Algebraic properties of isometries between groups of invertible elements in Banach algebras

We prove that an isometry T between open subgroups of the invertible groups of unital Banach algebras A and B is extended to a real-linear isometry up to translation between these Banach algebras. While a unital isometry between unital semisimple commutative Banach algebras need not be multiplicative, we prove in this paper that if A is commutative and A or B are semisimple, then (T(eA))⁻¹T is extended to an isometric real algebra isomorphism from A onto B. In particular, A−1 is isometric as a metric space to B−1 if and only if they are isometrically isomorphic to each other as metrizable groups if and only if A is isometrically isomorphic to B as a real Banach algebra; it is compared by the example of ?elazko concerning on non-isomorphic Banach algebras with the homeomorphically isomorphic invertible groups. Isometries between open subgroups of the invertible groups of unital closed standard operator algebras on Banach spaces are investigated and their general forms are given.     

关于Gelfand-Mazur型的两个定理

证明了两个Gelfand-Mazur型的定理，其一是：设A是一单位C^*- wa ovt ,AH≌R，且h∈AK时，e^h具有凸谱集。则A≌C。这一结果回答了Bhatt等人的问题，给出了他们的结果在实情形中的结论。其二，部分地回答了Bhatt等人的另一个问题。结果是：设A是一复单位厄米Banach^*- 代数，假设（i)对任意x∈AH，谱集σA(x)的内部是空集，且C\σA（x)是连通的，（ii)A没有非零零因子，则A同构到C。     

Questions concerning matrix algebras and invariance of spectrum

Proceedings - Mathematical Sciences - LetA andB be unital Banach algebras withA a subalgebra ofB. Denote the algebra of alln xn matrices with entries fromA byM n (A). In this paper we prove some...     

THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS

1. IntroductionGiven a locally compact abelian group G and a multiplier p on G, one can associate tothem the twisted group C*-algebra C*(G, p), which is the universal object for unitary prepresentations of G. C* (Zm, p) is said to be a noncommutative torns of rank m and denotedby A.. The multiplier p determines a subgroup S. of G, called its symmetry group, andthe multiplier p is called totally skew if the symmetry group S. is trivial. And A. is calledcompletely irrational if p is totally…     

Simplicity of the Lie Algebra of Skew Symmetric Elements of a Leavitt Path Algebra

For a field F of characteristic not 2 and a directed row-finite graph Γ let L(Γ) be the Leavitt Path Algebra with standard involution *. We study the lie algebra of K = K(L(Γ), *) of * ?skew-symmetric elements and find necessary and sufficient conditions for the Lie algebra [K, K] to be simple.     

LENTICULAR NONCOMMUTATIVE TORI

All C*-algebras of sections of locally trivial C* -algebra bundles over ∏i=1sLki(ni) with fibres Aw Mc(C) are constructed, under the assumption that every completely irrational noncommutative torus Aw is realized as an inductive limit of circle algebras, where Lki (ni) are lens spaces. Let Lcd be a cd-homogeneous C*-algebra over whose cd-homogeneous C*-subalgebra restricted to the subspace Tr × T2 is realized as C(Tr) A1/d Mc(C), and of which no non-trivial matrix algebra can be factored out.The lenticular noncommutative torus Lpcd is defined by twisting in by a totally skew multiplier p on Tr+2 × Zm-2. It is shown that is isomorphic to if and only if the set of prime factors of cd is a subset of the set of prime factors of p, and that Lpcd is not stablyisomorphic to if the cd-homogeneous C*-subalgebra of Lpcd restricted to some subspace LkiLki (ni) is realized as the crossed product by the obvious non-trivial action of Zki on a cd/ki-homogeneous C*-algebra over S2ni+1 for ki an integer greater than     

On the Equi-nuclearity of Roe Algebras of Metric Spaces

The authors define the equi-nuclearity of uniform Roe algebras of a family of metric spaces. For a discrete metric space X with bounded geometry which is covered by a family of subspaces {Xi}∞i=1, if {Cu*(Xi)}∞i=1 are equi-nuclear and under some proper gluing conditions, it is proved that Cu*(X) is nuclear. Furthermore, it is claimed that in general, the coarse Roe algebra C*(X) is not nuclear.     

