Abstract. The space POVM H (X) of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, is studied from the perspectives of classical and non-classical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C * -algebra C(X) ⊗ B(H) into B(H). This association is achieved by using an operator-valued integral in which non-classical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz Representation Theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps φ : C(X) ⊗ B(H) → B(H). The extremal and C * -extremal points of POVM H (X) are determined.
Abstract. We provide an explicit construction of representations in the discrete spectrum of two p-adic symmetric spaces. We consider GL n (F) × GL n (F) GL n (F) and GL n (F) GL n (E), where E is a quadratic Galois extension of a nonarchimedean local eld F of characteristic zero and odd residual characteristic. e proof of the main result involves an application of a symmetric space version of Casselman's Criterion for square integrability due to Kato and Takano.
Let F be a p-adic field (p = 2), let E be a quadratic Galois extension of F , and let n ≥ 2. We construct representations in the discrete spectrum of the p-adic symmetric space H\G, where G = GL2n(E) and H = U E/F (F ) is a quasi-split unitary group over F .
Let F be a p-adic field of characteristic zero and odd residual characteristic. Let Sp 2n (F) denote the symplectic group defined over F , where n ≥ 2. We prove that the Speh representations U(δ, 2), where δ is a discrete series representation of GL n (F), lie in the discrete spectrum of the p-adic symmetric space Sp 2n (F)\GL 2n (F).
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