K-POTENT PRESERVING LINEAR MAPS

A linear map $T$ from a Banach algebra $A$ into another $B$ preserves zero products if $T(a)T(b)=0$ whenever $a,b\in A$ are such that $ab=0$. This paper is mainly concerned with the question of whether every continuous linear surjective map $T\colon A\rig

Surjective linear maps preserving local spectra

In this article, we consider several local conditions under which linear mappings on algebras act like Lie n-centralizers and we study these linear mappings, Lie n-centralizers and n-commuting linear maps. With various examples, we compare Lie n-centralizers, n-commuting linear maps, and linear mappings acting similarly to Lie n-centralizers under different local conditions. In the following, we consider the mentioned concepts for linear mappings on standard operator algebras over a complex Banach space X with dim X ≥ 2 . As an application of the obtained results, we present some variants of Posner’s second theorem on unital standard operator algebras.

In this article we obtain 2 generalizations of the well known Gleason-Kahane-Zelazko Theorem. We consider a unital Banach algebra A, and a continuous unital linear mapping φ of A into Mn(C) – the n × n matrices over C. The first generalization states that if φ sends invertible elements to invertible elements, then the kernel of φ is contained in a proper two sided closed ideal of finite codimension. The second result characterizes this property for φ in saying that φ(Ainv) is contained in GLn(C) if and only if for each a in A and each natural number k: trace(φ(a)) = trace(φ(a)) .

By a von Neumann algebra M we mean a weakly closed, self-adjoint algebra of operators on a Hilbert space which contains I, the identity operator. A factor is a von Neumann algebra whose centre consists of scalar multiples of I.In all that follows ϕ:M → N will be a one to one, *-linear map from the von Neumann factor M onto the von Neumann algebra N such that both ϕ and ϕ−1 preserve commutativity. Our main result states that if M is not of type I2 then where is an isomorphism or an antiisomorphism, c is a non-zero scalar, and λ is a *-linear map from M into ZN, the centre of N.Our interest in this problem was aroused by several recent results. In [1], Choi, Jafarian, and Radjavi proved that if S is the real linear space of n × n matrices over any algebraically closed field, n ≧ 3, and ψ a linear operator on S which preserves commuting pairs of matrices, then either ψ(S) is commutative or there exists a unitary matrix U such thatfor all A in S. They proved an analogous result for the collection of all bounded self-adjoint operators on an infinite dimensional Hilbert space when ψ is one to one. Subsequently, Omladic [7] proved that if ψ:L(X) → L(X) is a bijective linear operator preserving commuting pairs of operators where X is a non-trivial Banach space, thenwhere U is a bounded invertible operator on X and A′ is the adjoint of A.

Linear maps preserving G-unitary operators in Hilbert space

Let $G$ be a locally compact amenable group. The goal of this paper is to investigate the problem of surjective zero products preserving maps from the Fourier algebra of $G$ into a completely contractive Banach algebra. We show that if $B$ is a completely contractive Banach algebra which is faithful and factors weakly, then every surjective completely bounded linear map from $A(G)$ into $B$ which preserves zero products is a weighted homomorphism. Moreover an equivalent condition is given for such a map to be a homomorphism. In particular, this result implies that if $B$ is a commutative C$^*$-algebra or a matrix space and $T:A(G)\rightarrow B$ is a continuous surjective linear map which preserves zero products, then $T$ is a weighted homomorphism and there is an equivalent condition for $T$ to be a homomorphism.

Let ℳ be a von Neumann algebra with a faithful normal trace τ, and let H∞ be a finite, maximal. subdiagonal algebra of ℳ. We prove that the Hilbert transform associated with H∞ is a linear continuous map from L1 (ℳ, τ) into L1.∞ (ℳ, τ). This provides a non-commutative version of a classical theorem of Kolmogorov on weak type boundedness of the Hilbert transform. We also show that if a positive measurable operator b is such that b log+b ∈ L1 (ℳ, τ) then its conjugate b, relative to H∞ belongs to L1 (ℳ, τ). These results generalize classical facts from function algebra theory to a non-commutative setting.

