On invariant subalgebras of the Fourier-Stieltjes algebra of a locally compact group. (English) Zbl 0787.22005
One of the most instructive Banach algebras attached to a locally compact group \(G\) is the commutative Fourier-Stieltjes algebra \(B(G)\), the dual space of the \(C^*\)-algebra \(C^*(G)\) of \(G\); it is spanned by the set \(P(G)\) of all continuous positive definite functions on \(G\). In [J. Funct. Anal. 11, 17-38 (1972; Zbl 0242.22010)], M. E. Walter proved the locally compact groups \(G_ 1\) and \(G_ 2\) to be isomorphic if and only if the algebras \(B(G_ 1)\) and \(B(G_ 2)\) are isometrically isomorphic. The carefully written paper under review describes the structure of a [translation-] invariant *-subalgebra \(A\) of \(B(G)\) in terms of a subgroup in \(G\).
A direct proof yields that in case \(A\) separates points of \(G\) and it is also conjugation-invariant with respect to the enveloping von Neumann algebra of \(C^*(G)\), it coincides with the Fourier algebra \(A(G)\) of \(G\). As a consequence, the locally compact group \(G\) is amenable if and only if there exists a bijection between the set of all closed normal subgroups of \(G\) and the set of all weak-*-closed invariant *-subalgebras \(A\) of \(G\) with \(A\neq\{0\}\); the proof uses the fact that in the amenable case, \(f\in P(G)\) is a limit on compacta of functions \(\varphi*\widetilde{\varphi}\), \(\varphi\in L^ 2(G)\), with \(\widetilde{\varphi}(x)= \overline {\varphi(x^{-1})}\), \(x\in G\). Also, if the discrete group \(G_ d\) corresponding to \(G\) is amenable, then \(B(G)\) is weak-*-dense in \(B(G_ d)\). The authors prove that not being the case for the nonamenable group \(SL(3,\mathbb{C})\), as it violates Reiter’s condition \((P_ 2^*)\).
Other general properties are established: If \(A\) is a norm-closed invariant *-subalgebra of \(A(G)\) and \(A\neq \{0\}\), then \(A=A(G/K)\) for some compact normal subgroup \(K\) of \(G\). The weak-*-closed \({\mathcal C}_ 0(G)\)-invariant *-subalgebra \(A\) of \(M^ 1(G)={\mathcal C}_ 0(G)^*\) [resp. the closed \(L^ \infty(G)\)-invariant *-subalgebra of \(L^ 1(G)\)], \(A\neq \{0\}\), is \(M^ 1(H)\) [resp. \(L^ 1(H)\)] for a closed [resp. open] subgroup \(H\) of \(G\). M. Takesaki and N. Tatsuuma obtained the very last result by different methods [Ann. Math., II. Ser. 93, 344-364 (1971; Zbl 0201.455)].
A direct proof yields that in case \(A\) separates points of \(G\) and it is also conjugation-invariant with respect to the enveloping von Neumann algebra of \(C^*(G)\), it coincides with the Fourier algebra \(A(G)\) of \(G\). As a consequence, the locally compact group \(G\) is amenable if and only if there exists a bijection between the set of all closed normal subgroups of \(G\) and the set of all weak-*-closed invariant *-subalgebras \(A\) of \(G\) with \(A\neq\{0\}\); the proof uses the fact that in the amenable case, \(f\in P(G)\) is a limit on compacta of functions \(\varphi*\widetilde{\varphi}\), \(\varphi\in L^ 2(G)\), with \(\widetilde{\varphi}(x)= \overline {\varphi(x^{-1})}\), \(x\in G\). Also, if the discrete group \(G_ d\) corresponding to \(G\) is amenable, then \(B(G)\) is weak-*-dense in \(B(G_ d)\). The authors prove that not being the case for the nonamenable group \(SL(3,\mathbb{C})\), as it violates Reiter’s condition \((P_ 2^*)\).
Other general properties are established: If \(A\) is a norm-closed invariant *-subalgebra of \(A(G)\) and \(A\neq \{0\}\), then \(A=A(G/K)\) for some compact normal subgroup \(K\) of \(G\). The weak-*-closed \({\mathcal C}_ 0(G)\)-invariant *-subalgebra \(A\) of \(M^ 1(G)={\mathcal C}_ 0(G)^*\) [resp. the closed \(L^ \infty(G)\)-invariant *-subalgebra of \(L^ 1(G)\)], \(A\neq \{0\}\), is \(M^ 1(H)\) [resp. \(L^ 1(H)\)] for a closed [resp. open] subgroup \(H\) of \(G\). M. Takesaki and N. Tatsuuma obtained the very last result by different methods [Ann. Math., II. Ser. 93, 344-364 (1971; Zbl 0201.455)].
