Representations of double coset hypergroups and induced representations

The principal goal of this paper is to investigate the representation theory of double coset hypergroups. IfK=G//H is a double coset hypergroup, representations ofK can canonically be obtained from those ofG. However, not every representation ofK originates from this construction in general, i.e., extends to a representation ofG. Properties of this construction are discussed, and as the main result it turns out that extending representations ofK is compatible with the inducing process (as introduced in [7]). It follows that a representation weakly contained in the left-regular representation ofK always admits an extension toG. Furthermore, we realize the Gelfand pair (where are a local field andR its ring of integers) as a polynomial hypergroup on ℕ₀ and characterize the (proper) subset of its dual consisting of extensible representations.     

Approximation in the mean by rational functions

In this paper we proved that for a large class of compact subsetsK in the complex plane,R(K) is dense inL ^q () if and only if the set of analytic bounded point evaluations forR ^q (K, ) is empty. As a consequence, we showed that this result is true for allK ifR(K) is replaced byA(K). Our main result includes the corresponding result of James Thomson for polynomials approximation as such a special case thatK is a disk.     

On the existence of (l,m,n)-spreads

We call a family ofm-flats of Pⁿ (K) an (l, m, n)-spread overK if it is a partition of the family of alll-flats. We prove two non-existence results for (l, m, n)-spreads in the case:K = GF(q) andl > 0.Alla memoria del Professor Giuseppe TalliniThis research was supported in part by M.U.R.S.T.     

Eine Winkelmetrik zur Begründung Euklidisch — Metrischer Ebenen

In a recent paper by H. KARZEL and R. STANIK, Abh. Math. Sem. Univ. Hamburg, a characterization of euclidean metric planes (the metric is defined there by a separable quadratic field extension L K of the base field K) by means of a congruence relation on segments is given. Here we develop a theory of congruence of angles for these planes to get an axiomatic foundation, which also works when L K is inseparable. The main result is of the following type: Let be an affine plane together with an equivalence relation onG ^x G, whereG denotes the set of lines — any equivalence class is then what we call an angle — such that some simple properties (W1) to (W3) of angles and one further condition (V) on equality of certain angles in quadrangles hold, then is pappian and results from a quadratic field extension L K.     

The dimension of the kernel in an intersection of starshaped sets

We establish the following Helly-type theorem: Let ${\cal K}$ be a family of compact sets in $\mathbb{R}^d$. If every d + 1 (not necessarily distinct) members of ${\cal K}$ intersect in a starshaped set whose kernel contains a translate of set A, then $\cap \{ K : K\; \hbox{in}\; {\cal K} \}$ also is a starshaped set whose kernel contains a translate of A. An analogous result holds when ${\cal K}$ is a finite family of closed sets in $\mathbb{R}^d$. Moreover, we have the following planar result: Define function f on $\{0, 1, 2\}$ by f(0) = f(2) = 3, f(1) = 4. Let ${\cal K}$ be a finite family of closed sets in the plane. For k = 0, 1, 2, if every f(k) (not necessarily distinct) members of ${\cal K}$ intersect in a starshaped set whose kernel has dimension at least k, then $\cap \{K : K\; \hbox{in}\; {\cal K}\}$ also is a starshaped set whose kernel has dimension at least k. The number f(k) is best in each case.Received: 4 June 2002     

Unramified Kummer extensions of prime power degree

We determine, for which xR^* the Kummer extension K(Pⁿx) is unramified over K, where K is a local field of char. 0 and res.char. p, _pnK, and R is the valuation ring of K. This builds on techniques of Hasse. Actually, a much more general result concerning cyclic Galois extensions of commutative rings is proved.     

Subgroups of the Spinor Group that Contain a Split Maximal Torus. II

In the first paper of the series, we proved the standardness of a subgroup H containing a split maximal torus in the split spinor group Spin(n,R) over a field K of characteristic different from 2 containing at least 7 elements under one of the following additional assumptions: (1) H is reducible, (2) H is imprimitive, (3) H contains a nontrivial root element. In the present paper, we complete the proof of a result announced by the author in 1990 and prove the standardness of all intermediate subgroups, provided that n=2l and . For an algebraically closed K, this follows from a classical result of Borel and Tits, and for a finite K this was proved by Seitz. Similar results for subgroups of the orthogonal groups SO(n,R) were previously obtained by the author not only for fields, but for any commutative semilocal rings R with residue fields large enough. Bibliography: 52 titles.     

