奈良教育大学学術リポジトリNEAR
Representation Theory of a Locally Compact Group Associated with a von Neumann Algebra
著者 KAWAKAMI Satoshi
journal or
publication title
奈良教育大学紀要. 自然科学
volume 36
number 2
page range 1‑8
year 1987‑11‑25
URL http://hdl.handle.net/10105/2056
Representation Theory of a Locally Compact Group Associated with a von Neumann Algebra
Satoshi Kawakami
(Department of Mathematics, Nara University of Education, Nara 630, Japan) (Received April 25, 1987)
Abstract
The aim of the present paper is to attempt to establish a representation theory of a locally compact group G associated with a von Neumann algebra 〃 by ex‑
tending several notions in the usual theory of unitary representations of G. Espe‑
cially, by considering a spectrum and an inner spectrum, besides a multiplicity and an associated measure, of a representation of G to M, it is shown that all quasi‑reg‑
ular representations of a countable abehan group G to a finite factor M are com‑
pletely classified up to conjugacy.
Introduction
Let G be a separable locally compact group and 〟 be a von Neumann algebra acting on a separable Hilbert space H. We denote by U(M) the group of all unitary elements of 〟 equipped with the strong operator topology. Then, in this paper, a continuos homomorphism of G into U(M) is called a representation of G to M.
When M is the full operator algebra B(H) of all bounded operators on H, this is just the definition of an ordinary continuos unitary representation of G on H. In the ordinary representation theory, fundamental are such notions as unitary equiva‑
lence or quasi‑equivalence of representations, a multiplicity of a representation, and a measure appearing in a decomposion of a representation. The ordinary representa‑
tions are often described by using these notions. In section 1, we generalize these fundamental notions to the ones associated with a von Neumann algebra M, which includes ordinary ones as a special case that 〟‑β (〟). Then, we shall study gener‑
al properties among such introduced notions as conjugacy, multiplicities, and meas‑
ures of representations, see Proposition 1. 3. However, in our situation, each con‑
jugacy class of representations is in general too small to be characterized by only using the above notions and so we need another invariant of conjugacy. Thus, in section 2, we introduce such new notions as a spectrum Sp (u) and an inner spec‑
trum ISp (u) relative to the von Neumann algebra M as an invariant of conjugacy for a representation u of G to M and we show in Theorem 2. 7 that all quasi‑regu‑
2 Satoshi Kawakami
lar representattions of a discrete countable abelian group G to a finite factor M are completely decided up to conjugacy by these invariants, applying Kleppners work
[4] on multiplier representations and Ocneanu's vanishing theorem [7] on the second cohomology.
The author would like to express his hearty thanks to Professors O.Takenouchi and T.Jimbo for their warm encouragements and instructive suggestions and to Doc‑
tors Y. Katayama and T. Kajiwara for their frequent and stimulating discussions.
1. General notions
In this section, we attempt to establish a representation theory associated with a von Neumann algebra by extending some notions in the usual theory of unitary rep‑
resentations. In order to do so, we shall define conjugacy, inner conjugacy, multi‑
plicities, canonical decompositions, and the associated measures of representations reト ative to a von Neumann algebra.
Let M denote a von Neumann algebra acting on a separable Hilbert space H and U (M) denote the group of all unitary operators of M equipped with the strong op‑
erator topology. Let G be a separable locally compact group. Then, a continuos homomorphism u of G into U(M) is called a representation of G to M When M is of type I, namely, M‑B(H) for some Hilbert space H, this u is just an usual uni‑
tary representation of G onガ.
Definition 1. 1. (equivalence of representations) Let w, (i‑1, 2) be two represen‑
tations of G to M. Then, u, is called conjugate to Ma often denoted by u,‑u2, if there exists an automorphism a of M such that u2厄) ‑a(u,(g)) for all g∈G.
When the above a may be taken as an inner automorphism of M, Ui is called in‑
nerly conjugate to wa often denoted by U/‑u2. It is clear that these satisfy the
axiom of equivalence relation among representations of G to M
For a representation u of G to M, we prepare some notations as follows.
