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(1)ili:~~k~'f:-I:"-f:ltll{iJfJEt~f'3~ NQ39, 200Sif-, pp.49-S! Research Reports of the School of Engineering, Kinki University NQ39, 2005, pp.49-S1. On extensions of unitary representations Kiyoshi IKESHOJI. Abstact A new proof is given for the theorem that every locally compact abelien group has the extension property.. Keywords: locally cpmpact group, unitary representation, positive-definite function, direct integral, extension 1. Introduction. Now we summarize the known results on this problem without proofs. (see [2]). In this short note, we shall investigate the extension problem for unitary representations of locally compact groups. The extension problem for unitary representations is the following. Let G be a locally compact group, H a closed subgroup of G and {p,5J p } a unitary representation of H. Then under what conditions on G, H and p, does there exist a unitary representation {7r, 5Jn} of G such that {p, -Dp} is unitarily equivalent to the restriction of some subrepresentation of {7r, 5Jn} to H? In this case, we shall say that p has a unitary extension 7r. Our general references on this problem is [2]. Let G, Hand p be as above. If p has a unitary extension 7r, we shall say that (G, H, p) is an extension triple. If (G, H, p) is an extension triple for all closed subgroups H of G and all unitary representations p of H, we shall say that G has the extension property. A closed subgroup H of G such that (G, H, p) is an extension triple for all unitary representations p of H will be called an extending subgroup of G.. Theorem A. Every locally compact abelien group has the extension property. Theorem C. Every compact group has the extension property. Theorem D. Every discrete group has the extension property. Theorem C-S. Every compact subgroup of a locally compact group is extending. Theorem 1. Every locally compact group containing a closed subgroup of finite index with the extension property has the extension property. Theorem L. (G, H, p) is an extension triple whenever p is a subrepresentation of the regular representation of H. Theorem Z. If G has the small invariant neighborhood property, then the centre of G is a extending subgroup. Department of Mechanical Engineering, School of Engineering, Kinki University. 49.

(2) 50. 2. Positive-definite function A function p defined on G is called positivedefinite if it satisfies the following inequality for every choice of C1, C2, ... ,Cn E C and gl, g2,··· ,gn E G,. I: CiCj p(g:;lgj) :::: 0 i,j If p is a positive-definite function on G, then it holds that. (1) p(g-l). = p(g). for all g E G,. (2) Ip(g)l:::; p(e) for all 9 E G and. (3) p is uniformly continuous on G if it is continuous at e. We denote by P(G) the set of all continuous positive-definite functions on G. The next result is very usefull dealing with continuous positivedefinite functions. (4) A continuous function p defined on G is positive-definite if and only if there exists uniquely up to unitary equivalence a cyclic unitary representation {11", SJrr } of G and a cyclic vector ~ E SJrr such that p(g). = (11"(g)~I~)Sj7r. to the restriction of { 11", .flo} to H. From assumptions it holds that q(h) = (p(h)~pl~p) = (11"(h)'fJI'fJ) for all h E H, where ~p is a corresponding cyclic vector to q and 'fJ is a vector in SJo. Set p(g) (11"(g)'fJI'fJ) for all 9 E G. Then p E P(G) and q(h) = p(h) for all h E H. That is (G, H, q) is an extension triple. 0. =. From Proposition 1 it is seen that the extension problem for unitary representations is equivalent to that for positive-definite functions. For example, the positive-definite version of Theorem L is as follows: Theorem 1'. (G, H, q) is an extension triple whenever q belongs to the Fourier-Algebra A(H) of H. 3. Direct integral decomposition We review the theory of the direct integral decompositions of unitary representations. Let {r, J..l} be a standard Borel space. Let {11",)" SJ')' }')' Er be a Borel field of unit ary representations of G, that is, the function , I---t (11"')' (g )~')' I 'fJ')') is Borel for all 9 E G, ~')', 'fJ')' E SJ')'. Then we set. for all 9 E G.. For each p E P(G) we denote by {11"p, ~p} a corresponding cyclic unitary representation of G and a cyclic vector as above. Let q be in P(H) for a closed subgroup H of G. If there exists p in P(G) such that q(h) = p(h) for all h E H, we call (G, H, q) an extension triple. Proposition 1. (see (1.5) in [2]) Let q be in. P (H). Then it holds that (G, H, q) is an extension triple if and only if (G, H, pq) is an extension triple.. Proof. Let q be in P(H) and suppose that (G, H, q) is an extension triple. Then there exists p in P(G) such that q(h) = p(h) for all h E H. Let {Pq, ~q}, {11"p' 'fJp} denote the cyclic unitary representations of Hand G corresponding to q and p respectively. Then we have (Pq(h)~ql~q) = (11"p(h)'fJpl'fJp) for all h E H. It means that {Pq, SJ p} is unitarily equivalent to the restriction of {11"p,SJrr} to H. Hence (G, H, pq) is an extension triple. Conversely suppose that (G, H, pq) is an extension triple. Then there exists a unitary representation {11", SJrr} of G and its subrepresentation {11", SJo} such that {Pq, SJ p} is unitarily equivalent. For all 9 E G,. ~,'fJ. E SJ it holds. The representation {11", SJ} of G is called the direct integral of the Borel field of unitary representations {11",)"SJ')'}')'Er. Conversely, let a unitary representation {11", SJ} of G is given. Let denote 2trr the von Neumann subalgebra of ~ (SJ), generated by {11"(g) : 9 EG}. Then for each maximal abelien subalgebra 9J1 of the commutant of 2t rr there exists a standard Borel space {r, J..l} such that 9J1 is *-isomorphic to L 00 (r , J..l) . Then there exists a Borel field of irreducible representations of G such that {11", SJ} is the direct integral of this field by the reduction theory. This is called an irreducible decomposition of {11", SJ} . Unless G is of type - I , maximal abelien subalgbra 9J1 of the commutant of 2trr is not unique up to conjugacy and so this irreducible decomposition is not unique. It is wellknown that all locally compact abelien groups and all compact groups are of type - I . The key to our new proof of Theorem A is the next Proposition..

