A note on k-potence preserving maps

Let M_n be the algebra of all n×n complex matrices and Γ_n the set of all k-potent matrices in M_n. Suppose ?:M_n→M_n is a map satisfying A-λB∈Γ_n implies ?(A)-λ?(B)∈Γ_n, where A, B∈M_n, λ∈C. Then either ? is of the form ?(A)=cTAT^-1, A∈M_n, or ? is of the form ?(A)=cTA^tT^-1, A∈M_n, where T∈M_n is an invertible matrix, c∈C satisfies c^k=c.     

Adding machines, kneading maps, and endpoints

Given a unimodal map f, let I=[c₂,c₁] denote the core and set E={(x₀,x₁,…)∈(I,f)|xi∈ω(c,f) for all i∈N}. It is known that there exist strange adding machines embedded in symmetric tent maps f such that the collection of endpoints of (I,f) is a proper subset of E and such that limk→∞Q(k)≠∞, where Q(k) is the kneading map.We use the partition structure of an adding machine to provide a sufficient condition for x to be an endpoint of (I,f) in the case of an embedded adding machine. We then show there exist strange adding machines embedded in symmetric tent maps for which the collection of endpoints of (I,f) is precisely E. Examples of this behavior are provided where limk→∞Q(k) does and does not equal infinity, and in the case where limk→∞Q(k)=∞, the collection of endpoints of (I,f) is always E.     

Topological classification of affine operators on unitary and Euclidean spaces

We study affine operators on a unitary or Euclidean space U up to topological conjugacy. An affine operator is a map f:U→U of the form f(x)=Ax+b, in which A:U→U is a linear operator and b∈U. Two affine operators f and g are said to be topologically conjugate if g=h^-1fh for some homeomorphism h:U→U.If an affine operator f(x)=Ax+b has a fixed point, then f is topologically conjugate to its linear part A. The problem of classifying linear operators up to topological conjugacy was studied by Kuiper and Robbin [Topological classification of linear endomorphisms, Invent. Math. 19 (2) (1973) 83-106] and other authors.Let f:U→U be an affine operator without fixed point. We prove that f is topologically conjugate to an affine operator g:U→U such that U is an orthogonal direct sum of g-invariant subspaces V and W,

•

the restriction g∣V of g to V is an affine operator that in some orthonormal basis of V has the form

(x₁,x₂,…,x_n)?(x₁+1,x₂,…,x_n-1,εx_n)     

Sums of commutators in non-commutative Banach function spaces

Let M be a II_∞-factor and denote by τ its normal faithful semi-finite trace. For any rearrangement invariant Köthe function space X on [0,+∞[, let X(M,τ) be the associated non-commutative Banach function space. This paper is concerned with ideals in M of the form I_X(M,τ)=M∩X(M,τ) that are contained in L^p(M,τ) for some p>0. It is proved that an element T in I_X(M,τ) is a finite sum of commutators of the form [A,B] with A∈I_X(M,τ) and B∈M if and only if the function belongs to X, where ν_T is the Brown spectral measure of T and t→λ_t(T) is the non-increasing rearrangement of the function λ→|λ| with respect to ν_T. This extends to general Banach function spaces a result obtained by Kalton for quasi-Banach ideals of compact operators and implies that the Dixmier's trace of a quasi-nilpotent element in L^1,∞(M,τ) is always zero.     

On Samelson products in p-localized unitary groups

The unitary group U(n) has elements εi∈π2i+1(U(n)) (0?i?n−1) of its homotopy groups in the stable range. In this paper we show that certain multi Samelson products of type 〈εi,〈εj,εk〉〉 are non-trivial. This leads us to the result that the nilpotency class of the group of the self homotopy set [SU(n),SU(n)] is no less than 3, if 4?n. Also by the power of generalized Samelson products, we can see the further result that, for a prime p and an integer n=pk, nil[SU(n),SU(n)]_(p)?3, if (1) p?7 or (2) p=5 and n≡0 or 1mod4.     

Iterative solution of nonlinear equations of accretive and pseudocontractive types

Let E be a real uniformly smooth Banach space. Let A:D(A)=E→2E be an accretive operator that satisfies the range condition and A⁻¹(0)≠∅. Let {λ_n} and {θ_n} be two real sequences satisfying appropriate conditions, and for z∈E arbitrary, let the sequence {x_n} be generated from arbitrary x₀∈E by x_n+1=x_n−λ_n(u_n+θ_n(x_n−z)), u_n∈Ax_n, n?0. Assume that {u_n} is bounded. It is proved that {x_n} converges strongly to some x∗∈A−1(0). Furthermore, if K is a nonempty closed convex subset of E and T:K→K is a bounded continuous pseudocontractive map with F(T):={Tx=x}≠∅, it is proved that for arbitrary z∈K, the sequence {x_n} generated from x₀∈K by x_n+1=x_n−λ_n((I−T)x_n+θ_n(x_n−z)), n?0, where {λ_n} and {θ_n} are real sequences satisfying appropriate conditions, converges strongly to a fixed point of T.     

