A strong duality theorem for separable locally compact groups

Author:
John Ernest

Journal:
Trans. Amer. Math. Soc. **156** (1971), 287-307

MSC:
Primary 22.60

DOI:
https://doi.org/10.1090/S0002-9947-1971-0281841-7

MathSciNet review:
0281841

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain a duality theorem for separable locally compact groups, where the group is regained from the set of factor unitary representations. Loosely stated, the group is isomorphic to the group of nonzero bounded, operator valued maps on the set of factor representations, which preserve unitary equivalence, direct sums, and tensor products. The axiom involving tensor products is formulated in terms of direct integral theory. The topology of *G* may be regained from the irreducible representations alone. Indeed a sequence in *G*, converges to *x* in *G* if and and only if converges strongly to for each irreducible representation of *G*. This result supplies the missing topological part of the strong duality theorem of N. Tatsuuma for type I separable locally compact groups (based on irreducible representations). Our result also generalizes this Tatsuuma strong duality theorem to the nontype I case.

**[1]**J. Dixmier,*Les algèbres d'opérateurs dans l'espace Hilbertien*, 2nd ed., Gauthier-Villars, Paris, 1969.**[2]**-,*Les*-*algèbres et leurs représentations*, Cahiers Scientifiques, fasc. 29, Gauthier-Villars, Paris, 1964. MR**30**#1404. MR**0171173 (30:1404)****[3]**J. A. Ernest,*A decomposition theory for unitary representations of locally compact groups*, Trans. Amer. Math. Soc.**104**(1962), 252-277. MR**25**#3383. MR**0139959 (25:3383)****[4]**J. A. Ernest,*Hopf-von Neumann algebras*, Proc. Conf. Functional Analysis (University of California, Irvine, Calif., 1966), Academic Press, London; Thompson, Washington, D. C., 1967, pp. 195-215. MR**36**#6956. MR**0223909 (36:6956)****[5]**G. W. Mackey,*Induced representations of locally compact groups*. II;*The Frobenius reciprocity theorem*, Ann. of Math. (2)**58**(1953), 193-221. MR**15**, 101. MR**0056611 (15:101a)****[6]**-,*The theory of group representations*, University of Chicago, Chicago, Ill., 1955 (mimeographed notes).**[7]**-,*Borel structure in groups and their duals*, Trans. Amer. Math. Soc.**85**(1957), 134-165. MR**19**, 752. MR**0089999 (19:752b)****[8]**M. E. Munroe,*Introduction to measure and integration*, Addison-Wesley, Reading, Mass., 1953. MR**14**, 734. MR**0053186 (14:734a)****[9]**M. Takesaki,*A duality in the representation theory of*-*algebras*, Ann. of Math. (2)**85**(1967), 370-382. MR**35**#755. MR**0209859 (35:755)****[10]**N. Tatsuuma,*A duality theorem for locally compact groups*, J. Math. Kyoto Univ.**6**(1967), 187-293. MR**36**#313. MR**0217222 (36:313)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
22.60

Retrieve articles in all journals with MSC: 22.60

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0281841-7

Keywords:
Separable locally compact group,
unitary representation,
factor representation,
Tatsuuma duality,
Borel options

Article copyright:
© Copyright 1971
American Mathematical Society