Positivity 3: 377–388, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.
Notes on Amenability of Commutative Fusion
Department of Mathematics, Tohoku University, Sendai, 980-77, Japan
(Received: 9 March 1998; accepted 15 November 1998)
Abstract. A commutative fusion algebra is proved to be amenable if and only if the associated
regular representation is bounded.
AMS subject classiﬁcations: 46L, 46N
Key words: amenability, commutative fusion algebra, hypergroup
Analytical theory of fusion algebras is developed by F. Hiai and M. Izumi in ,
where various equivalent formulations of amenability are worked out among other
In their deﬁnition, fusion algebras are assumed to admit a dimension function
from the outset, which enabled them to associate hypergroup structures to fusion
algebras and analytical properties of fusion algebras were explored in terms of
convolution products of hypergroups.
One of the prominent features of amenable fusion algebras is that the assumed
dimension function is uniquely determined by the fusion algebra itself, i.e., the
amenability is inherent without explicit references to dimension functions. For ex-
ample, the existence and uniqueness of dimension functions is proved by V.S. Sun-
der for ﬁnite-dimensional fusion algebras in .
In the present notes, we shall further clarify this point by showing that the
existence of amenable dimension functions is an internal property for commutat-
ive fusion algebras: A commutative fusion algebra allows an amenable dimension
function if and only if the associated regular representation is bounded.
2. Fusion Algebras
We here review the notation and terminologies on fusion algebra introduced in ,
. Since our main result holds under weaker assumptions than fusion algebras,
we organize axioms in a slightly different fashion from the original deﬁnitions.