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Suppose that H is a finite dimensional discrete quantum group and K is a Hilbert space. This paper shows that if there exists an action γ of H on L(K) so that L(K) is a modular algebra and the inner product on K is H-invariant, then there is a unique C*-representation 9 of H on K supplemented by the γ. The commutant of θ(H) in L(K) is exactly the H-invariant subalgebra of L(K). As an application, a new proof of the classical Schur-Weyl duality theory of type A is given.

Let T1,...,T, C B(H) be bounded operators on a Hilbert space NH such that TiT1 + . + TnTnT < I-H. Given a symmetry j on NH, i.e., j2 = j*j = I-, we define the j-symmetric commutant of {Ti,...,Tn to be the operator space {A C B(N): TiA = jATi, i = 1, ..., n}. In this paper we obtain lifting theorems for symmetric commutants. The result extends the Sz.-Nagy-Foias commutant lifting theorem (n = 1, j = I-H), the anticommutant lifting theorem of Sebestyen (n 1, j = -Ia), and the noncommutative commutant lifting theorem (j = Ia). Sarason's interpolation theorem for H' is extended to symmetric commutants on Fock spaces. 1. LIFTING THEOREM FOR SYMMETRIC COMMUTANTS Let F+ be the unital free semigroup on n generators s,... ., sr, and let e be its neutral element. For any o... sik E F+, we define its length 11 := k, and lel = 0. On the other hand, if Ti E B(H), i = 1,.. ., n, we denote T, :Til ... and Te := IH. Let us recall from [Pol], [Po2], and [Po4] a few results concerning the noncommutative dilation theory for n-tuples of operators. A sequence of operators T := [Ti,... , T], Ti E B(H), i = 1, ... , n, is called contractive (or row contraction) if TIT1* + *.. + TnTn < I-. We say that a sequence of isometries V := [VI, ... , 1V] on a Hilbert space IC D H is a minimal isometric dilation of T if the following properties are satisfied: (i VI VI* + + Vn Vn* < IK; (ii Vi* I = Ti* I i = 1, . . n; (iii) IC = VaEF+ VaH. Consider the full Fock space on n generators F2(Hn) := ClE Q Hm Secondary 30E05. The author was partially supported by NSF Grant DMS-9531954. ?2000 American Mathematical Society 1705 This content downloaded from 207.46.13.103 on Thu, 20 Oct 2016 04:32:19 UTC All use subject to http://about.jstor.org/terms