R (m) ‑the von Neumann subalgebra of M generated by the set {u 伝) ; g∈GI
Z(u) ‑R (ォ)'H# (ォ) ‑the center of i? (u)
N(u) ‑Z(u) fW‑the relative commutant of Z(u) in M
z(ォ) ‑N(ォ)′ nN(m) ‑the center of N(u)
Then, it is easy to show such inclusion relations to hold in general that 〟⊃Ⅳ(〟)⊃
R (u)⊃Z(m) and児(u)′⊃Z(u) ⊃Z(ォ). For a separable locally compact group G, G denotes the dual of G, namely, the set of unitary equivalence classes of all irreduci‑
ble representations of G, and G denotes the quasi‑dual of G, namely, the set of quasi‑equivalence classes of all factor representations of G, both of which may be assumed to be equipped with Mackey Borel structure [5]. Then, applying general
reduction theory of von Neumann algebras and representations ([1], [9]), we get the fallowings.
(A) Corresponding to the center Z (u) of R (u), there exists a standardカnite mea‑
sure fj. on G such that
Z{u) ≡L,∞ (a fi) and u= J冒uzdn(x),
(B) Corresponding to the abelian subalgebra Z (w) in R (u)′, there exists a stand‑
ardカnite measure space (F, v) such that
z(u) ≡L∞(r, u), n(u) ≡ J雷Ndレ(γ), and u≡ J雷uγdレ(γ).
(C) There exists a measurable map ￠; r‑+G, u♪ to null sets, such that ￠* (のis
equivalent to fi and uγ is quasi‑equivalent to uJ for each γ∈￠ Oc).
Definition 1. 2. (associated measure and multiplicity) In the above situation, the measures p. and v appearing in each factor decomposition of u are called the associa‑
ted measure of u with respect to the center Z(ォ) and to the relative center Z(u) respecively. Each representation gives rise to only an equivalence class of measures
in general. However, when M is a von Neumann algebra of finite type and a faith‑
ful normal normalized trace z of M is given, such measures 〟 and v are canonically de伝ned according to this trace r so that we denote // by z(u) and v by舌(ォ) in this case. Next, we shall give a family m (u) of multiplicities of u. For a factor repre‑
sentation u of G to a factor M of finite type, a multiplicity m (u) of u is defined by [M: R (w)];/ where [:] is Jones'index ([3]). We note that this definition does not contradict with an ordinary notion of a multiplicity when both 〟 and 〟 are factors of finite type I. For a representation u of G to a finite von Neumann algebra,
m(γ) ‑ [TV7:月(uγ)]7 is also defined forレーalmost all γ∈F because N7⊃R(uγ) is a pair of finite factors and so we write m (u) ‑ {m (γ))γ。r in this case.
Proposition 1. 3. Let u( (i‑l, 2) be two representations of a separable locally com‑
pact group G to a von Neumann algebra M. Then, using the above notations with
su飾es i‑l, 2, we get the followings.
(i) When G is of type I and M is a factor of type I, then, Z(m,) ‑Z(m,) (i‑l, 2) and the fallowings are mutually equivalent.
(a) u,≡u2 (b) ui…U2 (c) vi≡レ2 and m(u,) ‑m(u2) (ii) IfR (ォ;) ‑月(u2) ‑M, then, Ui≡u2 if and only ifHt芸fl*
(iii) Suppose that M is a finite von Neumann algebra with a faithful normal nor‑
malized trace z and u2‑a (uj) for some automorphism a of M. Then, x(ui) ≡r(u2) in general but if a keeps z invaγiant, r (u,) ‑z ¥u2). Moreover, a induces a canonical Borel isomorphism a of T, onto T2 up to null sets such that ￠i‑<t>2‑a,チ(u,) ‑d* (チ(ォ*)), and m(u,) ‑m(u2) ‑a.
Satoshi Kawakami
(iv) When G is a compact group and M is a type II, factor with the unique nom‑
alized trace z, the followings are mutually equivalent.
