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1.
First, we introduce the notion of f
I-sets and investigate their properties in ideal topological spaces. Then, we also introduce the notions of R
I
C-continuous, f
I-continuous and contra*-continuous functions and we show that a function f: (X,τ,I) to (Y,φ) is R
I
C -continuous if and only if it is f
I-continuous and contra*-continuous.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
2.
The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m
i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer
i such that limi→∞
f
r
i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension.
the National Natural Science Foundation of China. 相似文献
3.
Let X be a compact metric space and let Lip(X) be the Banach algebra of all scalar- valued Lipschitz functions on X, endowed with a natural norm. For each f ∈ Lip(X), σπ(f) denotes the peripheral spectrum of f. We state that any map Φ from Lip(X) onto Lip(Y) which preserves multiplicatively the peripheral spectrum:
σπ(Φ(f)Φ(g)) = σπ(fg), A↓f, g ∈ Lip(X)
is a weighted composition operator of the form Φ(f) = τ· (f °φ) for all f ∈ Lip(X), where τ : Y → {-1, 1} is a Lipschitz function and φ : Y→ X is a Lipschitz homeomorphism. As a consequence of this result, any multiplicatively spectrum-preserving surjective map between Lip(X)-algebras is of the form above. 相似文献
σπ(Φ(f)Φ(g)) = σπ(fg), A↓f, g ∈ Lip(X)
is a weighted composition operator of the form Φ(f) = τ· (f °φ) for all f ∈ Lip(X), where τ : Y → {-1, 1} is a Lipschitz function and φ : Y→ X is a Lipschitz homeomorphism. As a consequence of this result, any multiplicatively spectrum-preserving surjective map between Lip(X)-algebras is of the form above. 相似文献
4.
5.
Abstract
Let A be a unital simple C*-algebra of real zero, stable rank one, with weakly unperforated K
0(
A) and unique normalized quasi-trace τ, and let X be a compact metric space. We show that two monomorphisms φ, ψ : C(X)→A are approximately unitarily equivalent if and only if φ and ψ induce the same element in KL(C(X), A) and the two lineal functionals τ∘φ and τ∘ψ are equal. We also show that, with an injectivity condition, an almost multiplicative
morphism from C(X) into A with vanishing KK-obstacle is close to a homomorphism.
Research partially supported by NSF Grants DMS 93-01082 (H.L) and DMS-9401515(G.G). This work was reported by the first named
author at West Coast Operator Algebras Seminar (Sept. 1995, Eugene, Oregon) 相似文献
6.
V. Yu. Protasov 《Functional Analysis and Its Applications》2011,45(1):46-55
We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x
1 + x
2) ⊂ φ(x
1) + φ(x
2). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: X → Y such that Ax ∈ φ(x), x ∈ X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is
proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces. 相似文献
7.
We provide proper mapping-characterizations of some embedding-like properties weaker than -embedding. For instance, we show that a subset A of a space X is -embedded in X if and only if for every continuous map g: A → Y into a Banach space Y of weight w(Y) ⩽ λ, there exists a continuous set-valued mapping φ of X into the nonempty compact subsets of Y such that g is a selection for φ∣A (i.e., g(x) ∈ φ(x) for every x ∈ A). On the other hand, we show that a subset A is C*-embedded in X if and only if for every continuous set-valued mapping φ of X into the non-empty compact subsets of a Banach space Y, every continuous selection g: A → Y for φ∣A is continuously extendable to the whole of X. Combining both results we get the well-known mapping-characterization of -embedding which makes more transparent the relation ‘’. Other weak components of -embedding are described in terms of expansions and selections, possible applications are demonstrated as well. 相似文献
8.
