CC BY
22
MSC 22D10, 22D15, 20C32
Irreducible decompositions of unitary representations and extremal decompositions of positive definite functions on groups
© H. Shimomura
Tokyo, Japan
This paper concerns with positive definite functions (PDF) 0 on the usual locally compact groups. A function 0 has an extremal decomposition through an irreducible decomposition of the unitary representation corresponding to 0 (bv the Gelfand-Naimark-Segal construction method). However there are other ways to get extremal decompositions, for example via the Choqet theorem. So it is interesting to find conditions which distinguish the natural particular one from other decompositions. We describe a necessary and sufficient condition for the above problem as well as an interesting negative example related to [3], [4].
Keywords: locally compact groups, unitary representations, positive definite functions
§ 1. Introduction
The subject of this paper is a study of extremal decompositions of continuous positive definite functions (PDF) 0 on a locally compact group G, A function 0 has a unitary representation U of G corresponding to it through the Gelfand-Naimark-Segal construction method and after through an irreducible decomposition of U by the Mautner method, we have a natural extremal decomposition of 0, However, other disintegrations of 0 are possible using, for example, the Choquet theorem. Hence,
0
the natural one described above. We describe a necessary and sufficient condition for the question as well as an interesting negative example related to [3], [4].
§ 2. Irreducible decompositions of unitary representations and extremal decompositions of PDF
2.1. Presentation of the problem. At the beginning of this section, we outline
G
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group, and (H, U) be a continuous unitary representation of G with a normalized cyclic vector v, It is well-known that (H, U) can be decomposed irreducibly to {(Ha, UA)}AeR according to a factor decomposition of the ring M generated by U(g), g e G, and a maximal Abelian ring A of M (cf, [1], p. 6):
u(g) - E UA(g) for all g e G, with a weight function a (A) on R (cf, [2]), Then, we have
< U(g)v, v >h = / < UA(g)vA,VA >ha da(A) with v = vx^/da(A).
Jr ./-
It is also well-known that vA = 0 for a-a,e, A, and this enables us to make the following definition. Denote
p(A) = IIva||Ha ,
then
0(g) :=< U(g)v, v >h, 0a(g) := p(A)-1 < UA(g)vA,vA >ha, du(A) := p(A)da(A).
Note that ^ is a probability measure, and we have 0(e) = 0A(e) = 1, because v is normalized. Such a function is said to be normalized.
In any case, 0 and 0A are continuous PDF on G, and forther, 0A A e R, are extremal functions of a convex set of the normalized continuous PDFs by virtue of the irredueibilitv of (HA, UA). Hence, we arrive at an external decomposition of 0:
0(g) = / 0A(g)d^(A) for all g e G. (2.1)
Jr
Throughout this section, we refer to the disintegration (2,1) as a natural
v
0G
0(g) = 0A(g)d^(A) for all g e G,
Jx
is given such that
• (PI) the measurable space (X, B) is standard (which is Borel isomorphic to the usual measurable space on R), and ^ is a probabiltv measure on it (thus, on R),
• (P2) for ^-a.e, A 0A is a normalized, continuous and extremal PDF on G,
• (P3) 0A(g) is measurable with respeet to A for each fixed g e G,
We then ask
More precisely, let (H, U) be a continuous unitary representation of G with a normalized cyclic vector v corresponding to 0, and M be the ring generated by
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{U(g)}geG. Then we ask whether there exists a maximal Abelian ring A of M with the following properties: take the ring N generated by M and A, and using the center of N, decompose H to a generalized direct sum of Ha,A G R, with a weight function ct(A); then, according to the irreducible decomposition of U and the decomposition of v:
0a (g) =< UA(g)vA,vA >ha /||va||Ha for all g G G and for a-a.e. A,
and d^(A)= Hva||Ha da(A).
The main issue that we wish to discuss in this section is that particular question.
2.2. Main results. In what follows, we consider the problem (P), and assume that all the conditions for (P) are fulfilled. In particular, in virtue of the above assumptions on the measurability, we may assume that X = R, B = B(R), and ^ is a Borel probability measure on B(R), Moreover, as mentioned above, we assume that G is separable, and thus, take a dense, countable subgroup G0 := (gi, g2,..., gn .. .}■ Now, take a continuous unitary representation (H, U^f G with a normalized cyclic vector v G H that correspondes to 0 in (P) (for example, through the Gelfand-Naimark-Segal construction method). First, we consider a decomposition of H into a generalized direct sum (cf, [2], pp. 407-408), Set
Since the integrand in the above equality is non-negative, there exists a Borel set N with ^(N) = 0 that satisfies for all m G N,
From this point on, we let A run mainly through Nc, Take a countable space Hd consisting of linear combinations
for all g G G, and
we have
Take a dense countable subset of consisting of the vectors = (^m’k,..., )
k G N. It follows that
m m
m
for all ^ Nc.
