C^(*)-Isomorphisms Associated with Two Projections on a Hilbert C^(*)-Module

Motivated by two norm equations used to characterize the Friedrichs angle,this paper studies C^(*)-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections.A triple(P,Q,H)is said to be matched if is a Hilbert C^(*)-module,P and Q are projections on H such that their infimum P∧Q exists as an element of L(H),where L(H)denotes the set of all adjointable operators on H.The C^(*)-sub algebras of L(H)generated by elements in{P-P∧Q,Q-P∧Q,I}and{P,Q,P∧Q,I}are denoted by i(P,Q,H)and o(P,Q,H),respectively.It is proved that each faithful representation(π,X)of o(P,Q,H)can induce a faithful representation(π,X)of i(P,Q,H)such that π~(P−P∧Q)=π(P)−π(P)∧π(Q),π~(Q−P∧Q)=π(Q)−π(P)∧π(Q)..When(P,Q)is semi-harmonious,that is,R(P+Q) and R(2I−P−Q) are both orthogonally complemented in H,it is shown that i(P,Q,H)and i(I-Q,I-P,H)are unitarily equivalent via a unitary operator in L(H).A counterexample is constructed,which shows that the same may be not true when(P,Q)fails to be semi-harmonious.Likewise,a counterexample is constructed such that(P,Q)is semi-harmonious,whereas(P,I-Q)is not semi-harmonious.Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C^(*)-modules are also provided.

关于图的同构判定方法的探讨

对于两图的同构的判定方法进行较深入的探讨,给出判定两图同构和判定两图不同构的几种方法,并对其判定方法的优劣进行比较.

On Maps Preserving Unitarily Invariant Norms of the Spectral Geometric Mean

We consider maps on positive definite cones of von Neumann algebras preserving unitarily invariant norms of the spectral geometric means. The main results concern Jordan *-isomorphisms between C*-algebras, and show that they are characterized by the preservation of unitarily invariant norms of those operations.

浅谈同构在代数中的应用

本文通过介绍同构在向量空间、群、环、域上的一些应用,说明同构在代数学中的重要性.

Small into-isomorphism from L_∞ (A,μ) into L_∞ (B,v)

In this paper we shall assert that if T is an isomorphism of L∞(Ω1, A, μ) into L∞(Ω2, B, υ) satisfying the condition ‖T‖·‖T ?1‖?1+? for ?∈ $\left( {0,\frac{1}{5}} \right)$ , then $\frac{T}{{\parallel T\parallel }}$ is close to an isometry with an error less than 6ε in some conditions.

THE ■■-LEMMA UNDER SURJECTIVE MAPS

We consider the ■■-lemma for complex manifolds under surjective holomorphic maps.Furthermore,using Deligne-Griffiths-Morgan-Sullivan’s theorem,we prove that a product compact complex manifold satisfies the ■■-lemma if and only if all of its components do as well.

ON THE APPROXIMATELY SIMILARITY OF OPERATORS

The author shows that a rank-preserving *-isorporphism between separable C*-algebras with unity is approximately equivalent to the identity representation. Some applications are made for approximately similarity of n-tuples of operators.

A Characterization of *-Automorphism on B(H)

Let H be a Hilbert space and A be a standard ＊-subalgebra of B（H）. We show that a bijective map Ф ： A →A preserves the Lie-skew product AB - BA＊ if and only if there is a unitary or conjugate unitary operator U ∈A（H） such that Ф（A） = UAU＊ for all A ∈ A, that is, Фis a linear ＊ -isomorphism or a conjugate linear ＊-isomorphism.