(a) u,^u2 (b) u,‑u2 (c) t(ui)‑z(u2)
Proof. The statement ( i ) only asserts a slight modification of some results in ordinary representation theory. The statement (ii) follows immediately from the fact that Ui is quasi‑equivalent to w* which is equivalent to say ut=u2 in the case
〟(ォ,)‑R(u2)‑M, if and only if fi,三fi2 (see [1]). The statement (iii) may be easily obtained by routine arguments and so we omit the details. The statement
(iv) follows from the facts that月(m) is an atomic subalgebra of M when G is a compact group and that any automorphism α keeps the trace T Of 〟 invariant.
[Q.E.D.]
We note that the associated measures z(u),千(ォ) and the multiplicity m (u) are
ones of conjugacy invariants of a representation u of G to M in such a sense as described in the above statement (iii). When M is a factor of type I, namely in
usual cases, each equivalence class of representations is decided by the associated measure and multiplicity as shown in the statement ( i ). However, in our general situation, it may happen that a number of mutually non‑conjugate representations have the same associated measure and multiplicity and so we need other conjugacy invariants of representations of G to M in order to classify them.
2. Spectra of a representation
In this section, we shall introduce other conjugacy invariants, a spectrum and an inner spectrum, of representations and we shall show that some families of represen‑
tations may be classified up to conjugacy owing to this introduced notion.
Let G be a separable locally compact group and denote by X(G) the abelian group of all unitary characters of G. Let 〟 be a von Neumann algebra and denote by Aut(M) the group of all automorphisms of M and by Int(M) the group of all inner automorphisms of M. Then, Int (M) is a normal subgroup of Aut (M).
For a representation u of G to M and a∈Aut(M), x∈X(G) is called an eigen‑
value of a with u if the following relation (*) is satisfied.
(*) a(uk)) ‑くx, g)uk) for any g∈G.
Thus, for a representation u of G to M, we define the sets Sp (w) and ISp yu), called
a spedγum and an inner spectrum of u respectively, by
S♪(u) {all eigenvalues of some a∈Aut(M) with u), ISp (u) ‑ {all eigenvalues of some a^Int (M) with u)
Then, we get the following lemma.
Lemma 2.1. For a representation u of G to M, Sp (u) and ISp (u) are subgroups of X (G), both of which are invariant under conjugate equivalence of representations.
Proof. It follows immediately from the definitions of Sp(u) and IS♪ (u) that they are subgroups of X (G), and so we only show their invariance under conjugate equivalence of representations. Let ut (i‑l, 2) be mutually conjugate representa‑
tions of G to M, namely, u2‑β(uj) for some β∈Aut(M). Take x∈Sp(u;). Then, for
such ui and x with the relation (*) for some a∈AutCM), we get βoa。β一°′∈Aul
(M) and (βoaoβ ') (u2反)) ‑<x, g)u2(g) for any g∈G, wich implies x∈Spiu2).
Hence, we see Sp (ォ,)⊂Sp (u2) so that Sp (u/) ‑Sp (u2) by the symmetry. Moreover,
if the above a is taken in Int(M), namely, a‑Adv for some v∈M, we obtain βoa。
β l‑Ado(v) and β(V) ∈M. Hence, x∈ISp (ui) implies x∈ISp(u2) and so ISp (u,) ‑ ISp (u2). [Q. E. D.]
Example 2.2. Let G be a discrete countable abelian group and (Q, //) be a pro‑
bability Lebesgue measure space. We assume that G acts on Q freely and ergodical‑
1y and moreover that this action α keeps the measure 〟 invariant. Then, it is well known that the von Neumann crossed product M‑L∞ (fl n)× tG is isomorphic onto the unique hyperfinite type IIi factor R for any such action a of G on L∞(ォn).
