REGULAR REPRESENTATIONS OF

INFINITE–DIMENSIONAL GROUPS AND FACTORS. I

A. V. Kosyak and R. Zekri

Received: 26.08.99

Abstract. In this article we prove that the von Neumann algebra generated by the right regular representation of infinite-dimensional group of upper triangular matrices is factor if the corresponding Gaussian measure is quasi-invariant with respect to the left action of the initial group and when some technical conditions on the measure are valid.

Introduction. In Kyiv, researches on quasi-invariant measures on infinitedimensional groups was initiated by Yu. L. Daletsky [1]. In the big extent being influenced by Yu. L. Daletsky seminar, always organized with noncomparable enthusiasm, the first of authors was irreversibly attracted by quasi-invariant measures on infinite-dimensional groups and their representations.

This article is also devoted to the quasi-invariant measures on infinite-dimensi-onal groups, to the corresponding regular representations of these groups and factors, generated by these representations.

Apparently, an analog of the regular representation of infinite-dimensional groups (the current group) appeared for the first time in [2], [3]. An analog of the regular representation for any infinite-dimensional group G, using G− quasi-invariant measures on some complitions G̃ of such a group is defined in [4], [5], [6].

For locally-compact groups G, the following commutation theorem is known [7], 13.10.4 (see also [8], V. 3, Theorem 5):

The commutant of the right regular representation ρ of a locally compact group G is generated by operators of the left regular representation λ:

(𝔄_{G}^{ρ})′ = 𝔄_{G}^{λ}, where 𝔄_{G}^{ρ} = (ρ_{t} | t ∈ G)′′, 𝔄_{G}^{λ} = (λ_{t} | t ∈ G)′′,

moreover there is an intertwining operator J,

(Jf)(x) = (dh(x^{−1})/dh(x))^{1/2}\overline{f(x^{−1})},

such that Jρ_{t}J = λ_{t}, t ∈ G, and J𝔄_{G}^{ρ}J = 𝔄_{G}^{λ}, where dh is the left Haar measure.

In [9] the commutation theorem was proved for an analog of the regular representation of the current group, corresponding to the Wiener measure.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 22E65, 46L35.

Key words and phrases. Infinite dimensional group, regular representations, von Neumann algebra, factor.

1. Let us consider the group B_{0}^{∞} = lim ind_{n}B(n, ℝ) of finite upper triangular matrices of infinite order with units on the principal diagonal and the group B^{∞} of all such matrices:

B^{∞} = {I + x = I + ∑_{k<n} x_{kn}E_{kn}},

B_{0}^{∞} = {I + x ∈ B^{∞} | x − is finite},

where E_{kn}, 1 ≤ k, n < ∞ are matrix unities of infinite order. Let us define the Gaussian measure μ_{b} on the group B^{∞} in the following way:

dμ_{b}(x) = ⊗_{k<n}(b_{kn}/π)^{1/2} exp(−b_{kn}x_{kn}^{2})dx_{kn} = ⊗_{k<n}dμ_{b_{kn}}(x_{kn}),

where b = (b_{kn})_{k<n} is some set of positive numbers.

Let us denote by R and L the right and the left action of the group B^{∞} on itself: R_{s}(t) = ts^{−1}, L_{s}(t) = st, s, t ∈ B^{∞} and by Φ : B^{∞} → B^{∞}, Φ(I + x) = (I + x)^{−1} the inverse mapping. It is known [5] that

Lemma 1. μ_{b}^{R_{t}} ∼ μ_{b} ∀t ∈ B_{0}^{∞} for any set b = (b_{kn})_{k<n}.

Lemma 2. μ_{b}^{L_{t}} ∼ μ_{b} ∀t ∈ B_{0}^{∞} ⇔

S_{kn}^{L}(b) = ∑_{m=n+1}^{∞} b_{km}b_{nm}^{−1} < ∞ ∀k < n.

Lemma 3. μ_{b}^{L_{t}} ⊥ μ_{b} ∀t ∈ B^{∞}\e ⇔ S_{kn}^{L}(b) = ∞ ∀k < n.

Lemma 4. [10], [11]. If E(b) = ∑_{k<n} S_{kn}^{L}(b)b_{kn}^{−1} < ∞ then μ_{b}^{Φ} ∼ μ_{b}.

Remark. If μ_{b}^{Φ} ∼ μ_{b}, then μ_{b}^{L_{t}} ∼ μ_{b} ∀t ∈ B_{0}^{∞}.

Proof. It follows from the fact that the inversion Φ replace the right action by the left action: R_{t} ∘ Φ = Φ ∘ L_{t} ∀t ∈ B_{0}^{∞}. Indeed, μ^{f}(Δ) := μ(f^{−1}Δ) so (μ^{f})^{g} = μ^{f∘g}. Hence

μ_{b} ∼ μ_{b}^{R_{t}} ∼ (μ_{b}^{R_{t}})^{Φ} = μ_{b}^{R_{t}∘Φ} = μ_{b}^{Φ∘L_{t}} = (μ_{b}^{Φ})^{L_{t}} ∼ μ_{b}^{L_{t}} ∀t ∈ B_{0}^{∞}.
□

If μ_{b}^{R_{t}} ∼ μ_{b} and μ_{b}^{L_{t}} ∼ μ_{b} ∀t ∈ B_{0}^{∞} one can define in a natural way (see [4], [5]) an analogue of the right T^{R,μ_{b}} and the left T^{L,μ_{b}} regular representations of the group B_{0}^{∞} in the space H_{b} = L^{2}(B^{∞}, dμ_{b}):

T^{R,b}, T^{L,b} : B_{0}^{∞} → U(H_{b} = L_{2}(B^{∞}, dμ_{b})),
(T_{t}^{R,b}f)(x) = (dμ_{b}(xt)/dμ_{b}(x))^{1/2}f(xt),
(T_{t}^{L,b}f)(x) = (dμ_{b}(t^{−1}x)/dμ_{b}(x))^{1/2}f(t^{−1}x).