(3) 51. Proposition 2. Let {P')', 5)')' }')'Er be a Borel field of cyclic unitary representations of Hand. Proof. Let {7ri,5)i} be a unitary extension of 1. Set. {Pi'~} for each i E. E9. {7r,5)} =. L{7ri,5)i} i. If (G, H, P')') are extension triples and the representation space of unitary extension of P')' is 5)')' for a.e. ~ E r, then (G, H, p) is also an extension triple.. Then it is easily seen that {7r, 5)} is a unitary extension of {p, Jt}. D. Proof. Let {7r')' , 5)')'} be unitary extensions of {P')', 5)')'} and 0')' a cyclic vector for {P')', 5)')'} a.e. ~ E r. Let g E G, f.')',TJ')' E 5)')' and n E N, then by the cyclicity of {P')', 5),)" O')'} there exisit hi,hj E Hand ci,dj E C (i = 1,2,··· ,i(n), j = 1,2,··· ,j(n)) such that. In [1] Theorem A is proved by using Bochner's Theorem. Here we give a new proof of Theorem A based on the disintegration theory instead of Bochner's Theorem.. This implies that functions ~ ~ (7r')' (g )f.')' ITJ')') are Borel for all g E G, f.')', TJ')' E 5)')' a.e.~ E r and so the field of cyclic unitary representations {7r')',5)')'}')'Er of G is Borel. Hence we set. Since it holds for all g E G, f. = J~ f.')' dJ.l( ~ ),. TJ = J~ TJ')' dJ.l(~). E 5). (7r(g)f.ITJ) = [(7r,),(g)f.')'ITJ')'). dJ.l(~),. it follows that for all h E H. (7r(h)f.ITJ) = [(7r,),(h)f.')'ITJ')'). dJ.l(~). = [(P')'(h)f.')'ITJ')'). dJ.l(~). = (p(h)f.ITJ)·. 4. A new proof of Theorem A.. Proof. Let G be a locally compact abelien group and H a closed subgroup of G. Let {p, Jt} be a unitary representation of H. We shall show that there exists a unitary representation {7r, 5)} of G such that 7r is a unitary extension of p. Let. be the irreducible decomposition of {p, Jt}. Since H is abelien, Jt')' = C for all ~ E r. That is P')' belongs to the dual group fI for all ~ E r. Let denote by Hl.. = {X E G: X(h) = 1 for all h E H } the closed subgroup of G which is called the annihilator of H. By virtue of the Pontryagin duality theorem we have fI ~ G/Hl... Hence for each P')' E H, there exists uniquely X')' E G such that p')'(h) = X')'(h) for all h E H. This implies that (G, H, P')') is an extension triple for all ~ E r. Furthermore ~ = 5)')' = C for all ~ E r. Applying Proposition 2 to extension triples (G, H, p')')')'Er, it follows that (G, H, p) is also an extension triple. Let. It means that 7r is a unitary extension of. p.. D. Corollary. Let {p, Jt} be a unitary representation of H. Let E9. {p,Jt} = L{Pi'~} i. be a cyclic decomposition of {p, Jt}, where E 1. If (G, H, Pi) is an extension triple for all i EI, then (G, H, p) is an extension triple.. {Pi, Jti } is cyclic for each i. Consequently we may conclude that {p, Jt} has the unique unitary extension {7r, 5)} and this D completes the proof. references. [1] Hewitt,E.and K.A. Ross: Abstract Harmonic Analysis, vol. II. Springer, Berlin (1970). [2] McMullen,J .R. : Extensions of positivedefinite functions, Mem. Amer. Math. Soc. 117 (1972)..

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