Moreover, we get a canonical representation 〟 of G and a normal representation T
of L∞ (Q, fi) to M satisfying the covariance relation u 伝) T(のu ky‑T(ag(f)) and
M is generated by uig) and Tif) ig∈G, f∈L認(Q fi)). We note that R(ォ) is a
maximal abelian subalgebra (often abbreviated as masa) of M. On the other hand,
weget another representation UofG on the Hilbertspace U(Q fi) by (U(g)h) (山)‑
h(ag'(&>)) for g∈G and h(co)∈U(Q, u). Then, Hahn [2] and Packer [8] show that R(u) is a regular masa of M if and only if U is decomposed as a direct sum
of unitary characters of G, namely U芸∑ Hx. In this case, H is a subgroup of G,
and it is easy to show that Sp(ォ) ‑G and ISp(u) ‑H. However, it always ho一ds
that r(u) ‑the normalized Haar measure of (ラ and m(u) ‑ [m(x)}x。 where m(x)
‑1 for all x∈G. Therefore, we may catch the difference among such representa‑
tions u of G to R, up to conjugacy, by considering inner spectra ISp (u) of u.
In a more concrete case that G‑Z, the integer group, Q‑T, one‑dimensional torus, 〃二‑the normalized Lebesgue measure on ㍗ and αβ‑the action of irrational ro‑
tation associated with an irrational real number ♂, we get such a representation 的 of G to R as described in the above. In this situation, we see that, for two irration‑
al real numbers 6 and 6', ue芸Uff if and only if 6…0′ mod 2nZ.
Example 2.3. Let G be also a discrete countable abehan group and take a count‑
able subgroup H of G. Denote the product abelian group GxH by P. then, a multi‑
plier c of P is canonically denned by
Satoshi Kawakami
C (k′. */), (g2, X2)) ‑くxi. 82) for毎h Xi) ∈GxH‑P, and denote the degenerate part of the multiplier c by So namely,
sc‑ {/>∈p; c(♪ q) ‑c(q, p) for all q∈Pl.
Let K be the annihilator of H in G, namely, K‑H⊥‑ k∈G; <g, x)‑1 for all x∈H).
Then, by simple calculations, we see Sc‑K in this case. For a unitary character y of K, we get a c‑representation Uy of P by inducing y from K to P in the sense of Mackey [6], namely, Uy‑c‑ind芸y. Then, the following results are obtained by app‑
lying Kleppner's results [4J.
(1) f/, is a finite factor representation of P, namely, R (Uy) is a factor of finite type.
(2) R(Uy) is of type IIi if and only if His not closed in (ラ.
(3) RWy) is of type ! (サ<+‑) if and only if His closed in G, and in this case, the equality n‑ H¥ ‑ G/Ki holds.
(4) Any factor c‑representation of P is given in such a form of c‑ind是y for
some y∈K up to quasi一七quivalence.
Here, we denote by uy the restriction of Uy to the suubgroup G of P. Then, it is easy to examine the fallowings when R (」/,) is a factor of type II
(a) R (Uy) is isomorphic to the hyperfinite type II】 factor R. Thus, uy is a repre‑
sentation of G to R.
(b) R(uy) is a regular masa of R.
(c) Sp{uy) ‑//‑the closure of H in G.
(d) ISp (uy) ‑H.
(e) r(uy) is the normalized measure concentrated on H+タin (; and invariant under a translation by an element of H, where y is an extension of y to an unitary character of G.
(f) m(uy) ‑ {m(x)}x。rwhere F‑H+y and m(x) ‑1 for all x∈r
Letォbe a representation of a locally compact abelian group G to a factor M.
For x∈ISp(m), denote by Mix) the set of all elements v of M such that a‑Ad iv) satisfies the condition (*) a (u 也)) ‑くx, g)u 伝) for all g∈G,
Definition 2.5. (quasi‑regular representation) In the above situation, we call u a quasi‑regular representation of G to M if R (u) is a masa of M and M is generated by M(x) where x runs over ISp (u).
Example 2.6. Let A be the regular representation of a locally compact abehan
group G on Lr (G) and denote the full operator algebra B(L2(G)) by M. Then, this
A is a quasiイegular representation of G to M with Sp (A) ‑ISp (A) ‑G.
For a discrete countable abehan group G, we may decide the conjugacy classes of quasi‑regular representations of G to a finite factor M as in the next theorem.
Theorem 2.7. Let G be a discrete countable abelian group and M be a type D, factor on a separable Hilbert space with the canonical normalized trace r. Then, for
two quasi‑regular representations u{ (i‑l, 2) of G to M, u, is conjugate to u2 if and only if z (w;) ‑z (u2) and ISp (m;) ‑ISp (u2). More precisely, each conjugacy class of quasi‑regular representations u of G to M is decided by a countable subgroup H of G
and a unitary character y of H⊥ such that supp (t(m)) ‑H+タand ISp (ォ) ‑H.
