Kadison has shown that local derivations from a von Neumann algebra into any dual bimodule are derivations. In this paper we extend this result to local derivations from any -algebra into any Banach -bimodule . Most of the work is involved with establishing this result when is a commutative -algebra with one self-adjoint generator. A known result of the author about Jordan derivations then completes the argument. We show that these results do not extend to the algebra of continuously differentiable functions on . We also give an automatic continuity result, that is, we show that local derivations on -algebras are continuous even if not assumed a priori to be so.
We study the finite groups for which the set of irreducible complex character degrees consists of the two most extreme possible values, that is, and . We are easily reduced to finite -groups, for which we derive the following group theoretical characterization: they are the -groups such that is a square and whose only normal subgroups are those containing or contained in . By analogy, we also deal with -groups such that is not a square, and we prove that if and only if a similar property holds: for any , either or . The proof of these results requires a detailed analysis of the structure of the -groups with any of the conditions above on normal subgroups, which is interesting for its own sake. It is especially remarkable that these groups have small nilpotency class and that, if the nilpotency class is greater than , then the index of the centre is small, and in some cases we may even bound the order of .
The two main theorems proved here are as follows: If is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization of the family of finite -module complexes with fixed sequence of dimensions and an ``almost projective' complex , there exists a canonical vector space embedding
where is the pertinent product of general linear groups acting on , tangent spaces at are denoted by , and is identified with its image in the derived category .
Let be a smooth projective curve over a field . For each closed point of let be the coordinate ring of the affine curve obtained by removing from . Serre has proved that is isomorphic to the fundamental group, , of a graph of groups , where is a tree with at most one non-terminal vertex. Moreover the subgroups of attached to the terminal vertices of are in one-one correspondence with the elements of , the ideal class group of . This extends an earlier result of Nagao for the simplest case .
Serre's proof is based on applying the theory of groups acting on trees to the quotient graph , where is the associated Bruhat-Tits building. To determine he makes extensive use of the theory of vector bundles (of rank 2) over . In this paper we determine using a more elementary approach which involves substantially less algebraic geometry.
The subgroups attached to the edges of are determined (in part) by a set of positive integers , say. In this paper we prove that is bounded, even when Cl is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of , involving unipotent and elementary matrices.
In a fibration we show that finiteness conditions on force the homology Serre spectral sequence with -coefficients to collapse at some finite term. This in particular implies that as graded vector spaces, is ``almost' isomorphic to . One consequence is the conclusion that is elliptic if and only if and are.
We show that a simply connected homotopy associative and homotopy commutative mod -space with finitely generated mod cohomology is homotopy equivalent to a finite product of , , the three-connected cover and the homotopy fiber of the map for . Our result also shows that a connected -space in the sense of Sugawara with finitely generated mod cohomology has the homotopy type of a finite product of , and for .
We prove the uniqueness of Riemann solutions in the class of entropy solutions in for the system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global -stability of the Riemann solutions even in the class of entropy solutions in with arbitrarily large oscillation for the system. We apply our framework established earlier to show that the uniqueness of Riemann solutions implies their inviscid asymptotic stability under perturbation of the Riemann initial data, as long as the corresponding solutions are in and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions , piecewise Lipschitz in , for any 0$">.
for some integer , where the 's are roots of unity of . Moreover, there are two effective constants and such that the least integer (for a fixed ) is less than , where
Let be a bounded symmetric domain in a complex vector space with a real form and be the real bounded symmetric domain in the real vector space . We construct the Berezin kernel and consider the Berezin transform on the -space on . The corresponding representation of is then unitarily equivalent to the restriction to of a scalar holomorphic discrete series of holomorphic functions on and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the -space.
Let be an -primary ideal in a Gorenstein local ring (, ) with , and assume that contains a parameter ideal in as a reduction. We say that is a good ideal in if is a Gorenstein ring with . The associated graded ring of is a Gorenstein ring with if and only if . Hence good ideals in our sense are good ones next to the parameter ideals in . A basic theory of good ideals is developed in this paper. We have that is a good ideal in if and only if and . First a criterion for finite-dimensional Gorenstein graded algebras over fields to have nonempty sets of good ideals will be given. Second in the case where we will give a correspondence theorem between the set and the set of certain overrings of . A characterization of good ideals in the case where will be given in terms of the goodness in their powers. Thanks to Kato's Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set of good ideals in heavily depends on . The set may be empty if , while is necessarily infinite if and contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring in three variables over a field . Examples are given to illustrate the theorems.
Let be the Bessel operator with matricial coefficients defined on by
where is a diagonal matrix and let be an matrix-valued function. In this work, we prove that there exists an isomorphism on the space of even , -valued functions which transmutes and . This allows us to define generalized translation operators and to develop harmonic analysis associated with . By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.