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A Representation for l1 (S)

A Representation for l1 (S) A REPRESENTATION FOR /'(S) COLIN M. MCGREGOR Introduction In [1] the authors investigate properties of the Banach algebra ^(S) where S is a semigroup. Our aim in this paper is to restrict attention to finitely generated free semigroups and for each such S, with more than one generator, to construct a con- tinuous, faithful and strictly irreducible representation for ^{S) on £ . We indicate how the method extends to the free semigroup on a countably infinite set of generators. Notation For standard Banach algebra notation the reader is referred to Rickart [4], Let N denote the set of positive integers and / the space of complex sequences {a,} with norm Given KeN, FS(K+\) shall denote the free semigroup on th e symbols s s , ..., u 2 s , t. Write {wj} for th e enumeration of the words in FS(K+1) ordered by length l(w) = number of symbols in w (w e FS(K+1)) with those of a given length ordered lexicographically. Let {0<} be the sequence formed by th e symbols s s ,...»s , t as they are printe d out in th e sequence {wj}. lt 2 K Fo r example with K = http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A Representation for l1 (S)

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/8.2.156
Publisher site
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Abstract

A REPRESENTATION FOR /'(S) COLIN M. MCGREGOR Introduction In [1] the authors investigate properties of the Banach algebra ^(S) where S is a semigroup. Our aim in this paper is to restrict attention to finitely generated free semigroups and for each such S, with more than one generator, to construct a con- tinuous, faithful and strictly irreducible representation for ^{S) on £ . We indicate how the method extends to the free semigroup on a countably infinite set of generators. Notation For standard Banach algebra notation the reader is referred to Rickart [4], Let N denote the set of positive integers and / the space of complex sequences {a,} with norm Given KeN, FS(K+\) shall denote the free semigroup on th e symbols s s , ..., u 2 s , t. Write {wj} for th e enumeration of the words in FS(K+1) ordered by length l(w) = number of symbols in w (w e FS(K+1)) with those of a given length ordered lexicographically. Let {0<} be the sequence formed by th e symbols s s ,...»s , t as they are printe d out in th e sequence {wj}. lt 2 K Fo r example with K =

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1976

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