The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the Hausdorff-Young inequality, Young's inequality, the Hirschman-Beckner uncertainty principle, the Donoho-Stark uncertainty principle. We characterize the minimizers of the uncertainty principles. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's λ-lattices, modular tensor categories etc.

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The explicit description of irreducible homogeneous operators in the Cowen-Douglas class and the localization of Hilbert modules naturally leads to the definition of a smaller class of Cowen-Douglas operators possessing a flag structure. These operators are shown to be irreducible. It is also shown that the flag structure is rigid in that the unitary equivalence class of the operator and the flag structure determine each other. We obtain a complete set of unitary invariants which are somewhat more tractable than those of an arbitrary operator in the Cowen-Douglas class.2010 Mathematics Subject Classification. 47B32, 47B35. Key words and phrases. The Cowen-Douglas class, strongly irreducible operator, homogeneous operator, curvature, second fundamental form. 2.A new class of operators in B 2 (Ω) 2.1. Definitions. If T is an operator in B 2 (Ω), then there exists a pair of operators T 0 and T 1 in B 1 (Ω) and a bounded operator S such that T = T 0 S 0 T 1 . This is Theorem 1.49 of [8, page 48]. We show, the other way round, that two operators T 0 and T 1 from B 1 (Ω) combine with the aid of an arbitrary bounded linear operator S to produce an operator in B 2 (Ω).Proposition 2.1. Let T be a bounded linear operator of the form T 0 S 0 T 1 . Suppose that the two operators T 0 , T 1 are in B 1 (Ω). Then the operator T is in B 2 (Ω).Proof. Suppose T 0 and T 1 are defined on the Hilbert spaces H 0 and H 1 , respectively. Elementary considerations from index theory of Fredholm operators shows that the operator T is Fredholm and ind(T ) = ind(T 0 ) + ind(T 1 ) (cf. [2, page 360]). Therefore, to complete the proof that T is in B 2 (Ω), all we have to do is prove that the vectors in the kernel ker(T − w), w ∈ Ω, span the Hilbert space H = H 0 ⊕ H 1 .Let γ 0 and t 1 be non-vanishing holomorphic sections for the two line bundles E 0 and E 1 corresponding to the operators T 0 and T 1 , respectively. For each w ∈ Ω, the operator T 0 − w 0

Abstract. Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator A in L(H) is said to be strongly irreducible, if A ′ (T), the commutant of A, has no non-trivial idempotent. An operator A in L(H) is said to be a Cowen-Douglas operator, if there exists Ω, a connected open subset of C, and n, a positive integer, such thatIn the paper, we give a similarity classification of strongly irreducible Cowen-Douglas operators by using the K 0 -group of the commutant algebra as an invariant.

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Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator A in L(H) is said to be a Cowen-Douglas operator if there exist , a connected open subset of complex plane C, and n, a positive integer, such thatIn the paper, we give a similarity classification of Cowen-Douglas operators by using the ordered K-group of the commutant algebra as an invariant, and characterize the maximal ideals of the commutant algebras of Cowen-Douglas operators. The theorem greatly generalizes the main result in (Canada J. Math. 156(4) (2004) 742) by simply removing the restriction of strong irreducibility of the operators. The research is also partially inspired by the recent classification theory of simple AH algebras of Elliott-Gong in (Documenta Math. 7 (2002) 255; ଁ

Abstract. Let A be an AI algebra; that is, A is the C * -algebra inductive limit of a sequence

A C * -algebra A is said to have the ideal property if each closed two-sided ideal of A is generated by the projections inside the ideal, as a closed two sided ideal. C * -algebras with the ideal property are generalization and unification of real rank zero C * -algebras and unital simple C * -algebras. It is long to be expected that an invariant (see [Stev] and [Ji-Jiang], [Jiang-Wang] and [Jiang1]) , we call it Inv 0 (A) (see the introduction), consisting of scaled ordered total K-group (K(A), K(A) + , ΣA) Λ (used in the real rank zero case), the tracial state space T (pAp) of cutting down algebra pAp as part of Elliott invariant of pAp (for each [p] ∈ ΣA) with a certain compatibility, is the complete invariant for certain well behaved class of C * -algebras with the ideal property (e.g., AH algebras with no dimension growth). In this paper, we will construct two non isomorphic AT algebras A and B with the ideal property such that Inv 0 (A) ∼ = Inv 0 (B). The invariant to differentiate the two algebras is the Hausdorffifized algebraic K 1 -groups U (pAp)/DU (pAp) (for each [p] ∈ ΣA) with a certain compatibility condition. It will be proved in [GJL] that, adding this new ingredients, the invariant will become the complete invariant for AH algebras (of no dimension growth) with the ideal property.