Let 𝔄^{R,b} = (T_{t}^{R,b} | t ∈ B_{0}^{∞})′′ and 𝔄^{L,b} = (T_{s}^{L,b} | s ∈ B_{0}^{∞})′′ be the von Neumann algebras, generated by the right T^{R,b} and by the left T^{L,b} regular representation of the group B_{0}^{∞}.

Theorem 1. [11]. If E(b) < ∞, then μ_{b}^{Φ} ∼ μ_{b}. In this case the left regular representation is well defined and the commutation theorem holds: (𝔄^{R,b})′ = 𝔄^{L,b}. Moreover, an operator J_{μ_{b}}, given by (J_{μ_{b}}f)(x) = (dμ_{b}(x^{−1})/dμ_{b}(x))^{1/2} \overline{f(x^{−1})} is an intertwining operator: T_{t}^{L,μ_{b}} = J_{μ_{b}}T_{t}^{R,μ_{b}}J_{μ_{b}}, t ∈ B_{0}^{∞} and J_{μ_{b}}𝔄^{R,b}J_{μ_{b}} = 𝔄^{L,b}.

If μ_{b}^{L_{t}} ⊥ μ_{b} ∀t ∈ B^{∞}\e one can’t define the left regular representation of the group B_{0}^{∞}. Moreover the following theorem holds

Theorem 2. [5]. The right regular representation T^{R,b} : B_{0}^{∞} → U(H_{b}) is irreducible if and only if μ_{b}^{L_{s}} ⊥ μ_{b} ∀s ∈ B_{0}^{∞}\e.

Corollary 1. The von Neumann algebra 𝔄^{R,b} is type I_{∞} factor if μ_{b}^{L_{s}} ⊥ μ_{b} ∀s ∈ B_{0}^{∞}\e.

Let us assume now that μ_{b}^{L_{s}} ∼ μ_{b} ∀s ∈ B_{0}^{∞}. In this case the right regular representation and the left regular representation of the group B_{0}^{∞} are well defined.

Our aim is to study when the von Neumann algebra 𝔄^{R,b} is factor, i.e.

𝔄^{R,b} ∩ (𝔄^{R,b})′ = {λ𝕀|λ ∈ ℂ^{1}}.

Since T_{t}^{L,b} ∈ (𝔄^{R,b})′ ∀t ∈ B_{0}^{∞}, then 𝔄^{L,b} ⊂ (𝔄^{R,b})′ hence

(1)  𝔄^{R,b} ∩ (𝔄^{R,b})′ ⊂ (𝔄^{L,b})′ ∩ (𝔄^{R,b})′ = (𝔄^{R,b} ∪ 𝔄^{L,b})′.

The last relation shows that 𝔄^{R,b} is factor if the representation

B_{0}^{∞} × B_{0}^{∞} ∋ (t, s) → T_{(t,s)}^{R,L,b} = T_{t}^{R,b}T_{s}^{L,b} ∈ U(H_{b})

is irreducible.

Let us denote by 𝔄^{R,L,b} the von Neumann algebra generated by the right and the left regular representations:

𝔄^{R,L,b} = (T^{R,b}(B_{0}^{∞}) ∪ T^{L,b}(B_{0}^{∞}))′′ = (𝔄^{R,b} ∪ 𝔄^{L,b})′′.

Recall [12] that a non necessary bounded self-adjoint operator A is affiliated to the von Neumann algebra M (notation A η M) if exp(itA) ∈ M, ∀t ∈ ℝ^{1}.

Let us denote

S_{kn}^{R,L}(b) = ∑_{m=n+1}^{∞} \frac{b_{km}}{S_{nm}^{L}(b)}, k < n.

Theorem 3. The representation

B_{0}^{∞} × B_{0}^{∞} ∋ (t, s) → T_{(t,s)}^{R,L,b} = T_{t}^{R,b}T_{s}^{L,b} ∈ U(H_{b})

is irreducible if S_{kn}^{R,L}(b) = ∞, ∀k < n.

Using the inclusion (1), we have (see also [13], [14])

Corollary 2. The von Neumann algebra 𝔄^{R,b} is factor if S_{kn}^{R,L}(b)=∞, ∀k <n.

Proof of the Theorem 3. We will prove by induction that x_{kn} η 𝔄^{R,L,b} i.e. that operators x_{kn} of multiplication by the independent variables: H_{b} ∋ f(x) ↦ x_{kn}f(x) ∈ H_{b} in the space H_{b} = L^{2}(B^{∞}, dμ_{b}) are affiliated to the von Neumann algebra 𝔄^{R,L,b} if S_{kn}^{R,L}(b) = ∞, ∀k < n.