Recall that the spectrum, σ(T), of a linear operator T in a complex Banach space is the set of complex numbers λ such that T—λI does not have a densely defined bounded inverse. It is known [7, § 5.1] that σ(T) is a closed subset of the complex plane C. If T is not bounded, σ(T) may be empty or the whole of C. If σ(T) ≠ C and T is closed the spectral mapping theorem, is valid for complex polynomials p(z) [7, §5.7]. Also, if T is closed and λ ∉ σ(T), (T–λI)−1 is everywhere defined.

A a-weakly closed subspace ` of B(' X) is said to have Propett S_ if for any a-weakly closed subspace \5 of a von Neumann algebra -I, {.x E 5 0 A: R-(,(.x) (= ' for all rp E B(0.)) = X ' l, where R , is the right slice map associated with T. It is shown that semidiscrete von Neumann algebras have Property Sa, and various stability properties of the class of a-weakly closed subspaces with Property S0 are established. It is also shown that if (?111, G, a) is a W*-dynamical system such that 'k has Property S0 and C is compact abelian, then all of the spectral subspaces associated with a have Property S0. Some applications of these results to the study of tensor products of spectral subspaces and tensor products of reflexive algebras are given. In particular, it is shown that if L , is a commutative subspace lattice with totally atomic core, and L, is an arbitrary subspace lattice, then alg(i 0 E2) algC1 0 algL?2. Introduction. Slice maps, introduced by Tomiyama, have proved to be a powerful tool in studying tensor products of C*-algebras and von Neumann algebras (see, e.g., [23,24,26-281). In this paper we use slice maps to study tensor products of a-weakly closed subspaces of von Neumann algebras, particularly tensor products of spectral subspaces, and tensor products of reflexive algebras. If ')T and 6. are von Neumann algebras, then for each qp in X*, the predual of .X, there is a unique a-weakly continuous linear map R., from 9DZ 0 9 to 6 such that R,W,(a X b) = (a, q) b (a E 91R, b E DL)l called the right slice map associated with p. Similarly, for each 4 E 9L, there is a left slice map LI, from 9DTh0 6 to 9DT. If 5 C 9Th and Y C % are a-weakly closed subspaces, and 5 8 '5 denotes the a-weak closure of the algebraic tensor product of 5 and C, it is clear that if x E 0 0 then R (x) C 9 for all p E 9<X*. If (x E 5 0 : R ,(x) Efor all qg E ' } = 5 Received by the editors February 19, 1982 and, in revised form, July 20, 1982. 1980 Mathematics Subject Classification. Primary 46L10, 46L55, 46M05, 47D25.

For the Fr'{e}chet algebras $(A, (p_k))$ and $(B, (q_k))$ and $n in mathbb{N}$, $ngeq 2$, a linear map $T:A rightarrow B$ is called textit{almost $n$-multiplicative}, with respect to $(p_k)$ and $(q_k)$, if there exists $varepsilongeq 0$ such that$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_k(a_1) p_k(a_2)cdots p_k(a_n),$$for each $kin mathbb{N}$ and $a_1, a_2, ldots, a_nin A$. The linear map $T$ is called textit{weakly almost $n$-multiplicative}, if there exists $varepsilongeq 0$ such that for every $kin mathbb{N}$ there exists $n(k)in mathbb{N}$ with$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_{n(k)}(a_1) p_{n(k)}(a_2)cdots p_{n(k)}(a_n),$$for each $k in mathbb{N}$ and $a_1, a_2, ldots, a_nin A$.The linear map $T$ is called $n$-multiplicative if$$Ta_{1}a_{2} cdots a_{n} = Ta_{1} Ta_{2} cdots Ta_{n},$$for every $a_{1}, a_{2},ldots, a_{n} in A$.In this paper, we investigate automatic continuity of (weakly) almost $n$-multiplicative maps between certain classes of Fr'{e}chet algebras, including Banach algebras. We show that if $(A, (p_k))$ is a Fr'{e}chet algebra and $T: A rightarrow mathbb{C}$ is a weakly almost $n$-multiplicative linear functional, then either $T$ is $n$-multiplicative, or it is continuous. Moreover, if $(A, (p_k))$ and $(B, (q_k))$ are Fr'{e}chet algebras and $T:A rightarrow B$ is a continuous linear map, then under certain conditions $T$ is weakly almost $n$-multiplicative for each $ngeq 2$. In particular, every continuous linear functional on $A$ is weakly almost $n$-multiplicative for each $ngeq 2$.