(2.2)
i,j=i
n
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where n G N aj = pj + qjv/—"1 p^ qj G Q, i = 1,..., n, It is a dense subset of H, For A G N, introduce a scalar product < ■, ■ >A on Hd :
n m n,m
< (gj)v,^AU(gj)v >a := a & 0a(g-1#).
j=1 j=1 j,j=1
To see that it is well-defined, we only have to check that
nn
J^ajU(gj)v = 0 =^ ^ aj aj^(g-^j) = 0.
i=1 j,j=1
To this end, take a sequence £n>Sfc, k G N, that converges to a := (a1,...,an), because a belongs to Nn, Since each component of £n,sfc satisfies the equation (2,2), a
Consequently, we get a Hilbert space HA, A G Nc, after completing the quotient space of Hd by the null kernel of the scalar product, and have a natural map from Hd to Ha:
n
h := ^ ajU(gj)v —^ hA.
j=1
It follows directly,
n
11 hA H Hx 'y y aj aj 0A(gj gj).
j,j=1
Second, we go to the definition of a(A)-summabilitv in [2].
An F-familv consists of the vector fields {/a}a over R which fulfills the following conditions:
• (FI) for each A G Nc, /a g Ha,
• (F2) < /a, hA >Hx is a ^-measurable function of A for each h G Hd,
• (F3) IIMI is a ^-measurable function of A and J II/aHHadMA) <
It is easy to see that a vector field {hA}A derived from h G Hd by the natural maps (this will be denoted by {hA}A, h G Hd, from this point on) belongs to F, We also see that
llhllH = J IIhAII d^(A).
Now, we ask whether F-familv with a suitable definition of the F-integral, which
H
sum of HA with the weight function a(A) (cf, [2]), First, we can easily see the following claim,
• (F4) < /a, gA >Hx is ^-measurable for any {gA}A G F,
Next, let {/a}a G F, and put
L(h) := / < h-A, /a >ha du(A) for any {hA}A, h G Hd.
./R
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Then L(h) is a continuous linear functional on Hd that extends continuously to the entire space H, So we have a unique f G H such that
< h,f >H = < hA,fA >Ha dMA)
./R
Note that f = k, if {fA}A = {kA}A, k G Hd. We call f the F-integral of {fa}a, and require that it be equal to the a-integral in [2], The following equality is crucial to this requirement:
•(F5) Ilf IIH =f BfABH,<MA) for all {fA}A GF.
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Once, we ensure that (F5) is fullfied, this enables us to define the isometric map T from the Hilbert space F, equipped with the natural norm, to H, As the image of T contains a dense subset Hd, it is a surjeetion. Therefore, we find that with the F F H HA
if and only if the condition (F5) is fulfilled.
Theorem 2.1 For (F5) to hold, it is necessary and sufficient that the following condition (c.l) is fulfilled.
★ (c.l) Suppose that {fA}A G F satisfies
/ < hA, fA >ha du(A) = 0 for any {^a}a, h G Hd.
R
Then, we get fA = 0 for a-a.e. A.
We omit the proof because of space, but it is not difficult.
We remark that condition (c.l) has another expression. Namely, take a continuous unitary representation (KA,TA) of G with a cyclic vector tA such that
<Mg) =< T\(g)ÍA,ÍA >ka for all g G G.
As 0A is extremal and normalized, (KA,TA) is irreducible, and tA is a unit vector for A hA
h = ^2 (gi)v G Hd
i=1
by the natural map: Hd —^ HA, we get
n
'''y ^ TA(gí) ¿A
^aIIha ^ 0A(gj gi)
i,j=1
i=1
Ka
This enables us to define a unitary map from HA to KA : hA S™=i aTA^)^. It follows that KA and the space of linear combinations ^™=1 where n G N,
a G C, gj G G0, i = 1,..., n, play a similar role to H^d {hA}A, h G Hd, to obtain
2
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a(A)-summability in [2], Therefore, we now find that condition (c.l) is equivalent to the following condition (e,2),
★ (e,2) If a vector field {nA}A,n G KA, over R satisfies the following three hypotheses:
(e,2,l) < nA,TA(g)tA >Ka is ^-measurable for every g G G0,
(c.2.2) ||nA||KA is ^-measurable, and / ||nA|lKAd^(A) < ro, and
then, nA = 0 for ^-a.e, A,
Next we proceed to address the main problem in this subsection.