Using the explicite form of the generators A_{km}^{R,b} and A_{km}^{L,b} (see [5])

(2)  A_{km}^{R,b} = \frac{d}{dt}T_{I+tE_{km}}^{R,b}|_{t=0} = ∑_{r=1}^{k−1} x_{rk}D_{rm} + D_{km}, k < m,

(3)  A_{km}^{L,b} = −\frac{d}{dt}T_{I+tE_{km}}^{L,b}|_{t=0} = ∑_{r=m+1}^{∞} x_{mr}D_{kr} + D_{km}, k < m,

where D_{km} := ∂/∂x_{km} − b_{km}x_{km}, and denoting by < f_{n} | n = 1, 2, ... > the closure of the linear space, generated by the set of vectors {f_{n}}_{n=1}^{∞} in a Hilbert space H we will show, for example, that

Lemma 5. x_{12} η 𝔄^{R,L,b} if S_{12}^{R,L}(b) = ∞. Moreover x_{12}1 ∈< A_{1n}^{R,b}(A_{2n}^{R,b} − A_{2n}^{L,b})1 | 3 ≤ n >, if and only if S_{12}^{R,L}(b) = ∞. In this case we have also D_{1n}, D_{2n+1} η 𝔄^{R,L,b}, n = 2, 3, ...

Proof. Indeed, using formulas (2) and (3), we have

A_{1n}^{R,b}(A_{2n}^{R,b} − A_{2n}^{L,b}) = D_{1n}(x_{12}D_{1n} + D_{2n} − (D_{2n} + ∑_{m=n+1}^{∞} x_{nm}D_{2m}))

= x_{12}D_{1n}^{2} − D_{1n} ∑_{m=n+1}^{∞} x_{nm}D_{2m}.

So

(4)  \begin{matrix}min_{\{t_{n}\}} ∥(∑_{n=3}^{m} t_{n}A_{1n}^{R,b}(A_{2n}^{R,b} − A_{2n}^{L,b}) − x_{12})1∥_{H_{b}}^{2} ≍ \frac{1}{∑{m \atop n=3} \frac{b_{1n}}{b_{1n}+S_{2n}^{L}(b)}} → 0, \\ if m → ∞. \end{matrix}

To prove the last estimation we construct a family F_{m}^{b}, m = 1, 2, ... of Fourier-Wiener transform and use the following equality (see [5] p. 25)

(5)  min_{\{t_{n}\}}\{∑_{n=1}^{m} t_{n}^{2}a_{n} | ∑_{n=1}^{m} t_{n}b_{n} = 1\} = \frac{1}{∑{m \atop n=1} \frac{b_{n}^{2}}{a_{n}}} ,

see also [15], § 26 bibliography, p. 79:

min_{\{t_{n}\}}\{∑_{n=1}^{m} x_{n}^{2}c_{n} | ∑_{n=1}^{m} x_{n} = 1\} = \frac{1}{∑{m \atop n=1} \frac{1}{c_{n}}} .

REPRESENTATIONS OF INFINITE DIMENSIONAL GROUPS AND FACTORS 5

Let us denote by F_{kn}^{b} the one-dimensional Fourier transform, corresponding to the measure dμ_{kn}(x_{kn}) = dμ_{b_{kn}}(x_{kn}) = (b_{kn}/π)^{1/2} exp(−b_{kn}x_{kn}^{2})dx_{kn}:

L^{2}(ℝ^{1}, dx) ⟶^{F} L^{2}(ℝ^{1}, dy)
\sqrt{\frac{dμ_{b_{kn}}(x_{kn})}{dx_{kn}}} = U_{kn}^{b} ↑  ↑ U_{kn}^{b^{−1}} = \sqrt{\frac{dμ_{b_{kn}^{−1}}(y_{kn})}{dy_{kn}}} ,
L^{2}(ℝ^{1}, dμ_{b_{kn}}) ⟶^{F_{kn}^{b}} L^{2}(ℝ^{1}, dμ_{b_{kn}^{−1}})

where

\frac{dμ_{b_{kn}}(x_{kn})}{dx_{kn}} = \sqrt{\frac{b_{kn}}{π}} exp(−b_{kn}x_{kn}^{2}) and \frac{dμ_{b_{kn}^{−1}}(y_{kn})}{dy_{kn}} = \sqrt{\frac{1}{b_{kn}π}} exp(−\frac{y_{kn}^{2}}{b_{kn}}).

By definition F_{kn}^{b} = (U_{kn}^{b^{−1}})^{−1}F U_{kn}^{b}, where

(F f)(y) = \frac{1}{\sqrt{2π}} ∫_{ℝ^{1}} exp(iyx)f(x)dx,

so we have:

(F_{kn}^{b}f)(y_{kn}) = exp(\frac{y_{kn}^{2}}{2b_{kn}})\sqrt{\frac{b_{kn}}{2π}} ∫_{ℝ^{1}} f(x_{kn}) exp(iy_{kn}x_{kn}) exp(−\frac{b_{kn}x_{kn}^{2}}{2} )dx_{kn}.

