Group extensions and cohomology for locally compact groups. IV

Moore, Calvin C.

doi:10.1090/S0002-9947-1976-0414776-1

Group extensions and cohomology for locally compact groups. IV
HTML articles powered by AMS MathViewer

by Calvin C. Moore PDF

Trans. Amer. Math. Soc. 221 (1976), 35-58 Request permission

Abstract:

In this paper we shall apply the cohomology groups constructed in [14] to a variety of problems in analysis. We show that cohomology classes admit direct integral decompositions, and we obtain as a special case a new proof of the existence of direct integral decompositions of unitary representations. This also leads to a Frobenius reciprocity theorem for induced modules, and we obtain splitting theorems for direct integrals of tori analogous to known results for direct sums. We also obtain implementation theorems for groups of automorphisms of von Neumann algebras. We show that the splitting group topology on the two-dimensional cohomology groups agrees with other naturally defined topologies and we find conditions under which this topology is ${T_2}$. Finally we resolve several questions left open concerning splitting groups in a previous paper [13].

References

Louis Auslander and Calvin C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. 62 (1966), 199. MR 207910
Felix E. Browder, On the iteration of transformations in noncompact minimal dynamical systems, Proc. Amer. Math. Soc. 9 (1958), 773–780. MR 96975, DOI 10.1090/S0002-9939-1958-0096975-9
Alain Guichardet, Sur la cohomologie des groupes topologiques, Bull. Sci. Math. (2) 95 (1971), 161–176 (French). MR 307265
Barry Edward Johnson, Cohomology in Banach algebras, Memoirs of the American Mathematical Society, No. 127, American Mathematical Society, Providence, R.I., 1972. MR 0374934
G. Hochschild and J.-P. Serre, Cohomology of group extensions, Trans. Amer. Math. Soc. 74 (1953), 110–134. MR 52438, DOI 10.1090/S0002-9947-1953-0052438-8
Konrad Jacobs, Neuere Methoden und Ergebnisse der Ergodentheorie, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 29, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960 (German). MR 0123676
Robert R. Kallman, Groups of inner automorphisms of von Neumann algebras, J. Functional Analysis 7 (1971), 43–60. MR 0279596, DOI 10.1016/0022-1236(71)90043-7
George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 44536, DOI 10.2307/1969423
Louis Michel, Sur les extensions centrales du groupe de Lorentz inhomogène connexe, Nuclear Phys. 57 (1965), 356–385 (French). MR 0210813
Calvin C. Moore, Extensions and low dimensional cohomology theory of locally compact groups. I, II, Trans. Amer. Math. Soc. 113 (1964), 40–63; ibid. 113 (1964), 64–86. MR 171880, DOI 10.1090/S0002-9947-1964-0171880-5

Extensions and low dimensional cohomology theory of locally compact groups

113

30

Calvin C. Moore, Restrictions of unitary representations to subgroups and ergodic theory: Group extensions and group cohomology, Group Representations in Math. and Phys. (Battelle Seattle 1969 Rencontres), Lecture Notes in Phys., Vol. 6, Springer, Berlin, 1970, pp. 1–35. MR 0279232
Calvin C. Moore, Group extensions of $p$-adic and adelic linear groups, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 157–222. MR 244258
Calvin C. Moore, Group extensions and cohomology for locally compact groups. III, Trans. Amer. Math. Soc. 221 (1976), no. 1, 1–33. MR 414775, DOI 10.1090/S0002-9947-1976-0414775-X