Reviewer: J.-P.Pier (Luxembourg)
MSC:
| 22D10 | Unitary representations of locally compact groups |
| 43A35 | Positive definite functions on groups, semigroups, etc. |
| 43A07 | Means on groups, semigroups, etc.; amenable groups |
References:
| [1] | Arsac, G.: Sur l’espace de Banach engendré par les coefficients d’une représentation unitaire. Publ. Dép. Math., Lyon13, 1–101 (1976) · Zbl 0365.43005 |
| [2] | Barnes, B.A.: A note on separating families of representations. Proc. Am. Math. Soc.87, 95–98 (1983) · Zbl 0507.22006 · doi:10.1090/S0002-9939-1983-0677240-X |
| [3] | Bekka, M.B.: A characterization of locally compact amenable groups by means of tensor products. Arch. Math.52, 424–427 (1989) · Zbl 0672.22003 · doi:10.1007/BF01198348 |
| [4] | Boidol, J.: A Galois-correspondence for general locally compact groups. Pac. J. Math.120, 289–293 (1985) · Zbl 0579.22006 |
| [5] | Dixmier, J.:C *-algebras. Amsterdam: North-Holland 1977 · Zbl 0372.46058 |
| [6] | Dunkl, C.F., Ramirez, D.E.:C *-algebras generated by Fourier-Stieltjes transforms. Trans. Am. Math. Soc.164, 435–441 (1972) · Zbl 0211.15903 |
| [7] | Eymard, P.: L’algèbre de Fourier d’un groupe localement compact. Bull. Soc. Math. Fr.92, 181–236 (1964) · Zbl 0169.46403 |
| [8] | Fell, J.M.G.: Weak containment and induced representations of groups, II. Trans. Am. Math. Soc.110, 424–447 (1964) · Zbl 0195.42201 |
| [9] | Gelfand, I.M., Naimark, M.A.: Unitary representations of the classical groups. Tr. Mat. Inst. Sketlova36 (1950) |
| [10] | Granirer, E., Leinert, M.: On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebraB(G) and of the measure algebraM(G). Rocky Mt. J. Math.11, 459–472 (1981) · Zbl 0502.43004 · doi:10.1216/RMJ-1981-11-3-459 |
| [11] | Hauenschild, W.: Subhypergroups and normal subgroups. Math. Ann.256, 1–18 (1981) · Zbl 0465.22005 · doi:10.1007/BF01450938 |
| [12] | Helgason, S.: Lacunary Fourier series on noncommutative groups. Proc. Am. Math. Soc.9, 782–790 (1958) · Zbl 0091.10905 · doi:10.1090/S0002-9939-1958-0100234-5 |
| [13] | Kaniuth, E.: Weak containment and tensor products of group representations, II. Math. Ann.270, 1–15 (1985) · doi:10.1007/BF01455523 |
| [14] | Pier, J.P.: Amenable locally compact groups. New York: Wiley 1984 · Zbl 0597.43001 |
| [15] | Repka, J.: A Stone-Weierstrass theorem for group representations. Int. J. Math. Math. Sci1, 235–244 (1978) · Zbl 0418.22005 · doi:10.1155/S0161171278000277 |
| [16] | Takesaki, M., Tatsuuma, N.: Duality and subgroups. Ann. Math.93, 344–364 (1971) · doi:10.2307/1970778 |
| [17] | Tatsuuma, N.: Duality for normal subgroup. Algèbres d’operateurs et leurs applications en physique mathématique. In: Proc. Colloq. Marseille 1977. (Colloq. Int. CNRS, vol. 274, pp. 373–386) Paris: CNRS 1979 |
| [18] | Walter, M.E.:W *-algebras and non-abelian harmonic analysis. J. Funct. Anal.11, 17–38 (1972) · Zbl 0242.22010 · doi:10.1016/0022-1236(72)90077-8 |
| [19] | Zimmer, R.I.: Ergodic theory and semisimple groups. Boston: Birkhäuser 1984 · Zbl 0571.58015 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