Obviously, F_{kn}^{b}1 = 1, where 1 ∈ L^{2}(ℝ, μ_{b_{kn}}) is const.

Let us define for any m ∈ ℕ the Fourier-Wiener transform F_{m}^{b} = ⊗_{k≤m<n}F_{kn}^{b}. Denote by H_{m,b^{−1}} the space

H_{m,b^{−1}} = H_{B_{m},b} ⊗ H_{A_{m},b^{−1}} ⊗ H_{b}^{m},

where

H_{B_{m},b} = L^{2}(B_{m}, dμ_{B_{m},b}), B_{m} = {I + ∑_{k<n≤m} x_{kn}E_{kn}}, μ_{B_{m},b} = ⊗_{k<n≤m}μ_{b_{kn}},

H_{A_{m},b} = L^{2}(A_{m}, dμ_{A_{m},b}), A_{m} = {I + ∑_{k≤m<n} x_{kn}E_{kn}}, μ_{A_{m},b} = ⊗_{k≤m<n}μ_{b_{kn}},

H_{A_{m},b^{−1}} = L^{2}(A_{m}, dμ_{A_{m},b^{−1}}), μ_{A_{m},b^{−1}} = ⊗_{k≤m<n}μ_{b_{kn}^{−1}},

H_{b}^{m} = L^{2}(B^{m}, dμ_{b}^{m}), B^{m} = {I + ∑_{m<k<n} x_{kn}E_{kn}}, μ_{b}^{m} = ⊗_{m<k<n}μ_{b_{kn}}.

Lemma 6. F_{m}^{b}, m = 2, 3, ... is an isometrie between two spaces:

H_{b} = L^{2}(B^{∞}, dμ_{b}) = H_{B_{m},b} ⊗ H_{A_{m},b} ⊗ H_{b}^{m} and

H_{m,b^{−1}} = H_{B_{m},b} ⊗ H_{A_{m},b^{−1}} ⊗ H_{b}^{m}.

6 A. V. KOSYAK AND R. ZEKRI

In the sequal we shall need the follwing formulas for momentum of the Gaussian measure. Let F_{a}(y) = \sqrt{\frac{a}{π}} ∫_{ℝ^{1}} exp(ixy) exp(−ax^{2})dx. Since

F_{a}(y) = exp(−\frac{y^{2}}{4a}) = ∑_{n=0}^{∞}(−1)^{n} \frac{y^{2n}}{(4a)^{n}n!} = ∑_{n=0}^{∞} \frac{(−1)^{n}(2n)!}{(4a)^{n}n!} \frac{y^{2n}}{(2n)!},

we have

(6) \frac{d^{2n}F_{a}(0)}{dy^{2n}} = \sqrt{\frac{a}{π}} ∫ (ix)^{2n} exp(−ax^{2})dx = \frac{(−1)^{n}(2n)!}{(4a)^{n}n!} = \frac{(−1)^{n}(2n − 1)!!}{(2a)^{n}}.

Analogously

F_{a^{−1}}(y) = \frac{1}{\sqrt{aπ}} ∫ exp(ixy) exp(−\frac{x^{2}}{a})dx = exp(−\frac{ay^{2}}{4} )

= ∑_{n=0}^{∞}(−1)^{n} (\frac{a}{4})^{n} \frac{y^{2n}}{n!} = ∑_{n=0}^{∞} \frac{(−1)^{n}a^{n}(2n)!}{4^{n}n!} \frac{y^{2n}}{(2n)!},

so

(7)  \begin{aligned}\frac{d^{2n}F_{a^{−1}}(0)}{dy^{2n}} & = \frac{1}{\sqrt{aπ}} ∫ (ix)^{2n} exp(−\frac{x^{2}}{a})dx \\ & = \frac{(−1)^{n}a^{n}(2n)!}{4^{n}n!} = (−1)^{n} (\frac{a}{2})^{n} (2n − 1)!!. \end{aligned}

It is easy to see that for any m = 2, 3, ... we have

(8)  Ã_{pq}^{R,b} := F_{m}^{b}A_{pq}^{R,b}(F_{m}^{b})^{−1} = ∑_{r=1}^{p−1} x_{rp}y_{rq} + y_{pq}, p ≤ m < q,
(9)  Ã_{pq}^{L,b} := F_{m}^{b}A_{pq}^{L,b}(F_{m}^{b})^{−1} = y_{pq} + ∑_{r=q+1}^{∞} x_{qr}y_{pr}, p ≤ m < q.

Particulary, for m = 2 we have:

Ã_{1n}^{R,b} = F_{2}^{b}A_{1n}^{R,b}(F_{2}^{b})^{−1} = y_{1n}, Ã_{2n}^{R,b} = F_{2}^{b}A_{2n}^{R,b}(F_{2}^{b})^{−1} = x_{12}y_{1n} + y_{2n},

Ã_{2n}^{L,b} = F_{2}^{b}A_{2n}^{L,b}(F_{2}^{b})^{−1} = y_{2n} + ∑_{r=n+1}^{∞} x_{nr}y_{2r}, 3 ≤ n.