Subgroups of a spinor group containing a maximal split torus. I

This paper studies subgroups of the split spinor group G=Spin (n, k),n 6, over a field K containing a maximal split torus T=T(n,K)assuming that charK2and ||K ||7.The main result of the first pan of the paper is the reduction of the proof that subgroups containing T are standard to the study of subgroups not containing root unipotent elements.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 49–75, 1991.     

Invariant subspaces for operators in a general II<Subscript>1</Subscript>-factor

Let ℳ be a von Neumann factor of type II₁ with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ _T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace affiliated with ℳ, such that the Brown measure of is concentrated on B and the Brown measure of is concentrated on ℂ∖B. Moreover, is T-hyperinvariant and the trace of is equal to μ _T(B). In particular, if T∈ℳ has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T∈ℳ the limit exists in the strong operator topology, and the projection onto is equal to 1_[0,r](A), for every r>0. Supported by The Danish National Research Foundation.     

Diagonals of positive semigroups

LetE be a Dedekind complete complex Banach lattice and letD denote the diagonal projection from the spaceL _r (E) onto the centerZ(E) ofE. Let {T(t)} _t0 be a positive strongly continuous semigroup of linear operators with generatorA. The first main result is that if the spectral bounds(A) equals to zero, then the functionD(T(t)) is a center valuedp-function. The second main result is that if for >0 the diagonalD(R(, A)) of the resolvent operatorR(, A) is strictly positive, then (D(R(, A))) ^–1 is a center valued Bernstein function. As an application of these results it follows that the order limit lim₀D(R(,A)) exists inZ(E) and equals the order limit lim _m D((R(, A)) ^m ) for any >0.     

极分解定理的注记

设H和K是Hilbert空间.首先,对给定的算子T∈B（H,K）,刻画了集合U_T={U∈B（H,K）：U是部分等距算子并且T=U（T^*T）^1/2}. 其次,对部分等距算子U∈B（H,K）,还给出集合T_U={T∈B（H,K）：N（T）=N（U）,R（T）=R（U）,T=U（T^*T）^1/2}的刻画. 最后,作为主要结果的应用得到了相关结论.     

Topological projective planes and uniform valuations

The valuation topology of any uniformly valued ternary field (K, T, v) can be extended to the projective plane II over (K, T) making it a topological projective plane in the sense of Salzmann. Appealing to Prieß-Crampe's celebrated fixed point theorem for ultrametric spaces, our result allows us to present a wide variety of new, totally disconnected, compact and non-compact topological projective planes.Dedicated to Professor S. Prieß-Crampe on the occasion of her 60th birthday     

The K_2-groups over Finaite Commutative Rings

The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R and finite commutative local ring Ri (1 < i < m), where R Ri and K2(R) = K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.     

Norms in quadratic fields and their relations to non commuting 2×2 matrices I

LetA be a fixed 2×2 integral matrix with irrational characteristic roots. LetB be an arbitrary 2×2 integral matrix. It was previously shown that -det (AB-BA)=norm, where is in the field of the characteristic roots ofA. It is now shown that the 's corresponding to varyingB's can be chosen to form a fractional ideal in this field.     

Cotangent functors of curve singularities

Let R be a complete analytic k-algebra of dimension one, where k is a field of characteristic 0. For a Noetherian normalization , the cotangent functors T_i(R/A, R) are of finite length. It is shown that if R is smoothable, then for all j, where denotes the complementary module. This result is used to give estimates for the length of the torsion of the module of differentials _R/k.     

有限交换环上线性群的Carter子群

令Ｒ为有限交换局部环，Ｋ为其剩余类域．本文研究了Ｒ上一般线性群ＧＬｎＲ的Ｃａｒｔｅｒ子群的存在性及结构．得到的结果是：若ｃｈａｒＫ为奇数或Ｋ＝Ｆ２，ＧＬｎＲ中存在唯一的Ｃａｒｔｅｒ子群的共轭类，即Ｓｙｌｏｗ－２子群的正规化子；若ｃｈａｒＫ＝２且｜Ｋ｜＞２，ＧＬｎＲ中不含Ｃａｒｔｅｒ子群．