Theorem 2.2 Let 0 be a continuous, normalized PDF on a separable locally compact group G, and suppose that a disintegration of 0 is given such that every condition in (P) is satisfied. Then the disintegration is natural, if and only if the condition (c.2) is fulfilled.
Proof. Since the proof of necessity is not difficult, we go to the sufficiency. Given a disintegration in (P), take a continuous unitary representation (H, U) of G with a cyclic vector v and take another irreducible ones (KA,UA) and normalized vectors wA G KA such that
0(g) =< U(g)v,v >H, and 0A(g) =< UA(g)wA,wA >Ka for all A G R and g G G.
H
generalized direct sum of KA, and everv U(g)v has an expression:
first, for all g G G0, and second, for all g G G, As < UA(g)xA,yA >KA is measurable
make sense (cf, [2]), and is equal to U(g), because they coincides on Hd,
Therefore, the rest of the proof involves examining the ring A that is a center of the generalized direct sum. Recall that M is the ring generated by U(g), g G G, We only need to ensure that A is a maximal Abelian ring in M'. Once this is assured,
We know that A = {Pz| z G L£°(R)} (cf. [2]). Now, take any P G M' H A' which is a projection. Then, < PPBU(g)v,v >H is an additive function of B G B for any
R
for all g G G0,
for any xa = Ua(Zi)wa yA = Ua((2)wa Ci,C2 G ^^d 1 ^ ||Ua(g)|, so
the ring N ^^^^rated by M and A ^^fefies NnN' = A, and it follows that the disintegration is natural.
Take any function z G L£°(R), and define Pz on H by
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g G G, where we use PB instead of PXB for the sake of simplicity. We readily see that PBv = 0 implies < PPBU(g)v,v >H = 0, In other words, the additive function is absolutely continuous with respect to Hence, some wA(g) G L*(R) exists such that
and that wA(g) = ^A(g 1) for any g G G and for ^-a.e, A, Consequently, there exists a Borel set N1 with ^(N1) = 0 such that
Again, there exists a Borel set N2 with ^(N2) = 0 such that
1 — Re 0A(gk) ^ HpaIIK — Re ^A(gk) for al 1 k G N and A G N|.
Proceeding in a similar manner, we find that there exists a Borel set N3 with MN3) = 0 such that for any A G N3, wA(-) is a PDF on G0, Using g := e in (2,4) produces
for ^-a.e, A. Therefore, we conclude that there exists a Borel set N4 with ^(N4) = 0 such that (2,5) and the following inequalities hold:
Expanding both side of the inequality
for all B G B(R). (2.3)
Pb(/ — U(g))v||H > ||PPb(I — U(g))vj|H,
[{2 — ^A(g)- <Mg)}<MA) > f {2|paBK — ^a(g)- ^a(g 1)}<MA) > °. (2.4)
J B J B
It follows from (2,3) that
wA(gfc) = wA(gfc 1) for all k G N and A G N[.
Hence, it follows from (2,4) that,
I {1 — Re0A(gfc)№(A) ^ I (||pa||Ha — Re^A(gfc)№(A) for all k G N.
IM^ = ^A(e)
(2.5)
1 — Re 0A(g) ^ wA(e) — Re wA(g) for all g G G0,
(2.6)
n
n
(2.7)
i,j=1
ij=1
for all n G N a G C, gi G G0, i = 1,..., n.
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In what follows, let A run through only N4, unless otherwise stated, and let us examine the continuity of wA(-). As 0A(g) is a continuous function of g, for each n G N, there exists a symmetric neighbourhood Un,A = Un of e such that Un+1 C Un and
n -1 ^ 1 — Re0A(g) for all g G Un.
Take any g G G and fa it. Further, take Yn G G0 R gUn for each n G N. It follows that 7m17n C U^ for all n, m ^ N + 1, and that
2
N^A(e) ^ 2^A(e)(1 — Re 0A(7m17n))
^ 2^A(e)(^A(e) — Re^A(7m17n))
^ I^A(Tm) — ^A(7n)|.
