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1.
Let A be a unital C*-algebra of real rank zero and B be a unital semisimple complex Banach algebra. We characterize linear maps from A onto B preserving different essential spectral sets and quantities such as the essential spectrum, the (left, right) essential spectrum, the Weyl spectrum, the index and the essential spectral radius.  相似文献   

2.
In J. Math. Anal. Appl. 12 (1995) 258–265, Araujo et al. proved that for any linear biseparating map  from C(X) onto C(Y), where X and Y are completely regular, there exist ω in C(Y) and an homeomorphism h from the realcompactification vX of X onto vY, such that
The compact version of this result was proved before by Jarosz in Bull. Canad. Math. Soc. 33 (1990) 139–144. In Contemp. Math., Vol. 253, 2000, pp. 125–144, Henriksen and Smith asked to what extent the result above can be generalized to a larger class of algebras. In the present paper, we give an answer to that question as follows. Let A and B be uniformly closed Φ-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically a weighted isomorphism, that is, there exist ω in B and a lattice and algebra isomorphism ψ between A and B such that
(a)=ωψ(a) for all aA.
We then assume that every universally σ-complete projection band in A is essentially one-dimensional. Under this extra condition and according to a result from Mem. Amer. Math. Soc. 143 (2000) 679 by Abramovich and Kitover, any linear biseparating map from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, the latter result is a generalization of the aforementioned theorem by Araujo et al. since we also prove that every universally σ-complete projection band in the uniformly closed Φ-algebra C(X) is essentially one-dimensional.  相似文献   

3.
Let 𝒜 and ? be two factor von Neumann algebras. In this article, we prove that a nonlinear bijective map Φ?:?𝒜?→?? satisfies Φ(X*?Y?+?YX*)?=Φ(X)*Φ(Y)?+?Φ(Y)Φ(X)* (?X,?Y?∈?𝒜), if and only if Φ is a *-ring isomorphism. In particular, if 𝒜 and ? are type I factors, then Φ is a unitary isomorphism or conjugate unitary isomorphism.  相似文献   

4.
Let C be an Abelian group. An Abelian group A from a class X of Abelian groups is said to be C H-definable in X if, for any group BX, the isomorphism Hom(C,A) ≅ Hom(C,B) implies that AB. If every group from X is C H-definable in X, then X is called an C H-class. In this paper, we study conditions under which a class of completely decomposable torsion-free Abelian groups is an C H-class, where C is a vector group.  相似文献   

5.
Maps completely preserving spectral functions   总被引:1,自引:0,他引:1  
Let X,Y be infinite dimensional complex Banach spaces and A,B be standard operator algebras on X and Y, respectively. In this paper, we show that surjective maps completely preserving certain spectral function Δ(·) from A to B are isomorphisms, where Δ(·) stands for any one of 13 spectral functions σ(·), σl(·), σr(·), σl(·)∩σr(·), σ(·), ησ(·), σp(·), σc(·), σp(·)∩σc(·), σp(·)∪σc(·), σap(·), σs(·), and σap(·)∩σs(·).  相似文献   

6.
In this paper, it is proved that every surjective linear map preserving identity and zero products in both directions between two nest subalgebras with non-trivial nests of any factor von Neumann algebra is an isomorphism; and that every surjective weakly continuous linear map preserving identity and zero Jordan products in both directions between two nest subalgebras with non-trivial nests of any factor von Neumann algebra is either an isomorphism or an anti-isomorphism.  相似文献   

7.
Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y, respectively. For a bounded linear operator A on X, the peripheral spectrum σπ(A) of A is the set σπ(A)={zσ(A):|z|=maxωσ(A)|ω|}, where σ(A) denotes the spectrum of A. Assume that Φ:AB is a map the range of which contains all operators of rank at most two. It is shown that the map Φ satisfies the condition that σπ(BAB)=σπ(Φ(B)Φ(A)Φ(B)) for all A,BA if and only if there exists a scalar λC with λ3=1 and either there exists an invertible operator TB(X,Y) such that Φ(A)=λTAT-1 for every AA; or there exists an invertible operator TB(X,Y) such that Φ(A)=λTAT-1 for every AA. If X=H and Y=K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the Jordan skew semi-triple product BAB are also characterized. Such maps are of the form A?UAU or A?UAtU, where UB(H,K) is a unitary operator, At denotes the transpose of A in an arbitrary but fixed orthonormal basis of H.  相似文献   

8.
Let X and Y be two infinite dimensional real or complex Banach spaces, and let φ: ?(X)?→??(Y) be an additive surjective mapping that preserves semi-Fredholm operators in both directions. In the complex Hilbert space context, Mbekhta and ?emrl [M. Mbekhta and P. ?emrl, Linear maps preserving semi-Fredholm operators and generalized invertibility, Linear Multilinear Algebra 57 (2009), pp. 55–64] determined the structure of the induced map on the Calkin algebra. In this article, we show the following: given an integer n?≥?1, if φ preserves in both directions ? n (X) (resp., 𝒬 n (X)), the set of semi-Fredholm operators on X of non-positive (resp., non-negative) index, having dimension of the kernel (resp., codimension of the range) less than n, then φ(T)?=?UTV for all T or φ(T)?=?UT*V for all T, where U and V are two bijective bounded linear, or conjugate linear, mappings between suitable spaces.  相似文献   

9.
The map F:XY is refinable if for each >0 there is an -map f from X onto Y that is -close to F. The closed set A in X is N-elementary if each neighborhood U of A contains a neighborhood V such that the natural homomorphism N(U) → N(V) has finitely generated image. If X is a compact ANR, then every closed subset is N-elementary for every N.Suppose F:XY is a refinable map between compacta. Then:If B is a compactum in Y such that F-1B is N-elementary in X then F induces an isomorphism from N(B) to N(F-1B). In particular, if X is an ANR, then F induces isomorphisms N(B) N (F-1B) and N(F-1B) N(B).If X=S3 and Y=S3/A, then A is cellular.If X is a finite-dimensional ANR, then Y is an ANR if one of the following is true: (1) Y is LC1, (2) F-1(y) is locally connected for each y Y, (3) F-1(y) is approximately 1-connected for each y Y or (4) for each >0 the f in the above definition can be chosen to be monotone.Applications are also made to generalized manifolds and ANR's in 2-dimensional manifolds.  相似文献   

10.
Let ? be a zero-product preserving bijective bounded linear map from a unital algebra A onto a unital algebra B such that ?(1)=k. We show that if A is a CSL algebra on a Hilbert space or a J-lattice algebra on a Banach space then there exists an isomorphism ψ from A onto B such that ?=kψ. For a nest algebra A in a factor von Neumann algebra, we characterize the linear maps on A such that δ(x)y+xδ(y)=0 for all x,yA with xy=0.  相似文献   

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