Proof. It is sufficient to show that any quasi‑regular representation of G to M is obtained in such a form as described in Example 2.4. Let u be a quasi‑regular representation of G to M and denote ISp (w) by H, For two unitary elements v and w of M such that v and w normalize the masa月(w) of M and‑Ad(f)‑Ad(w) on R (m), there exists a unitary element z in R (u) with v‑zw. Indeed, for such v and
w, wv∈月(u)′nM‑R(a). Thus, for each x∈H, taking v(x) and w(x) in M(x), we
see AdOOc)) ‑Ad(w(x)) on月(w) and so we may findzGO ∈U(R(u)) such that v Gc) ‑2(x)w (x). Here, for each x∈H, take and fix an element w (x) in Mix). Then, we get M(x) ‑U(R(u))w(x) and, for any x, and x2 in H, there exists z(x]t x2) ∈ U(R(u)) such that wCOw(x2) ‑z(xh x2)vo(xhx2) because both of w{x,)w(x2) and w (%iX2) lie in M(x,x2). Moreover, z (xlt x2) satisfies the condition of U(R (u))‑
valued 2‑cocycle on the group H.
Since 〟 is countably generated by the assumption, the standard Hubert space L2(M, r) is separable. By simple calculations, we get that, for x∈H and z∈U(R
(ォ)), z(zw(x)) ‑0 ifズ≠1. Thus, w (x) (x∈H) are mutually orthogonal vectors as
elements in L (M, r). Therefore, the cardinal number of H must be at most counta‑
ble.
Hence, we may apply the vanishing theorem on the second cohomology as in
Ocneanu's work [7, Therrem 7. 6] to the action aェ‑Ad (w 00) of the (at most) cou‑
ntable abelian group H on R (u) which is centrally free, and so we see that there exists b(x) ∈U(R (u)) such that
z (xj, x2) ‑aェ,(& (x2))*b (x,)*b Ocjxz) for xlt x2∈H・
Set v(x) ‑b(x)w (x), then, for x,, x2∈H, we have
v (xj) v (x2) ‑b (xi) w (xi) b (x2) w (x2) ‑b (x,)axl(b (x2)) w (xj) w (x2)
‑b (x,) α∫′(b (x2)) z (.ズi, x2) w (x,x2) ‑b (x,x2) w {x,x2) ‑v (x,x2).
This implies that v is a representation of H to M such that M is generated by u 伝) and v(x)厄∈G and x∈H) and
Satoshi Kawakami
v(x)uk) ‑くx, g)uk)v(x) forg∈G andズ∈H.
Set P‑Gx:H and U(g, x)ここuk)v(x). Then, we get, for (g′, x;),厄2, x2) in P,
U反′, x,) Uk2, X2) ‑U鹿′)v(x,)uk2)v(x2) ニーu k′)くxi, g2)u 伝;) V (X,) V (x2)
‑くxt, g2)u鹿′gi) V (X,エ2)
‑くx,, g2)U((g,, x,)厄 x2)).
This implies that U is a c‑representation of P such that U(P) ‑M is a type n factor, where c is a multiplier of P as described in Example 2.4. Thus, applying the results in Example 2.4, we get the desired conclusion. [Q. E. D.J
Remark. As shown in the above proof, each quasi‑regular representation u of a discrete countable abelian group G to a finite factor M is conjugate to the restriction of c‑ind芸y to G for a (at most) countable subgroup H of G and some y∈K where K
‑HL. Moreover, such a員nite factor M which has a quasi‑regular representation of G must be the hyper伝nite type Hi factor R or the finite type ! factor M(ォ, C). It seems to be a hard task to classify all representations of a locally compact group to a factor, up to conjugacy, in more general situations. We hope that some new tech‑
niques will be developed in future in order to solve them.
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[3] Jones, V.F.R言Index for sub factors, Invent. Math., 72 (1983), 1‑25.
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