h ∈ l 1 (G, A, T ) such that h l 1 (G,A,T ) ≤ ξ E + 2δ and ρ d (h)ξ 0 − ξ E ≤ δ( ξ E + 2δ); taking δ small enough now proves the claim.
From here onward we assume that A is a C * -algebra and that the operators T y are isometries. We recall that we assumed that they preserve the involution: T y (a * ) = (T y a) * , a ∈ A, y ∈ G. Before discussing the symmetry of L 1 (G, A, T ) we shall first look at the discretised version.
So let D : A → B(H) be a faithful * -representation of A on some Hilbert space H. We define a map Proposition 3.6. The map Q is an isometric * -isomorphism of ℓ 1 (G d , A, T ) onto a closed * -subalgebra of l 1 (G d )⊗B(H).
Proof. The proof rests on the isometric identification l 1 (G, E) = l 1 (G)⊗ E, which holds for any Banach space E [10, Ch. VIII.1.10]. It follows that for where l v = y∈G (T y n vy )m y −1 . Hence Similarly one computes that Q intertwines the involutions. In fact Thus Q is a * -homomorphism. Since Q is an isometry, the image of Q is a closed subalgebra of l 1 (G d )⊗ B(H).
Because symmetry is inherited by closed subalgebras, we obtain the following consequence.
Corollary 3.7. Let G be a discrete rigidly symmetric group, A a C * -algebra. Then l 1 (G d , A, T ) is a symmetric Banach * -algebra. Especially any of its algebraically irreducible representations is preunitary.
Theorem 3.8. If G is rigidly symmetric as a discrete group, and if A is a C *algebra, then L 1 (G, A, T ) is symmetric.
Proof. We shall verify that a non-trivial algebraically irreducible representation ρ : L 1 (G, A, T ) → End(E) is preunitary. We know that its discretised version ρ d : l 1 (G d , A, T ) → End(E) is preunitary too. So let H be the Hilbert space, π d the * -representation of l 1 (G d , A, T ) on it, and U the intertwining operator according to the definition 3.1. Since E is a complete space with respect to the norm as well as with respect to E , these norms are equivalent. Furthermore, h → π d (h)Uξ 0 = Uρ d (h)ξ 0 is bounded from l 1 (G d , A, T ) to H. These two facts show that U : E → H is bounded.
From [30,Satz 3] we know that there is a unitary representation π : G d → B(H) and a * -representation (again denoted by the same letter) π : A → B(H) such that Since ρ : G → B(E) is a continuous representation and U a continuous intertwining operator it follows that by means of π the group G acts continuously on the image U(E) in H. As this subspace is dense in H and π is a bounded representation we infer that this action is continuous on the whole space H. This allows to define a representation π of L = L 1 (G, A, T ) on H by It is easily checked that this formula defines a * -representation of L on H, and that U intertwines π and ρ. Corollary 3.9. If G is rigidly symmetric as a discrete group then the twisted algebra L 1 (G, Ruc(G), T ) is a symmetric Banach * -algebra.
is not symmetric. Hence in the above theorem it is necessary to assume that A is a C * -algebra and not only a symmetric Banach- * -algebra. See the example in section 5. (iii) A special role in the theory of symmetry of group algebras is played by the group of affine mappings of the real line, the "ax + b"-group, denoted E. This group has a symmetric L 1 -algebra [31], but its discretised version is not symmetric [23]. We do not know about the rigid symmetry of the continuous group.

Spectral Invariance of CD reg
We shall show that the spectrum of an element of CD reg is the same no matter if it is considered as a bounded operator on L 2 (G) or as an element of CD reg . To this end we define two representations of L 1 (G, Ruc(G), T ) and first show that they are weakly equivalent. The first representation is R : L 1 (G, Ruc(G), T ) → CD reg ⊂ B(L 2 (G), which we call the canonical representation. The second one is the D-regular representation λ D : L 1 (G, Ruc(G), T ) → B(L 2 (G, L 2 (G)), acting on L 2 (G, L 2 (G)). Proof. We identify L 2 (G, L 2 (G)) with L 2 (G × G). Let R ω be the extension of R from L 2 (G) to L 2 (G × G) by letting the operators R(f ), f ∈ L, act in the first coordinate only, i.e., for ξ ∈ L 2 (G × G) Next we define a candidate for an intertwining operator between the D-regular representation and the card(G)-multiple R ω of the canonical representation by Then on the one hand we have On the other hand Consequently, (11) λ D (f )(Sξ) = SR ω (f )ξ for all f ∈ L and ξ ∈ L 2 (G × G). Since S is unitary on L 2 (G × G), λ D and R ω are equivalent, whence R ω (f ) = λ D (f ) for all f ∈ L.
Now for ξ ∈ L 2 (G × G) write ξ z (x) = ξ(x, z): For the converse inequality let ϕ ∈ L 2 (G) be an element of norm one. Embed Corollary 4.2. Let G be an amenable group, which is rigidly symmetric as a discrete group. Then for f ∈ L r L (f * f ) = R(f ) 2 B(L 2 (G)) . Proof. We imposed the amenability on G to have that the largest C * -norm, denoted . * , on L is just given by the D-regular representation, since the representation D of Ruc is a maximal representation [30,Satz 6]. Therefore Since L is symmetric Ptaks theorem [38] asserts Theorem 4.3. Let G be an amenable group, which is rigidly symmetric as a discrete group. Then for an operator A ∈ CD reg sp CD (A) = sp B(L 2 (G)) (A).
Proof. Since A = R(f ) for some f ∈ L, this follows from the above corollary by an application of Hulanicki's Lemma [15].

An example of Non Symmetry
A special role in the theory of symmetry of group algebras is played by the group of affine mappings of the real line, the "ax + b"-group. This group has a symmetric L 1 -algebra [33], but its discretised version is not symmetric [23]. We do not know about the rigid symmetry of the continuous group. We shall consider the connected component of the identity.
The multiplication is (a, b) · (a ′ , b ′ ) = (aa ′ , ab ′ + b) and the action on R: (a, b) : The left Haar measure is da a 2 db. Proof. We identify the normal subgroup N of translations with R, and by means of the exponential map identify R with the subgroup of dilations. Explicitly: Let ω : (s, x) → e s x from R × R → R. The action of R on C 0 (R) is given by u → u s := u(ω(s, ·)). The additive group R acts on L 1 (R, C 0 (R)) bỹ T (s)f (x)(·) = e −s f (ω(−s, x))(ω(s, ·)) ∈ C 0 (R), f ∈ L 1 (R, C 0 (R)), s, x ∈ R.
Here e s is the modulus of the action ω(s, ·) on R, with respect to translation invariant Lebesgue measure dx. From [35,Theorem 5] we know that the twisted algebra L 1 (R, L 1 (R, C 0 (R)),T ) is not symmetric. So the theorem will be proved if we show that this Banach- * algebra is isomorphic to a closed * -subalgebra of L 1 (E, Ruc(E), T ).
Now we can embed C 0 (R) → Ruc(E) by extending a function, already defined on the dilations to a function on E which is independent of the translation coordinate of a group element. Restricting the left translation to this subspace is just the action we denoted T ′ . So we obtain an embedding of L 1 (E, C 0 (R), T ′ ) into L 1 (E, Ruc(E), T ).
Proof. By the above theorem there is a selfadjoint operator in CD reg (E), with a non-real spectrum. So some operator in CD reg (E) + Cid is invertible in B(L 2 (E)) but not in CD reg (E) + Cid.