Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

M.J. Crabb / Journal of Functional Analysis 236 (2006) 630–633 633
Remark. The set M ∪{0} forms an inverse monoid under the multiplication given by
(A ∪ gB)gh if A ∪ gB ∈ L,
AgBh =
0 otherwise.
This is called the McAlister monoid on X and is the Rees quotient of the free inverse monoid on
X by the ideal generated by all elements xy −1 and x −1 y with x,y ∈ X and x = y [4]. Barnes [1]
constructs for any inverse semigroup a representation on Hilbert space; this is used here to
generate C ∗ M. Weaver’s example in [7] is also generated by an inverse semigroup of partial
isometries. Related results on l 1 -algebras are described in [3,4].
Acknowledgment
I thank Professor W.D. Munn for introducing me to the notion of the McAlister monoid and
for help with setting out this paper.
References
[1] B.A. Barnes, Representations of the l 1 -algebra of an inverse semigroup, Trans. Amer. Math. Soc. 218 (1976) 361–
396.
[2] F. Bonsall, J. Duncan, Complete Normed Algebras, Springer-Verlag, Berlin, 1973.
[3] M.J. Crabb, The l 1 -algebra of a free inverse monoid, Glasgow University, preprint.
[4] M.J. Crabb, W.D. Munn, The contracted l 1 -algebra of a McAlister monoid, Glasgow University, preprint.
[5]J.Dixmier,SurlesC ∗ -algèbres, Bull. Soc. Math. France 88 (1960) 95–112.
[6] J. Dixmier, C ∗ -algèbres et leurs représentations, Gauthier–Villars, Paris, 1969.
[7] N. Weaver, A prime C ∗ -algebra that is not primitive, J. Funct. Anal. 203 (2003) 356–361.