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Petersburg Department of Steklov Institute of Mathematics

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PREPRINT 13/1997

S.A. Evdokimov, I.N. Ponomarenko, A.M. Vershik
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Algebras in plancherel duality
and *C*-algebras

This preprint was accepted July 30, 1997.

Contact:
` S.A. Evdokimov, `
` I.N. Ponomarenko, `
` A. M. Vershik `

Abstract:

In this paper we discuss algebras in Plancherel duality, i.e.
a special class of pairs of semisimple finite-dimensional
algebras with involution being in a nondegenerate duality as
vector spaces. This class arised more than twenty years ago as
a generalization of the Krein-Tanaka duality and Hopf algebras.
We present new axiomatics of algebras in Plancherel duality according
to the properties of the corresponding pairing. It is proved that the
pairing is a Plancherel one iff it is positive, homogeneous and isometric.
It turns out that the above class provides a natural framework
for the algebraic approach to combinatorics connected with the
notion of *C*-algebra.
For an arbitrary *C*-algebra (possibly non-commutative)
a positivity
condition generalizing the Krein condition in commutative case,
is defined.
We show that the class of positive *C*-algebras includes those
arising in algebraic combinatorics from
association schemes (possibly non-commutative). It is proved
that the category of positive *C*-algebras is equivalent to
the category of pairs of algebras in Plancherel duality one
of which being commutative.

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