Minus partial order and linear preservers

In this paper, we investigate the properties of minus partial order in unital rings. We generalize several results well known for matrices and bounded linear operators on Banach spaces. We also study linear maps preserving the minus partial order in unital semisimple Banach algebras with essential socle.     

Quasidiagonal representations of nilpotent groups

We show that every unitary representation of a discrete solvable virtually nilpotent group G is quasidiagonal. Roughly speaking, this says that every unitary representation of G   approximately decomposes as a direct sum of finite dimensional approximate representations. In operator algebraic terms we show that C^?(G)$C^{?}(G)$ is strongly quasidiagonal.     

A solution to a problem on invertible disjointness preserving operators

We construct an invertible disjointness preserving operator on a normed lattice such that is not disjointness preserving.

     

Random spectral theorems of self-adjoint random linear operators on complete complex random inner product modules

The purpose of this article is to give two random spectral theorems of self-adjoint random linear operators on complete complex random inner product modules.     

C*-bialgebra Defined as the Direct Sum of UHF Algebras

Let 𝒜₀(*) denote the direct sum of a certain set of uniformly hyperfinite (UHF) algebras, and let 𝒜(*) ≡ C ⊕ 𝒜₀(*). We introduce a non-cocommutative comultiplication Δ_? on 𝒜(*), and give an example of comodule-C*-algebra of the C*-bialgebra (𝒜(*), Δ_?). With respect to Δ_?, we define a nonsymmetric tensor product of *-representations of UHF algebras and show tensor product formulas of Gel'fand–Na\u?mark–Segal (GNS) representations by product states.     

On the Moore-Penrose inverse, EP Banach space operators, and EP Banach algebra elements

The main concern of this note is the Moore-Penrose inverse in the context of Banach spaces and algebras. Especially attention will be given to a particular class of elements with the aforementioned inverse, namely EP Banach space operators and Banach algebra elements, which will be studied and characterized extending well-known results obtained in the frame of Hilbert space operators and C^∗-algebra elements.     

Dual Toeplitz operators on orthogonal complement of the harmonic Dirichlet space

In this paper, we study some algebraic and spectral properties of dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space of the unit disk.     

A boolean transfer principle from L*-Algebras to AL*-Algebras

Just as Kaplansky [4] has introduced the notion of an AW*-module as a generalization of a complex Hilbert space, we introduce the notion of an AL*-algebra, which is a generalization of that of an L*-algebra invented by Schue [9, 10]. By using Boolean valued methods developed by Ozawa [6–8], Takeuti [11–13] and others, we establish its basic properties including a fundamental structure theorem. This paper should be regarded as a continuation or our previous paper [5], the familiarity with which is presupposed. MSC: 03C90, 03E40, 17B65, 46L10.     

Group inverse and core inverse in Banach and C*-algebras

Abstract

In this paper, we have focused our study on the acute perturbation of the group inverse for the elements of Banach algebra with respect to the spectral radius. We also give perturbation analysis for the core inverse in C*- algebra. The perturbation bounds for the core inverse under some conditions are presented. Additionally, this paper extends the results obtained in [11, 14].     

*-representations on Banach *-algebras

We study notions of -bounded linear functionals and represent-
able functionals on Banach *-algebras. An equivalence between these two is established for general Banach *-algebras. In particular, we characterize -bounded linear functionals on Banach *-algebras with approximate identity and isometric involution. In addition, we prove a result on representation of -bounded positive linear functionals in terms of cyclic vectors for the corresponding *-representation.

     

The stable rank of full corners in C*-algebras

We give a treatment of Rieffel's theory of stable rank for C*-algebras in terms of left invertibility of generalized nonsquare matrices, and prove that if is a full projection in a unital C*-algebra , then the stable rank of the corner is at least as large as the stable rank of .