Representations of the Banach algebra ℓ1(S)

McLean, R. G.; Kummer, Hans

doi:10.1007/bf02573128

Cited by 3 publications

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“…Definition.A representation n of a monoid S on a Hilbert space H will be called formally real is there is an orthonormal basis {e,|ie/} of H for which (n(x)e s \e r } is real for all x6S and all r,sel.It is known that if the *-representations of S are separating then the Banach *-algebra <f'(s) is *-semisimple. This follows from Theorem 3.4 of [1] (a direct proof may be found R. G. MCLEAN in[3]). The following lemma shows that under suitable conditions a separating set of representations of S gives rise to a separating set of representations of £ l (S).Lemma.…”

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confidence: 82%

“…Then si is a self-conjugate unital subalgebra of /"(S) which separates points of S so by Theorem 1 of [3], if / is a non-zero element of ^1(S), then there is a ge si with xeS It follows that there is a ne0l with 7i(/)#0.…”

Section: Proof Let Si Be the Set Of All Functions / : S->f Withmentioning

confidence: 99%

“…for some representation n of S on a Hilbert space H and some vectors <f),\li sH, where n is a finite direct sum of representations from 3$. Then si is a self-conjugate unital subalgebra of /"(S) which separates points of S so by Theorem 1 of [3], if / is a non-zero element of ^1(S), then there is a ge si with…”

Section: Proof Let Si Be the Set Of All Functions / : S->f Withmentioning

confidence: 99%

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On a problem of Barnes and Duncan

McLean

1991

Proceedings of the Edinburgh Mathematical Society

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Consider the free monoid on a non-empty set P, and let R be the quotient monoid determined by the relations:Let R have its natural involution * in which each element of P is Hermitian. We show that the Banach »-algebra /'(R) has a separating family of finite dimensional *-representations and consequently is *-semisimple. This generalizes a result of B. A. Barnes and J. Duncan (J. Fund. Anal. 18 (1975), 96-113.) dealing with the case where P has two elements. Mathematics subject classification (1985 Revision): 43A65Consider the free monoid on a non-empty set P, and let R be the quotient monoid determined by the relations:We equip R with its natural involution * in which each element of P is Hermitian. When P contains exactly two elements Barnes and Duncan [2] have shown that the Banach *-algebra ^i(R) has a separating family of finite dimensional *-representations. We show that this result is in fact true for an arbitrary P. It follows that t l (R) is •-semisimple. Let S be a monoid i.e. a semigroup with an identity element 1. By a representation n of S we shall mean a bounded map n from S into the set of all bounded linear operators on a (real or complex) Hilbert space H such that TI(1) is the identity operator and n(xy) = n{x)n(y) for all x,yeS. Each representation of S has a unique extension to a representation of the Banach algebra t l (S). Tensor products and direct sums may be formed in a similar way to those of group representations. Definition.A representation n of a monoid S on a Hilbert space H will be called formally real is there is an orthonormal basis {e,|ie/} of H for which (n(x)e s \e r } is real for all x6S and all r,sel.

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confidence: 82%

Section: Proof Let Si Be the Set Of All Functions / : S->f Withmentioning

confidence: 99%

Section: Proof Let Si Be the Set Of All Functions / : S->f Withmentioning

confidence: 99%

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On a problem of Barnes and Duncan

McLean

1991

Proceedings of the Edinburgh Mathematical Society

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“…One of these is by MCLEAN and KUMMER [43] and contains a completely different proof of Theorem 3.3 (see Section 3) for the case in which S is a discrete semigroup. For a commutative semigroup 5 it is quite easy to determine the complex homomorphisms of the algebra l l (S).…”

Section: The Algebra L L (S)mentioning

confidence: 97%

Measure algebras on semigroups

Baker¹

1990

The Analytical and Topological Theory of Semigroups

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On the l¹-algebra of certain monoids

Crabb

Munn

1998

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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The monoids considered are the free monoid Mx and the free monoid-with-involution MIx on a nonempty set X. In each case, relative to a simply-defined involution, an explicit construction is given for a separating family of continuous star matrix representations of the l1-algebra of the monoid and it is shown that this algebra admits a faithful trace. The results are based on earlier work by M. J. Crabb et al. concerning the complex semigroup algebras of Mx and MIx.

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Representations of the Banach algebra ℓ1(S)

Cited by 3 publications

References 2 publications

On a problem of Barnes and Duncan

On a problem of Barnes and Duncan

Measure algebras on semigroups

On the l¹-algebra of certain monoids

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Representations of the Banach algebra ℓ1(S)

Cited by 3 publications

References 2 publications

On a problem of Barnes and Duncan

On a problem of Barnes and Duncan

Measure algebras on semigroups

On the l1-algebra of certain monoids

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On the l¹-algebra of certain monoids