So

\begin{aligned}Ã_{1n}^{R,b}(Ã_{2n}^{R,b} − Ã_{2n}^{L,b}) & = y_{1n}(x_{12}y_{1n} + y_{2n} − (y_{2n} + ∑_{r=n+1}^{∞} x_{nr}y_{2r})) \\ & = x_{12}y_{1n}^{2} − y_{1n} ∑_{r=n+1}^{∞} x_{nr}y_{2r}. \end{aligned}

Using (6) and (7) we have finaly, if ∑_{n=3}^{m} t_{n}My_{1n}^{2} = ∑_{n=3}^{m} t_{n}2^{−1}b_{1n} = 1,

∥ [∑_{n=3}^{m} t_{n}A_{1n}^{R,b}(A_{2n}^{R,b} − A_{2n}^{L,b}) − x_{12}] 1∥_{H_{b}}^{2}

= ∥ [∑_{n=3}^{m} t_{n}(x_{12}y_{1n}^{2} − y_{1n} ∑_{r=n+1}^{∞} x_{nr}y_{2r}) − x_{12}] 1∥_{H_{2,b^{−1}}}^{2}

= ∥ ∑_{n=3}^{m} t_{n} [x_{12}(y_{1n}^{2} − My_{1n}^{2}) − y_{1n} ∑_{r=n+1}^{∞} x_{nr}y_{2r}] 1∥_{H_{2,b^{−1}}}^{2}

= ∑_{n=3}^{m} t_{n}^{2} [∥x_{12}∥^{2}∥y_{1n}^{2} − My_{1n}^{2}∥^{2} + ∥y_{1n}∥^{2}∥ ∑_{r=n+1}^{∞} x_{nr}y_{2r}∥^{2}]

= ∑_{n=3}^{m} t_{n}^{2} [\frac{2(2^{−1}b_{1n})^{2}}{2b_{12}} + \frac{b_{1n}}{8} ∑_{r=n+1}^{∞} \frac{b_{2r}}{b_{nr}} ] ≍ ∑_{n=3}^{m} t_{n}^{2}b_{1n}(b_{1n} + S_{2n}^{L}(b)).

Using (5) we have

δ_{12,m} = min_{t_{n}} ∥ [∑_{n=3}^{m} t_{n}A_{1n}^{R,b}(A_{2n}^{R,b} − A_{2n}^{L,b}) − x_{12}] 1∥_{H_{2,b}}^{2} ≍ \frac{1}{∑{m \atop n=3} \frac{b_{1n}}{b_{1n}+S_{2n}^{L}(b)}} .

So lim_{m→∞} δ_{12,m} = 0 if and only if ∑_{n=3}^{∞} \frac{b_{1n}}{b_{1n}+S_{2n}^{L}(b)} ≍ ∑_{n=3}^{∞} \frac{b_{1n}}{S_{2n}^{L}(b)} = S_{12}^{R,L}(b) = ∞. Finally we have

x_{12} η 𝔄^{R,L,b} and D_{1n}, D_{2n+1} = A_{2n+1}^{R} − x_{12}D_{1n+1} η 𝔄^{R,L,b}, n > 2.

We prove that we have convergence

A_{m_{1},m_{2}} = ∑_{m_{1}}^{m_{2}} t_{n}D_{1n}(A_{2n}^{R,b} − A_{2n}^{L,b}) → x_{12}, if m_{2} → ∞

in the strong resolvent sence. This means that

exp(itA_{m_{1},m_{2}}) → exp(itx_{12}) ∀t ∈ ℝ^{1}.

By Theorem VIII, 25 of [16], it suffices to show the convergnce A_{m_{1},m_{2}}f → x_{12}f for any f ∈ D, where D is a common essential domain for all the operators A_{m_{1},m_{2}} and A = x_{12}. For the role of D we choose a dense set consisting of finite linear combinations of arbitrary monomials

x^{α} = x_{12}^{α_{12}}x_{13}^{α_{13}}x_{23}^{α_{23}} · ... · x_{1n}^{α_{n−1n}} · ..., α_{ij} = 0, 1, ..., i < j.

Obviously D is the common essential domain for the operators A_{m_{1},m_{2}} and A, since D consists of analytic vectors for the operators A_{m_{1},m_{2}} and A. The function f ∈ D is cylindrical, so for some m_{1} this function f does not depend on the variables x_{1n}, ..., x_{n−1n} for n ≥ m_{1} and we have f = f_{m_{1}} ⊗ 1. Since

\begin{aligned}Ã_{1n}^{R,b}(Ã_{2n}^{R,b} − Ã_{2n}^{L,b})f̃ & = (x_{12}y_{1n}^{2} − y_{1n} ∑_{r=n+1}^{∞} x_{nr}y_{2r})f̃_{m_{1}} ⊗ 1^{m_{1}} \\ & = x_{12}f̃_{m_{1}} ⊗ y_{1n}^{2}1^{m_{1}} − f̃_{m_{1}} ⊗ (y_{1n} ∑_{r=n+1}^{∞} x_{nr}y_{2r}1^{m_{1}}), \end{aligned}

and

\begin{aligned}Ã_{1n}^{R,b}(Ã_{2n}^{R,b} − Ã_{2n}^{L,b})1 & = (x_{12}y_{1n}^{2} − y_{1n} ∑_{r=n+1}^{∞} x_{nr}y_{2r})1_{m_{1}} ⊗ 1^{m_{1}} \\ & = x_{12}1_{m_{1}} ⊗ y_{1n}^{2}1^{m_{1}} − 1_{m_{1}} ⊗ (y_{1n} ∑_{r=n+1}^{∞} x_{nr}y_{2r}1^{m_{1}}), \end{aligned}

we have

min_{{t_{n}}} ∥( ∑_{n=m_{1}}^{m_{2}} t_{n}A_{1n}^{R,b}(A_{2n}^{R,b} − A_{2n}^{L,b}) − x_{12})f∥_{H_{b}}^{2}