Osamu Hatori Takeshi Miura Rumi Shindo Hiroyuki Takagi 《Rendiconti del Circolo Matematico di Palermo》2010,59(2):161-183
Let $
A
$
A
and ℬ be unital semisimple commutative Banach algebras. It is shown that if surjections S,T: $
A
$
A
→ ℬ with S(1)=T(1)= 1 and α ∈ ℂ \ {0} satisfy r(S(a)T(b) − α)= r(ab− α) for all a,b ∈ $
A
$
A
, then S=T and S is a real algebra isomorphism, where r(a) is the spectral radius of a. Let I be a nonempty set, A and B be uniform algebras. Let ρ, τ: I → A and S,T: I → B be maps satisfying σ
π
(S(p)T(q)) ⊂ σ
π
(ρ(p) τ(q)) for all p,q ∈ I, where σ
π
(f) is the peripheral spectrum of f. Suppose that the ranges ρ(I), τ(I) ⊂ A and S(I),T(I) ⊂ B are closed under multiplication in a sense, and contain peaking functions “enough”. There exists a homeomorphism ϕ: Ch(B)→Ch(A) such that S(p)(y)= ρ(p)(ϕ(y)) and T(p)(y)= τ(p)(ϕ(y)) for every p ∈ I and y ∈ Ch(B), where Ch(A) is the Choquet boundary of A. 相似文献
9.
Let [A, a] be a normed operator ideal. We say that [A, a] is boundedly weak*-closed if the following property holds: for all Banach spaces X and Y, if T: X → Y** is an operator such that there exists a bounded net (T
i
)
i∈I
in A(X, Y) satisfying lim
i
〈y*, T
i
x
y*〉 for every x ∈ X and y* ∈ Y*, then T belongs to A(X, Y**). Our main result proves that, when [A, a] is a normed operator ideal with that property, A(X, Y) is complemented in its bidual if and only if there exists a continuous projection from Y** onto Y, regardless of the Banach space X. We also have proved that maximal normed operator ideals are boundedly weak*-closed but, in general, both concepts are different.
相似文献
10.
LetX be a real linear normed space, (G, +) be a topological group, andK be a discrete normal subgroup ofG. We prove that if a continuous at a point or measurable (in the sense specified later) functionf:X →G fulfils the condition:f(x +y) -f(x) -f(y) ∈K whenever ‖x‖ = ‖y‖, then, under some additional assumptions onG,K, andX, there esists a continuous additive functionA :X →G such thatf(x) -A(x) ∈K. 相似文献
11.
Fang Liping 《数学学报(英文版)》1998,14(1):139-144
Letf be a holomorphic self-map of the punctured plane ℂ*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI
0(f)={z ∈ ℂ*:f
n
(z) → 0,n → ∞} andI
∞(f)={z ∈ ℂ*:f
n
(z) → 0,n → ∞}. We try to find the relation betweenI
0(f),I
∞(t) andJ(f). It is proved that both the boundary ofI
0(f) and the boundary ofI
∞)f) equal toJ(f),I
0(f) ∩J(f) ≠ θ andI
∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI
0(f) andI
∞(f) are not doubly-bounded.
Supported by the National Natural Science Foundation of China 相似文献
12.
Jin Chuan HOU Xiu Ling ZHANG 《数学学报(英文版)》2006,22(1):179-186
Let X be an infinite-dimensional complex Banach space and denote by B(X) the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B(X) onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B(X) such that either Φ(T) = cATA^-1 + ψ(T)I for all T, or Φ(T) = cAT*A^-1 + ψ(T)I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero. 相似文献
13.
Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖
X
and ‖.‖
Y
denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖
Y
= ‖fg‖
X
, for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖
X
= ‖Tf Tg + α‖
Y
, f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ {1, −1} and a clopen subset K of c(B) such that for each f ∈ A,
$
Tf\left( y \right) = \left\{ \begin{gathered}
\eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\
- \frac{\alpha }
{{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\
\end{gathered} \right.
$
Tf\left( y \right) = \left\{ \begin{gathered}
\eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\
- \frac{\alpha }
{{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\
\end{gathered} \right.
相似文献
14.
T. Noiri 《Acta Mathematica Hungarica》2003,99(4):315-328
A subset A of a topological space X is said to be β-open [1] if A ⊂ Cl (Int (Cl (A))). A function f : X → Y is said to be β-irresolute [4] if for every β-open set V of Y, f
-1(V) is β-open in X. In this paper we introduce weak and strong forms of β-irresolute functions and obtain several basic properties of such functions.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
15.
