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 country collage in East Lansing, Michigan, on June 30-July three, 2008. This quantity additionally incorporates a kaleidoscope of papers in response to numerous of the extra very important talks provided on the assembly. It offers 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 broad references to past paintings, this quantity permits the reader to speedy achieve an outline of present job within the box and turn into accustomed to some of the fresh sub-branches of its improvement

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**Extra resources for Advances in P-adic and Non-archimedean Analysis: Tenth International Conference June 30-july 3, 2008 Michigan State University East Lansing, Michigan**

**Sample text**

M TS (M) Let S be a regular sequence. For every maximal ideal M of A A is equivalent to an containing I(S), the K-Banach algebra quotient norm of M absolute value extending this of K. 1: ideal. For every regular ultraﬁlter U, J (U) is a univalent maximal Remark: Consider a maximal ideal M of A and suppose that there exists φ ∈ M ultm (A, . ) which does not lie in the closure of M ulta (A, . ) (which obviously implies that K is not spherically complete). Then M is not univalent A and therefore the K-Banach algebra quotient norm of the ﬁeld is not equivalent M to its norm .

N]q ! 2. An analogue of Jackson derivative 1 . We assume here that Qp ⊂ K and that q ∈ K is such that |q − 1| < |p| p−1 . f (X + 1) − f (X) For f ∈ K{X} let us put ∇q (f )(X) = . Hence, one deﬁnes q X (q − 1) a continuous linear endomorphism ∇q of K{X}, called the second kind of Jackson 1 1 f and since ∇q (X) = X , one sees derivative. One has ∇q (f ) ≤ |q − 1| q (q − 1) 1 that ∇q = . |q − 1| We have deﬁned previously the restricted power series Ψ0,q (X) = 1 and n ≥ n−1 [X − j]q , with [X − j]q = 1, Ψn,q (X) = j=0 q X−j − 1 and showed that they form q−1 an orthonormal basis of K{X}.

Is the Banach algebra Pq which is a completion of Opq an integral ring ? Linked to the theory of q-diﬀerence equations, it should be interesting to study the spectrum of the elements of Pq and naturally that of the elements of the other algebras of operators we have encountered until now. 2. The quantum plane algebras Pc,q , for |q − 1| < |p| p−1 . We assume in this subsection that K is an extension of the p-adic ﬁeld Qp and −1 that q ∈ K is such that |q − 1| < |p| p−1 For any restricted formal power series f , one has τ1 ◦ Zq (f )(X) = Zq (f )(X + 1) = q X+1 Xf (X + 1) = qZq ◦ τ1 (f )(X) =⇒ τ1 ◦ Zq = qZq ◦ τ1 (C6 ).