≍ min_{{t_{n}}} ∥( ∑_{n=m_{1}}^{m_{2}} t_{n}A_{1n}^{R,b}(A_{2n}^{R,b} − A_{2n}^{L,b}) − x_{12})1∥_{H_{b}}^{2} → 0 if m_{2} → ∞.

In addition

∑_{n=0}^{∞} \frac{t^{n}∥A_{m_{1},m_{2}}^{n}f∥}{n!} < ∞ ⇔ ∑_{n=0}^{∞} \frac{t^{n}∥A_{m_{1},m_{2}}^{n}1∥}{n!} < ∞ for some t > 0.

This means that it suffices to prove that 1 is analytic vector for the operators A_{m_{1},m_{2}}. It is evident taking into account (6) and (7). Analogously, any vector f ∈ D is analytic for the operator A. □

Lemma 7. x_{23} η 𝔄^{R,L,b} if S_{23}^{R,L}(b) = ∞. Moreover x_{23}1 ∈< D_{2n}(A_{3n}^{R,b} − A_{3n}^{L,b})1 | 4 ≤ n >, if and only if S_{23}^{R,L}(b) = ∞.

Proof. Using Fourier-Wiener image Ã_{pn}^{R,b} = F_{3}^{b}A_{pn}^{R,b}(F_{3}^{b})^{−1}, and Ã_{pn}^{L,b} = F_{3}^{b}A_{pn}^{L,b}(F_{3}^{b})^{−1}, 1 ≤ p ≤ 3, p < n, we have (see (8) and (9)):

Ã_{1n}^{R,b} = y_{1n},  A_{1n}^{L,b} = y_{1n} + ∑_{r=n+1}^{∞} x_{nr}y_{1r},

Ã_{2n}^{R,b} = x_{12}y_{1n} + y_{2n},  A_{2n}^{L,b} = y_{2n} + ∑_{r=n+1}^{∞} x_{nr}y_{2r},

Ã_{3n}^{R,b} = x_{13}y_{1n} + x_{23}y_{2n} + y_{3n},  A_{3n}^{L,b} = y_{3n} + ∑_{r=n+1}^{∞} x_{nr}y_{3r}.

REPRESENTATIONS OF INFINITE DIMENSIONAL GROUPS AND FACTORS 9

Hence

\begin{aligned}D̃_{2n}(Ã_{3n}^{R,b} − Ã_{3n}^{L,b}) & = (Ã_{2n}^{R,b} − x_{12}Ã_{1n}^{R,b})(Ã_{3n}^{R,b} − Ã_{3n}^{L,b}) \\ & = y_{2n}(x_{13}y_{1n} + x_{23}y_{2n} − ∑_{r=n+1}^{∞} x_{nr}y_{3r}). \end{aligned}

So if ∑_{n=4}^{m} t_{n}My_{2n}^{2} = ∑_{n=4}^{m} t_{n}2^{−1}b_{2n} = 1, then

∥ [∑_{n=4}^{m} t_{n}D_{2n}(A_{3n}^{R,b} − A_{3n}^{L,b}) − x_{23}] 1∥_{H_{b}}^{2}

= ∥ [∑_{n=4}^{m} t_{n}(y_{2n}(x_{13}y_{1n} + x_{23}y_{2n} − ∑_{r=n+1}^{∞} x_{nr}y_{3r})) − x_{23}] 1∥_{H_{3,b^{−1}}}^{2}

= ∥ ∑_{n=4}^{m} t_{n} [x_{13}y_{1n}y_{2n} + x_{23}(y_{2n}^{2} − My_{2n}^{2}) − y_{2n} ∑_{r=n+1}^{∞} x_{nr}y_{3r}] 1∥_{H_{3,b^{−1}}}^{2}

= ∑_{n=4}^{m} t_{n}^{2} [\frac{2^{−1}b_{1n}2^{−1}b_{2n}}{2b_{13}} + \frac{2(2^{−1}b_{2n})^{2}}{2b_{23}} + \frac{b_{2n}}{8} ∑_{r=n+1}^{∞} \frac{b_{3r}}{b_{nr}} ]

≍ ∑_{n=4}^{m} t_{n}^{2}b_{2n}(b_{1n} + b_{2n} + S_{3n}^{L}(b)).