Rumi Shindo 《Central European Journal of Mathematics》2010,8(1):135-147
Let A and B be uniform algebras. Suppose that α ≠ 0 and A
1 ⊂ A. Let ρ, τ: A
1 → A and S, T: A
1 → B be mappings. Suppose that ρ(A
1), τ(A
1) and S(A
1), T(A
1) are closed under multiplications and contain expA and expB, respectively. If ‖S(f)T(g) − α‖∞ = ‖ρ(f)τ(g) − α‖∞ for all f, g ∈ A
1, S(e
1)−1 ∈ S(A
1) and S(e
1) ∈ T(A
1) for some e
1 ∈ A
1 with ρ(e
1) = 1, then there exists a real-algebra isomorphism $
\tilde S
$
\tilde S
: A → B such that $
\tilde S
$
\tilde S
(ρ(f)) = S(e
1)−1
S(f) for every f ∈ A
1. We also give some applications of this result. 相似文献
16.
J. Feldman 《Israel Journal of Mathematics》1965,3(2):99-103
A short proof of the Levy continuity theorem in Hilbert space.
In the theory of the normal distribution on a real Hilbert spaceH, certain functionsφ have been shown by L. Gross to give rise to random variablesφ∼ in a natural way; in particular, this is the case for functions which are “uniformly τ-continuous near zero”. Among such
functions are the characteristic functionsφ of probability distributionsm onH, given byφ(y)=∫e
i(y,x)dm(x). The following analogue of the Levy continuity theorem has been proved by Gross: Letφ
j be the characteristic function of the probability measurem
j onH, Then necessary and sufficient that ∫f dm
j → ∫f dm for some probability measurem and all bounded continuousf, is that there exists a functionφ, uniformly τ-continuous near zero, withφ
j∼ →φ∼ in probability.φ turns out, of course, to be the characteristic function ofm. In the present paper we give a short proof of this theorem.
Research supported by National Science Foundation Grant GP-3977. 相似文献
17.
Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σπ (T) of T is defined by σπ (T) = z ∈ σ(T): |z| = maxw∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σπ (φ(S)ϕ(T)) = σπ (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A
1
TA
2
−1 and ϕ(T) = A
2
TA
1
−1 for some bijective bounded linear operators A
1; A
2 of X onto Y, or of the form φ(T) = B
1
T*B
2
−1 and ϕ(T) = B
2
T*B
−1 for some bijective bounded linear operators B
1;B
2 of X* onto Y.
相似文献
18.
For topological spaces X and Y and a metric space Z, we introduce a new class N( X ×Y, Z ) \mathcal{N}\left( {X \times Y,\,Z} \right) of mappings f: X × Y → Z containing all horizontally quasicontinuous mappings continuous with respect to the second variable. It is shown that, for
each mapping f from this class and any countable-type set B in Y, the set C
B
(f) of all points x from X such that f is jointly continuous at any point of the set {x} × B is residual in X: We also prove that if X is a Baire space, Y is a metrizable compact set, Z is a metric space, and f ? N( X ×Y, Z ) f \in \mathcal{N}\left( {X \times Y,\,Z} \right) , then, for any ε > 0, the projection of the set D
ε
(f) of all points p ∈ X × Y at which the oscillation ω
f
(p) ≥ ε onto X is a closed set nowhere dense in X. 相似文献
19.
Ralph deLaubenfels 《Israel Journal of Mathematics》1993,81(1-2):227-255
We show that, whenA generates aC-semigroup, then there existsY such that [M(C)] →Y →X, andA|
Y
, the restriction ofA toY, generates a strongly continuous semigroup, where ↪ means “is continuously embedded in” and ‖x‖[Im(C)]≡‖C
−1
x‖. There also existsW such that [C(W)] →X →W, and an operatorB such thatA=B|
X
andB generates a strongly continuous semigroup onW. If theC-semigroup is exponentially bounded, thenY andW may be chosen to be Banach spaces; in general,Y andW are Frechet spaces. If ρ(A) is nonempty, the converse is also true.
We construct fractional powers of generators of boundedC-semigroups.
We would like to thank R. Bürger for sending preprints, and the referee for pointing out reference [37]. This research was
supported by an Ohio University Research Grant. 相似文献
20.
We obtain asymptotic representations as t ↑ ω, ω ≤ + ∞, for all possible types of P
ω(Y
0, λ
0)-solutions (where Y
0 is zero or ±∞ and −∞ ≤ λ0 ≤ +∞) of nonlinear differential equations y
(n) = α
0
p(t)φ(y), where α
0 ∈ {−1, 1}, p: [a, ω[→]0,+∞[ is a continuous function, and φ is a continuous regularly varying function in a one-sided neighborhood of Y
0. 相似文献
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