Hence, {^A(7n)}n converges, and the limit is independent of the choice {7n}n. We denote the limit by the same letter wA(g), because it is an extension of wA(-) from G0
Next, given e > 0, take n G N such that
2
^A(e) < e.
n1
Then, for any g' G gUn we can take the above {7^}m from the set gUn, Thus, we find that
2
e > ----7^A(e) ^ ) — ^A(g)|.
n — 1
Letting m —> to, we obtain
2
e >-----7^A(e) ^ |^A(g) — ^A(g)1,
n — 1
and this demonstrates the continuity of wA(-).
Finally, for each n G ^d {ai}n=1 C C, let F be the set consisting of (g1,..., gn), gi G G
nn
'y ] ai 0A(gi gj) ^ ^ ] ai ^A(gi gj) 0.
i,j = 1 i,j=1
Gn G0n F = Gn
functions, wA and 0A — wA are continuous PDFs on G. By the extremal assumption we thus have
wA(g) = wA(e)0A(g) for al 1 g G G and A G N4.
It follows that regarding wA(e) = w(A) as an essentially bounded measurable function
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A
< PwU(g)v, v >h ^A(e) < Ua(g)wA,WA >ka du(A)
^A(e)0A(g) d^(A)
^A(g) du(A)
=< PU(g)v, v >h .
In other words, P = Pw G A Therefore, we have A = M' R A', and A is a maximal
2.3. Characters on the infinite permutation group, and their disintegrations. We begin bv reviewing briefly Thoma’s result on characters. Let S0 be the infinite permutation group of the finite permutations on N. Note that eaeh g G S0
g
cycles. For each n ^ 2, let rn(g) be the number of the cycles with length n in the g
By a character 0 of S0 we mean that it is a PDF on S0 (equipped with the discrete topology) such that 0(e) = 1 and 0(ghg-1) = 0(h) for any g, h G S0, Clearly, the set of the characters forms a convex set. The extremal point of the convex set is said to be indecomposable.
Finally, let ¿+(Z) be the set of the sequence {A}i=-0 such that ^ ^ 0 for all i G Z, two sequences {^¿}+=^ ^d {^-i}+°^ are both decreasing for i G N, and
With that background, we are now ready to describe Thoma’s result (cf, [4]),
Theorem 2.3 [4] Given any indecomposable character 0 on S0, there exists a unique {$ K+Ooo G (Z) such, that
Conversely, given {(5i}i=0rc> G ^+(Z), the right hand side of the above equality gives an indecomposable character.
S0
Obata, However, we use a somewhat different notation from that of [3] to avoid inconsistencies with our previous notations,
Obata [3] had a disintegration of the indecomposable characters. Take the direct Z0 Z A G Z0
from N to Z. With each A, we have a partition rA := {A-1({j})}jeZ of N. Define the Young subgroup HA by
M'
□
Ei=°!o A = 1
0(g) = g G S
HA := S(A 1({j})) (restricted direct product),
jez
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where S(A 1({j})) is a group of the permutations that leaves every element in A-1({j})c invariant. Take a one-dimensional representation xA °f HA defined by
XA(g) := sgn g-, according to the unique expression of g = g+g0g- with
g+ G S (uj>1A 1({j})) , go G S(A 1({0})), g- G S (Uj>1A 1({-j})) .
Take the induced representation U(rA; xA) := ind(xA; HA | S0). It is irreducible, if all the cardinals of the sets A-1({n}), n G N, are infinite. In any case, it is a cyclic unitary representation for any A G Z“, and
0 ( ) := f Xa(g), if g G ha,
A g | 0, otherwise
is a normalized PDF that corresponds to U(rA; xA) with a cyclic vector. Finally, we introduce a probability measure ^ ^ for each ft := {^i}+=-“ G ¿+(Z) on the
standard Borel space (Z“, B(Z“)) as a product measure of countable copies of v on Z such that v({i}) = ft* for all i G Z. Note that the set of A G Z“ such that
|A-1({n}| < ro is of zero measure, so that U(rA; xA) is irreducible for ^-a.e, A. In
other word, 0A is extremal for the same A.