Using (5) and the inequality b_{1n} < b_{2n} for large n, since S_{12}^{L}(b) = ∑_{n=3}^{∞} \frac{b_{1n}}{b_{2n}} < ∞ we have

\begin{aligned}δ_{23,m} & = min_{t_{n}} ∥ [∑_{n=4}^{m} t_{n}D_{2n}(A_{3n}^{R,b} − A_{3n}^{L,b}) − x_{23}] 1∥_{H_{3,b}}^{2} ≍ \frac{1}{∑{m \atop n=4} \frac{b_{2n}}{b_{1n}+b_{2n}+S_{3n}^{L}(b)}} \\ & < \frac{1}{∑{m \atop n=4} \frac{b_{2n}}{2b_{2n}+S_{3n}^{L}(b)}} → 0, m → ∞ ⇔ S_{23}^{R,L}(b) = ∞. \end{aligned}

By analogy with the proof of the Lemma 6 we conclude that we have convergence

∑_{m_{1}}^{m_{2}} t_{n}D_{2n}(A_{3n}^{R,b} − A_{3n}^{L,b}) → x_{23}

in the strong resolvent sence. □ Lemma 8. x_{13} η 𝔄^{R,L,b} if S_{13}^{R,L}(b) = ∞ and S_{23}^{R,L}(b) = ∞. Moreover x_{13}1 ∈< D_{1n}(A_{3n}^{R,b} − A_{3n}^{L,b} − x_{23}D_{2n})1 | 4 ≤ n >, if and only if S_{13}^{R,L}(b) = ∞. Proof. Indeed,

\begin{matrix}D̃_{1n}(Ã_{3n}^{R,b} − Ã_{3n}^{L,b} − x_{23}D̃_{2n}) = y_{1n}(x_{13}y_{1n} + x_{23}y_{2n} + y_{3n} \\ − (y_{3n} + ∑_{r=n+1}^{∞} x_{nr}y_{3r}) − x_{23}y_{2n}) = y_{1n}(x_{13}y_{1n} − ∑_{r=n+1}^{∞} x_{nr}y_{3r}). \end{matrix}

10 A. V. KOSYAK AND R. ZEKRI

So if ∑_{n=4}^{m} t_{n}My_{1n}^{2} = ∑_{n=4}^{m} t_{n}2^{−1}b_{1n} = 1, then

∥ [∑_{n=4}^{m} t_{n}D_{1n}(A_{3n}^{R,b} − A_{3n}^{L,b} − x_{23}D_{2n}) − x_{13}] 1∥_{H_{b}}^{2}

= ∥ [∑_{n=4}^{m} t_{n}(x_{13}y_{1n}^{2} − y_{1n} ∑_{r=n+1}^{∞} x_{nr}y_{3r})) − x_{13}] 1∥_{H_{3,b^{−1}}}^{2}

= ∥ [∑_{n=4}^{m} t_{n}(x_{13}(y_{1n}^{2} − My_{1n}^{2}) − y_{1n} ∑_{r=n+1}^{∞} x_{nr}y_{3r}] 1∥_{H_{3,b^{−1}}}^{2}

= ∑_{n=4}^{m} t_{n}^{2} [∥x_{13}∥^{2}∥y_{1n}^{2} − My_{1n}^{2}∥^{2} + ∥y_{1n}∥^{2}∥ ∑_{r=n+1}^{∞} x_{nr}y_{3r}∥^{2}]

= ∑_{n=4}^{m} t_{n}^{2} [\frac{2(2^{−1}b_{1n})^{2}}{2b_{13}} + \frac{b_{1n}}{8} ∑_{r=n+1}^{∞} \frac{b_{3r}}{b_{nr}} ] ≍ ∑_{n=4}^{m} t_{n}^{2}b_{1n}(b_{1n} + S_{3n}^{L}(b)).

Using (5) we have

\begin{matrix}δ_{13,m} = min_{t_{n}} ∥ [∑_{n=4}^{m} t_{n}D_{1n}(A_{3n}^{R,b} − A_{3n}^{L,b} − x_{23}D_{2n}) − x_{13}] 1∥_{H_{b}}^{2} \\ ≍ \frac{1}{∑{m \atop n=4} \frac{b_{1n}}{b_{1n}+S_{3n}^{L}(b)}} → 0, m → ∞ ⇔ S_{13}^{R,L}(b) = ∑_{n=4}^{∞} \frac{b_{1n}}{S_{3n}^{L}(b)} = ∞. \end{matrix}

By analogy with the proof of the Lemma 6 we conclude that we have convergence

∑_{m_{1}}^{m_{2}} t_{n}D_{1n}(A_{3n}^{R,b} − A_{3n}^{L,b} − x_{23}D_{23}) → x_{13}

in the strong resolvent sence.

□

Finally we have proved the basis of induction. If S_{12}^{R,L}(b) = ∞, S_{13}^{R,L}(b) = ∞, S_{13}^{R,L}(b)=∞ then

x_{12}, x_{13}, x_{23} η 𝔄^{R,L,b}, and D_{kn} η 𝔄^{R,L,b}, k < n, k ≤ 3. The induction step. Let us suppose that for some p holds

(10)  (x_{kn})_{k<n≤p} η 𝔄^{R,L,b}, and D_{kn} η 𝔄^{R,L,b}, k < n, k ≤ p.