A set of linear operators from one Banach space to another is collectively compact if and only if the union of the images of the unit ball has compact closure. Semi-groups S = {T(t): t ^ 0} of bounded linear operators on a complex Banach space into itself and in which every operator T(t), t > 0 is compact are considered. Since Tit, + t 2) = T(tJT(tJ for each operator in the semi-group, it would be expected that the theory of collectively compact sets of linear operators could be profitably applied to semi-groups. 1* Introduction* Let X be a complex Banach space with unit ball Xλ and let [X, X] denote the space of all bounded linear operators on X equipped with the uniform operator topology. The semi-group definitions and terminology used are those of Hille and Phillips [6]. Let S be a semi-group of vector-valued functions T: [0, oo)—>[X, X]. It is assumed that T(t) is strongly continuous for t ^ 0. If lim^ || T(t)x - T(to)x || = 0 for each t0 ^ 0, xeX and if there is a constant M such that the || T(t) || ^ M for each t ^ 0, then S — {T(t): t ^> 0} is called an equicontinuou s semi-group of class Co. The infinitesimal generator A of the semi-group S is defined by Ax = lim— [T(s)x - x] s->o S whenever the limit exists. The domain D(A) of A is a dense subset of X consisting of just those elements x for which this limit exists. A is a closed linear operator having resolvents R(X) which, for each complex number λ with the real part of λ greater than zero, are given by the absolutely summable Riemann-Stieltjes integral

It is shown that in every complex Banach space E with open unit ball D C E there is a closed C-linear subspace Cc such that V C\ D is the orbit of the origin 0 E E under the group Aut(D) of all biholomorphic automorphisms of D. In particular two complex Banach spaces are isometrically equivalent if and only if their open unit balls are biholomorphically equivalent.Suppose E, F are complex Banach spaces and U E E is an open subset.Then a mapping/: U -* Fis called holomorphic if the Frchet derivative of/ at x (denoted by f'(x)) exists for every x G U. (For details compare [6].)In this paper we show that the Banach space E is completely determined by the holomorphic structure of its (open) unit ball; more precisely, Theorem.The complex Banach spaces E and are isometrically isomorphic if and only if there is a biholomorphic mapping from the unit ball of E onto the unit ball of .This answers a question L. A. Harris raised in [3, p. 15].For the proof let be a fixed complex Banach space with unit ball DEE.Let 0 denote the set of all holomorphic vector fields on D i.e. of all expressions X = f(z)d/dz where/: D -* E is a holomorphic mapping and z denotes the coordinate of D. Then D is a complex Lie algebra in the product [/W,^)]=(.'W/(z)-/'(z)g(z)).For every integer v the Lie algebra D contains the linear subspace ^v = l h(z)~~ h: E ^> E is a. homogeneous polynomial of degree v + 1 >.Obviously [$r,$J C SJL^ for every v and \i.Since D is a bounded domain in E, the set g of all complete holomorphic vector fields on D has the following properties (compare [7]):(0) Every h(z)d/dz G g is uniquely determined by h(0) and h'(0) (Cartan's uniqueness theorem).

We give an expression for the most general continuous linear input-output map that takes the space of (Lebesgue) integrable functions into itself. This expression is a function-space limit of an integral. As an application, a representation result is given for an important family of linear maps that take the space of bounded measurable functions into itself and satisfy a certain continuity condition, and it is noted that no such result holds for a general continuous linear maps that takes the space into itself and meets the continuity condition. Some related material concerning engineering education is also given.

Let E be a complex Banach space and F be a complex Banach algebra. We will be interested in the subspace ℙ_f({}^n E,F) of P({}^n E,F) generated by the collection of functions \varphi^n(n\in ℕ, \varphi \in L(E,F)) where \varphi^n(x) = (\varphi (x))^n for each x\in E .