Finite-dimensional left ideals in some algebras

associated with a locally compact group

Author:
M. Filali

Journal:
Proc. Amer. Math. Soc. **127** (1999), 2325-2333

MSC (1991):
Primary 43A10, 22D15

DOI:
https://doi.org/10.1090/S0002-9939-99-04793-0

Published electronically:
April 9, 1999

MathSciNet review:
1487366

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a locally compact group, let be its group algebra, let be its usual measure algebra, let be the second dual of with an Arens product, and let be the conjugate of the space of bounded, left uniformly continuous, complex-valued functions on with an Arens-type product. We find all the finite-dimensional left ideals of these algebras. We deduce that such ideals exist in and if and only if is compact, and in (except those generated by right annihilators of ) and if and only if is amenable.

**[1]**J. W. Baker and M. Filali,*Minimal ideals in group algebras and their biduals*, Math. Proc. Cambridge Philos. Soc.**120**(1996), 475-488. CMP**96:12****[2]**J. F. Berglund, H. D. Junghenn and P. Milnes,*Analysis on Semigroups: Function Spaces, Compactifications, Representations*, Wiley, New York, 1989. MR**91b:43001****[3]**M. Filali,*The ideal structure of some Banach algebras*, Math. Proc. Cambridge Philos. Soc.**111**(1992), 567-576. MR**93e:22005****[4]**M. Filali,*Linear equations in*, Proc. Roy. Soc. Edinburgh Sect. A**123A**(1993), 1001-1009. MR**95j:39035****[5]**M. Filali,*Minimal left ideals in the group algebra and its second conjugate*, Mathematica Japonica (to appear).**[6]**E. E. Granirer,*The radical of*, Proc. Amer. Math. Soc.**41**(1973), 321-324. MR**48:4646****[7]**F. P. Greenleaf,*Invariant means on topological groups and their applications*, Van Nostrand, New York, 1969. MR**40:4776****[8]**E. Hewitt and K. A. Ross,*Abstract harmonic analysis I*, Springer-Verlag, Berlin,, 1963, 1970. MR**28:158****[9]**B. Sz.-Nagy,*Uniformly bounded linear transformations in Hilbert space*, Acta Math. (Szeged)**11**(1947), 152-157.

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Additional Information

**M. Filali**

Affiliation:
Department of Mathematical Sciences, University of Oulu, SF 90570 Finland

Email:
mfilali@cc.oulu.fi

DOI:
https://doi.org/10.1090/S0002-9939-99-04793-0

Keywords:
Locally compact group,
Arens product,
representation,
amenable,
$U$-invariant,
finite-dimensional left ideal

Received by editor(s):
October 28, 1997

Published electronically:
April 9, 1999

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1999
American Mathematical Society