On local automorphisms of group algebras of compact groups

We show that with few exceptions every local isometric automorphism of the group algebra of a compact metric group is an isometric automorphism.

     

Local automorphisms of operator algebras on Banach spaces

In this paper we extend a result of Semrl stating that every 2-local automorphism of the full operator algebra on a separable infinite dimensional Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we obtain the same conclusion for the much larger class of Banach spaces with Schauder bases. The proof rests on an analogous statement concerning the 2-local automorphisms of matrix algebras for which we present a short proof. The need to get such a proof was formulated in Semrl's paper.

     

A note on supercyclic vectors of Hilbert space operators

Let H be an infinite dimensional complex Hilbert space and T be a bounded linear operator on H. We show that if there exists $x\in H$ such that the closure of $\{\alpha T^{n}x:\alpha \in \mathbb{C},n?0\}$ is H, then there is a subsequence ${(n_k)}_{k=1}^{\infty }$ such that the closed linear span of $\{T^{n_k}x:k?1\}$ is not the whole space H.     

An algebraic invariant for Jordan automorphisms on B(H): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H)the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that φ is a surjective map from B(H) onto itself. If for everyλ∈ {-1, 1, 2, 3, 1/2, 1/3} and A, B ∈ B(H), A - λB ∈ I(H) (→)φ(A) - λφ(B) ∈ I(H), then φis a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that φ(A) = TAT-1 for all A ∈ B(H), or φ(A) = TA*T-1 for all A ∈ B(H); if, in addition, A - iB ∈ I(H) (→)φ(A) - iφ(B) ∈ I(H), here i is the imaginary unit, then φ is either an automorphism or an anti-automorphism.     

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.     

A note on GK dimension of skew polynomial extensions

Let be a finitely generated commutative domain over an algebraically closed field , an algebra endomorphism of , and a -derivation of . Then if and only if is locally algebraic in the sense that every finite dimensional subspace of is contained in a finite dimensional -stable subspace.

Similarly, if is a finitely generated field over , a -endomorphism of , and a -derivation of , then if and only if is an automorphism of finite order.

     

Local automorphisms of standard operator algebras

Let X be an infinite-dimensional separable real or complex Banach space and A a closed standard operator algebra on X. Then every local automorphism of A is an automorphism. The assumptions of infinite-dimensionality, separability, and closeness are all indispensable.     

一类基于量子程序理论的序列效应代数

空间上的算子理论是量子力学的基本数学框架之一.Hilbert空间效应代数是指小于等于单位算子的正算子集合.我们引入了Hilbert空间效应代数的一类子序列效应代数,并讨论了其上序列积的基本运算性质.我们发现：由于代数结构的不同,这类新的序列效应代数与现有效应代数上的运算性质有很大差异.     

Martin's Axiom is consistent with the existence of nowhere trivial automorphisms

Martin's Axiom does not imply that all automorphisms of are somewhere trivial. An alternate method for obtaining models where every automorphism of is somewhere trivial is explained.

     

Uniform primeness of the Jordan algebra of symmetric operators

In this note we establish the best possible constant for the general lower estimate for the Jacobson - McCrimmon operator on the algebra of symmetric operators acting on a Hilbert space.

     

A note on locally Lipschitzian functions

An oversight in a paper of Correa and Lemaréchal (this journal, 1993) is noted; a counterexample is given.     

Sporadic groups and automorphisms of linear spaces

This article is a contribution to the study of the automorphism groups of finite linear spaces. In particular we look at almost simple groups and prove the following theorem: Let G be an almost simple group and let 𝒮 be a finite linear space on which G acts as a line‐transitive automorphism group. Then the socle of G is not a sporadic group. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 353–362, 2000     

A note on Weil index

Let F be a non-archimedean local field of characteristic 0 and(?)a nontrivial additive character.Weil first defined the Weil indexγ(a,(?))(a∈F~*)in his famous paper,from which we know thatγ(a,(?))γ(b,(?))=γ(ab,(?))γ(1,(?))(a,b)andγ(a,(?))~4 =(-1,-1),where(a,b)is the Hilbert symbol for F.The Weil index plays an important role in the theory of theta series and in the general representation theory.In this paper,we establish an identity relating the Weil indexγ(a,(?))and the Gauss sum.     

A note on Faber operator

For a rectifable Jordan curve Γ with complementary domainsD and D,Anderson conjectured that the Faber operator is a bounded isomorphism between the Besov spaces Bp(1 p ∞) of analytic functions in the unit disk and in the inner domain D,respectively.We point out that the conjecture is not true in the special case p=2,and show that in this case the Faber operator is a bounded isomorphism if and only if Γ is a quasi-circle.     

The hypercentral structure of the group of unitriangular automorphisms of a free metabelian Lie algebra

We describe the hypercentral structure of the group of unitriangular automorphisms of a free metabelian Lie algebra over an arbitrary field. Using it, we prove that this group admits no faithful representation by matrices over a field provided that the algebra rank is at least four.     

A note on Browder spectrum of operator matrices

When A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the Hilbert space H⊕K of the form . In this note, it is shown that the following results in [Hai-Yan Zhang, Hong-Ke Du, Browder spectra of upper-triangular operator matrices, J. Math. Anal. Appl. 323 (2006) 700-707]