Generalizations of weakly peripherally multiplicative maps between uniform algebras

Let A and B be uniform algebras on first-countable, compact Hausdorff spaces X and Y, respectively. For f∈A, the peripheral spectrum of f, denoted by σπ(f)={λ∈σ(f):|λ|=‖f‖}, is the set of spectral values of maximum modulus. A map T:A→B is weakly peripherally multiplicative if σπ(T(f)T(g))∩σπ(fg)≠∅ for all f,g∈A. We show that if T is a surjective, weakly peripherally multiplicative map, then T is a weighted composition operator, extending earlier results. Furthermore, if T₁,T₂:A→B are surjective mappings that satisfy σπ(T₁(f)T₂(g))∩σπ(fg)≠∅ for all f,g∈A, then T₁(f)T₂(1)=T₁(1)T₂(f) for all f∈A, and the map f?T₁(f)T₂(1) is an isometric algebra isomorphism.     

Properties of convergence for the q-Meyer-König and Zeller operators

In this paper, we discuss properties of convergence for the q-Meyer-König and Zeller operators Mn,q. Based on an explicit expression for Mn,q(t²,x) in terms of q-hypergeometric series, we show that for qn∈(0,1], the sequence (Mn,qn(f))_n?1 converges to f uniformly on [0,1] for each f∈C[0,1] if and only if limn→∞qn=1. For fixed q∈(0,1), we prove that the sequence (Mn,q(f)) converges for each f∈C[0,1] and obtain the estimates for the rate of convergence of (Mn,q(f)) by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. We also give explicit formulas of Voronovskaya type for the q-Meyer-König and Zeller operators for fixed 0<q<1. If 0<q<1, f∈C¹[0,1], we show that the rate of convergence for the Meyer-König and Zeller operators is o(qn) if and only if

     

An embedding theorem for partial algebras and the free completion of a free partial algebra within a primitive class

Let U be any nontrivial primitive class of partial algebras, i.e. there existsA ∈ U with |A|≥2, and U is closed with respect to homomorphic images (in the weak sense), subalgebras (on closed subsets) and cartesian products of U-algebras, and let U _f denote the—also nontrivial and primitive—class of all full U-algebras. Then every U-algebra with at least two elements is a relative algebra of some U _f -algebra. For any U-algebraAsetU _A =U _i εI({i}×(A ^K _i—domf _i ^A )), where (K _i) _i εI is the type under consideration. Furthermore let F(N, U) denote any U-algebra U-freely generated by some setN (and let F (M, U _f ) be similarly defined). Then for every nonempty setM there exists a setN satisfyingM ?N such that there exists a bijective mapping σ:U _{F(N, U)} →N ?M satisfying σ((i, α)) ? α(K _i ) for all (i, α) ∈U _{F (N, U)}, and, for the structureg=(g _i)_iεI defined by ,g _i : =f _i ^{F(N, U)} ∪ {(α, σ((i, α))) | (i, α ∈U _{F(N, U)}} id _M induces an isomorphism betweenF(M, U _f ), and (F(N, U)g).     

k-Potence preserving maps without the linearity and surjectivity assumptions

Let M_n be the space of all n × n complex matrices, and let Γ_n be the subset of M_n consisting of all n × n k-potent matrices. We denote by Ψ_n the set of all maps on M_n satisfying A − λB ∈ Γ_n if and only if ?(A) − λ?(B) ∈ Γ_n for every A,B ∈ M_n and λ ∈ C. It was shown that ? ∈ Ψ_n if and only if there exist an invertible matrix P ∈ M_n and c ∈ C with c^k−1 = 1 such that either ?(A) = cPAP⁻¹ for every A ∈ M_n, or ?(A) = cPA^TP⁻¹ for every A ∈ M_n.     

Coincidences of projections and linear n-valued maps of tori

We prove that the Nielsen fixed point number N(φ) of an n-valued map φ:X?X of a compact connected triangulated orientable q-manifold without boundary is equal to the Nielsen coincidence number of the projections of the graph of φ, a subset of X×X, to the two factors. For certain q×q integer matrices A, there exist “linear” n-valued maps Φn,A,σ:Tq?Tq of q-tori that generalize the single-valued maps fA:Tq→Tq induced by the linear transformations TA:Rq→Rq defined by TA(v)=Av. By calculating the Nielsen coincidence number of the projections of its graph, we calculate N(Φn,A,σ) for a large class of linear n-valued maps.     

