By Martin Berz, Khodr Shamseddine

This quantity comprises the complaints of the 10th overseas convention on p-adic and Non-Archimedean research, held at Michigan kingdom collage in East Lansing, Michigan, on June 30-July three, 2008. This quantity additionally includes a kaleidoscope of papers in keeping with numerous of the extra vital talks provided on the assembly. It presents a state-of-the-art connection to a few of crucial fresh advancements within the box. via a mix of survey papers, learn articles, and vast references to previous paintings, this quantity permits the reader to fast achieve an summary of present job within the box and develop into accustomed to a number of the contemporary sub-branches of its improvement

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3. The family K{X}. (j) (X − 1)q is an orthonormal basis of j≥0 24 12 BERTIN Bertin DIARRA Diarra (j) −††− Let (Dq )j≥0 be the elementary q-diﬀerence operators deﬁned by (0) (j) setting Dq = id, Dq = (hq − id)(j) = (hq − id) · · · (hq − q (j−1) id), j ≥ 1. j−1 (j) − • − For the integers n and j ≥ 0, one has Dq (X n ) = (q n − q ) · X n = =0 (j) (q n − 1)(j) · X n , with Dq (X n ) = 0, if j ≥ n + 1. For 0 ≤ j ≤ n, one has (q n − 1)(j) n = [ ]q . Since (X n )n≥0 is an orthonormal basis of K{X}, one sees that j (q j − 1)(j) (j) Dq = sup |(q n − 1)(j) | = |(q j − 1)(j) |.

7] A. Escassut. Ultrametric Banach Algebra. , Singapore (2003). [8] A. Escassut and N. Ma¨ınetti. 79-61, (1998). [9] A. Escassut and N. Ma¨ınetti. On Ideals of the Algebra of p-adic Bounded Analytic Functions on a Disk Bulletin of the Belgian Mathematical Society, Special issue for the Proceedings of the 9-th International Conference on p-adic Functional Analysis. [10] A. Escassut and N. Ma¨ınetti,. About the ultrametric Corona problem Bulletin des Sciences Math´ ematiques 132, p. 382-394 (2008) [11] A.

One then sees that the algebra Wc,q . is isomorphic to the quantum Weyl algebra Aq . m αi Ψi,q (X) ∈ K[q X ], then one has −††− It is readily seen that if Q = i=0 m m Q(Zq ) = αi βi,q (Zq ). Since Q(Zq )(1) = i=0 m αi βi,q (Zq )(1) = i=0 obtains Q(Zq ) = sup |αi |. αi Ψi,q (X), one i=0 0≤i≤m Notice that if the element f of K{X} is such that f = 1, then the operator mf of multiplication by f is an isometry and this because the norm on K{X} is multiplicative. As a consequence each operator βi,q (Zq ), beeing the operator of multiplication by Ψi,q (X), is an isometry.

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