We prove that the property (10) holds also for p + 1 if S_{kp+1}^{R,L}(b) = ∞, k < p + 1. Lemma 9. x_{kp+1} η 𝔄^{R,L,b}, k < p + 1 if S_{kp+1}^{R,L}(b) = ∞, k < p + 1. Moreover x_{kp+1} ∈< D_{kn}(A_{p+1n}^{R,b} − A_{p+1n}^{L,b} − ∑_{r=k+1}^{p−1} x_{rp+1}D_{rn})1 | p + 1 < n > if and only if S_{kp+1}^{R,L}(b) = ∞, k < p + 1. In this case we have also D_{p+1q} η 𝔄^{R,L,b}, p + 1 < q.

REPRESENTATIONS OF INFINITE DIMENSIONAL GROUPS AND FACTORS 11

Proof. Let us approximate the variable x_{pp+1}. Since Ã_{p+1n}^{R} − Ã_{p+1n}^{L} =

(x_{1p+1}y_{1n} + x_{2p+1}y_{2n} + ... + x_{p−1p+1}y_{p−1n} + y_{p+1n} − (y_{p+1n} + ∑_{r=n+1}^{∞} x_{nr}y_{p+1n})),

we have

∥ [ ∑_{n=p+1}^{m} t_{n}D_{pn}(A_{p+1n}^{R,b} − A_{p+1n}^{L,b}) − x_{pp+1}] 1∥_{H_{b}}^{2}

= ∥ [ ∑_{n=p+1}^{m} t_{n}y_{pn}(∑_{r=1}^{p} x_{rp+1}y_{rn} − ∑_{r=n+1}^{∞} x_{nr}y_{p+1r}) − x_{pp+1}] 1∥_{H_{p+1,b^{−1}}}^{2}

= ∥ [ ∑_{n=p+1}^{m} t_{n}(y_{pn} ∑_{r=1}^{p−1} x_{rp+1}y_{rn} + x_{pp+1}(y_{pn}^{2} − My_{pn}^{2})

−y_{pn} ∑_{r=n+1}^{∞} x_{nr}y_{p+1r})] 1∥_{H_{p+1,b^{−1}}}^{2}

= ∑_{n=p+1}^{m} t_{n}^{2} [∥y_{pn}∥^{2} ∑_{r=1}^{p−1} ∥x_{rp+1}∥^{2}∥y_{rn}∥^{2} + ∥x_{pp+1}∥^{2}∥y_{pn}^{2} − My_{pn}^{2}∥^{2}

+∥y_{pn}∥^{2} ∑_{r=n+1}^{∞} ∥x_{nr}∥^{2}∥y_{p+1r}∥^{2}]

= ∑_{n=p+1}^{m} t_{n}^{2} [∑_{r=1}^{p−1} \frac{2^{−1}b_{pn}2^{−1}b_{rn}}{2b_{rp+1}} + \frac{2(2^{−1}b_{pn})^{2}}{2b_{pp+1}} + \frac{b_{pn}}{8} ∑_{r=n+1}^{∞} \frac{b_{p+1r}}{b_{nr}} ]

≍ ∑_{n=p+1}^{m} t_{n}^{2}b_{pn}(∑_{r=1}^{p} b_{rn} + S_{p+1n}^{L}(b)).

Using (5) and inequalities b_{rn} < b_{pn}, r < p for large n, n > m_{1}, since S_{rp}^{L}(b) = ∑_{n=p+1}^{∞} \frac{b_{rn}}{b_{pn}} < ∞, we have:

\begin{matrix}δ_{pp+1,m_{1}m_{2}} = min_{\{t_{n}\}} ∥ [ ∑_{n=m_{1}}^{m_{2}} t_{n}D_{pn}(A_{p+1n}^{R,b} − A_{p+1n}^{L,b}) − x_{pp+1}] 1∥_{H_{b}}^{2} \\ ≍ \frac{1}{∑{m_{2} \atop m_{1}} \frac{b_{pn}^{2}}{b_{pn}(∑_{r=1}^{p} b_{rn}+S_{p+1n}^{L}(b))}} < \frac{1}{∑{m_{2} \atop m_{1}} \frac{b_{pn}}{pb_{pn}+S_{p+1n}^{L}(b)}} → 0, if m_{2} → ∞. \end{matrix}

Let for some k, 1 < k ≤ p the variables x_{rp+1}, k < r ≤ p are approximated. We show that in this case one can approximate the variables x_{kp+1}. Indeed, by the same argument we have

x_{kp+1} ∈< D_{kn}(A_{p+1n}^{R,b} − A_{p+1n}^{L,b} − ∑_{r=k+1}^{p−1} x_{rp+1}D_{rn})1 | p + 1 < n >

if and only if S_{kp+1}^{R,L}(b) = ∞, k < p + 1.

□

Acknowledgments. The first of authors thanks Prof. Richard Zekri for his kind invitation to Institut de Mathématiques de Luminy (Marseille, France) in spring 1995 where this work was began. The interest towards the factor-representations of infinite-dimensional groups appeared after discussions with Prof. A. M. Vershik.

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Institute of Mathematics, Ukrainian National Academy of Sciences, 3 Tereshchenkivs’ka, Kyiv, 01601, Ukraine

E-mail address: kosyak@imath.kiev.ua

Université de la Mediterrannée et Institut de Mathématiques de Luminy, Faculté des sciences de Luminy, 163, Avenue de Luminy, 13288, Marseille Cedex 09, France

E-mail address: zekri@iml.univ-mrs.fr