An Erd?s-Ko-Rado theorem for partial permutations

Let [n] denote the set of positive integers {1,2,…,n}. An r-partial permutation of [n] is a pair (A,f) where A⊆[n], |A|=r and f:A→[n] is an injective map. A set A of r-partial permutations is intersecting if for any (A,f), (B,g)∈A, there exists x∈A∩B such that f(x)=g(x). We prove that for any intersecting family A of r-partial permutations, we have .It seems rather hard to characterize the case of equality. For 8?r?n-3, we show that equality holds if and only if there exist x₀ and ε₀ such that A consists of all (A,f) for which x₀∈A and f(x₀)=ε₀.     

Elementary maps on nest algebras

Let A₁, A₂ be algebras and let M:A₁→A₂, M^∗:A₂→A₁ be maps. An elementary map of A₁×A₂ is an ordered pair (M,M^∗) such that

     

Maps preserving the nilpotency of products of operators

Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:

(a)

There is a bijective bounded linear or conjugate-linear operator S:X→X such that ? has the form A?S[f(A)A]S^-1.

(b)

The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S⁻¹.

If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra M_n of n × n complex matrices, then there exist a map f:M_n→C?{0}, a field automorphism ξ:C→C, and an invertible S ∈ M_n such that ? has one of the following forms:

     

On bilinear maps determined by rank one matrices with some applications

Let M_n be the algebra of all n×n matrix over a field F, A a rank one matrix in M_n. In this article it is shown that if a bilinear map ? from M_n×M_n to M_n satisfies the condition that ?(u,v)=?(I,A) whenever u·v=A, then there exists a linear map φ from M_n to M_n such that . If ? is further assumed to be symmetric then there exists a matrix B such that ?(x,y)=tr(xy)B for all x,y∈M_n. Applying the main result we prove that if a linear map on M_n is desirable at a rank one matrix then it is a derivation, and if an invertible linear map on M_n is automorphisable at a rank one matrix then it is an automorphism. In other words, each rank one matrix in M_n is an all-desirable point and an all-automorphisable point, respectively.     

Beta-expansion and continued fraction expansion

For any real number β>1, let ε(1,β)=(ε₁(1),ε₂(1),…,εn(1),…) be the infinite β-expansion of 1. Define . Let x∈[0,1) be an irrational number. We denote by kn(x) the exact number of partial quotients in the continued fraction expansion of x given by the first n digits in the β-expansion of x. If is bounded, we obtain that for all x∈[0,1)?Q,

     

Lelek maps and n-dimensional maps from compacta to polyhedra

For a natural number m?0, a map from a compactum X to a metric space Y is an m-dimensional Lelek map if the union of all non-trivial continua contained in the fibers of f is of dimension ?m. In [M. Levin, Certain finite-dimensional maps and their application to hyperspaces, Israel J. Math. 105 (1998) 257-262], Levin proved that in the space C(X,I) of all maps of an n-dimensional compactum X to the unit interval I=[0,1], almost all maps are (n−1)-dimensional Lelek maps. Moreover, he showed that in the space C(X,Ik) of all maps of an n-dimensional compactum X to the k-dimensional cube Ik (k?1), almost all maps are (n−k)-dimensional Lelek maps. In this paper, we generalize Levin's result. For any (separable) metric space Y, we define the piecewise embedding dimension ped(Y) of Y and we prove that in the space C(X,Y) of all maps of an n-dimensional compactum X to a complete metric ANR Y, almost all maps are (n−k)-dimensional Lelek maps, where k=ped(Y). As a corollary, we prove that in the space C(X,Y) of all maps of an n-dimensional compactum X to a Peano curve Y, almost all maps are (n−1)-dimensional Lelek maps and in the space C(X,M) of all maps of an n-dimensional compactum X to a k-dimensional Menger manifold M, almost all maps are (n−k)-dimensional Lelek maps. It is known that k-dimensional Lelek maps are k-dimensional maps for k?0.     

Asymptotically stable sets and the stability of ω-limit sets

Let C be the collection of continuous self-maps of the unit interval I=[0,1] to itself. For f∈C and x∈I, let ω(x,f) be the ω-limit set of f generated by x, and following Block and Coppel, we take Q(x,f) to be the intersection of all the asymptotically stable sets of f containing ω(x,f). We show that Q(x,f) tells us quite a bit about the stability of ω(x,f) subject to perturbations of either x or f, or both. For example, a chain recurrent point y is contained in Q(x,f) if and only if there are arbitrarily small perturbations of f to a new function g that give us y as a point of ω(x,g). We also study the structure of the map Q taking (x,f)∈I×C to Q(x,f). We prove that Q is upper semicontinuous and a Baire 1 function, hence continuous on a residual subset of I×C. We also consider the map given by x?Q(x,f), and find that this map is continuous if and only if it is a constant map; that is, only when the set is a singleton.