Theorem 2.4 [3] Let 0 = 0^ he an indecomposable character corresponding to ft G (Z) with ft0 = 0. Then, we have
0^(g) =
In what follows, we examine whether Obata’s disintegration is natural or not. We retain the notation in the previous subsections.
So, given ft G (Z), ft0 = 0, we have the indecomposable character 0 = 0^, and
the probability measure ^ on Z“, and the disintegration described in Theorem
2.4. The representation space HA of U(rA; xA) consists of the C-valued functions on S“
(1) f (gh) = XA(h)f (g) for all g G ^d h G Ha,
(2) ^ |f(g)|2 < geG/H
As usual, we regard |f (g)| as a function on S/HA in virtue of (1). The representation U(rA; xA) acts by translations:
U(rA; XA)(g)f O = f (g-1-) for all f G HA, g G S“.
Now, we deform (HA, U (rA; xA )), using a sect ion s = sA of the natural map n; S“ —> S“/Ha. Fomy f G HA put
F(X) := f (s(X)) for all X G S“/Ha.
0A(g) (A) for a 11 g G S“
oo
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We readily see that F G £2(S“/HA), and the map W : f —> F is isometric.
Moreover, for any F G £2(S“/HA), a function f defined by
f (g) := F(n(g))XA(s(n(g))-1g)
satisfies f(s(X)) = F(X) for any X G S“/HA and f(gh) = xA(h)f(g) for all
g G S“ and h G HA, In other words, W is a unitary operator. We set
UA(g):= W ◦ U(rA; XA)(g) ◦ W-^d K := £2(S“/Ha).
Then, it easily follows that
(UA(g)F)(X) = F(g-1X)xa Mg-1X)-1g-1SA(X)) ,
and the cyclic vector FA,e G KA corresponding to fA,e G HA defined by
f ( ) i XA(g), if g G HA,
A,e g | 0, otherwise,
is a function (up to scalar factor) such that
( 1, if X = Ha,
Fa e(X) := { ’
, | 0, otherwise.
Therefore, for the present purpose, we only have to examine the following question (Q):
(Q) Let Ga G £2(S“/Ha)(= Ka) and [ ||GaHKa(A) < ro.
Suppose that
< Ga, U\(g)FA,e >ka = GA(gHA) xa (sA(HA)-1g-1SA(gHA))
A
I GA(gHA) xa (sA(HA)-1g-1SA(gHA)) d^^(A) = 0 for all g G S“.
Jz^
Then, does it implies that GA = 0 for ^-a.e, A ?
The next theorem is a partial answer to the question (Q),
Theorem 2.5 For ft = {fti}+=-“ with ft0 = 0, if we have either fti = ftj = 0 or
ft-i = ft-j = 0 for some different i,j G N, then (Q) is negative, so that Obata’s
disintegration of 0^ is not natural.
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Proof. Suppose that the first assumption holds. We take a transposition p := (i, j) on Z. This transposition acts from the left on Z“ such that (pA)n = p(An), where An is the n-th component of A G Z“, Note that
p^ = ^, HpA = Ha and XpA = Xa.
Define a function GA G £2(S“/HA) by
GA(X) := }(A) ■ [^{i}(A1) — ^{j}(A1^ ,
using the indicator function ^{^ of the set {i}. Then,
i |GaIIK d^(A) = 2fti,
Jz
which is easily checked, and furthermore,
GA(gHA) XA (sA(HA)-1g-1 sA(gHA^ = ^{weZ^|gHA=HW}(A) X
x [^{i}(A1) - ^{j}(A1)] XA(g-1).
Therefore, the above function is measurable, and it has the opposite sign and the
A pA
/ Ga№a) xa (sA(HA)-1g-1SA(gHA)) d^(A) = 0 for all g G S“.
Jz^
While, we have GA = 0 for ^-a.e. A, and this demonstrates the proof. The second case is similar, □
References
1, F, I, Mautner, Unitary representation of locally compact group, I, Ann, Math,, 1950, vol. 51, 1-25.
2, J. von Neumann. Rings of operators. Reduction theory, Ann. Math., 1949, vol. 50, 401-485.
3. N. Obata. Integral expression of some indecomposable characters of the infinite symmetric group in term of irreducible representations, Math. Ann., 1989, Band 287, 369-375.
4. E. Thoma. Die unzerlegbaren positiv-definiten, Klassenfunktionen der abzahlbar unendlichen symmetrischen Gruppe, Math. Z., 1964, Band 85, No. 1, 40-61.
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