Maps preserving zero products

K-POTENT PRESERVING LINEAR MAPS

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.

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 $ngeq 2$.

Analytical expressions for power spectral density issued from one-dimensional continuous piecewise linear maps with three slopes

We characterize the pairs of Moscatelli Frechet and spaces such that: 1) Every continuous linear map from E into F sends bounded sets in E into precompact ones in F. 2) Every bounded map from E into F sends some 0-neighbourhood in E into a precompact set in F.

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.

Surjective linear maps preserving local spectra

On spaces of continuous linear mappings between locally convex spaces

Linear maps preserving G-unitary operators in Hilbert space

Continuous extension of functions from countable sets

The paper begins with an introduction to the cohomology theory for Banach algebras needed in the remainder of the paper. A perturbation of the multiplication π of a Banach algebra A is a continuous associative multiplication ρ which is near ρ in the usual norm for bilinear maps. We consider the question of whether such a multiplication is necessarily of the form ρ(a, b) = W−1(WaWb) where W is a bounded linear map from A into A near the identity. A cohomological condition is given for this to hold. A linear map T from one Banach algebra A into another B is “almost” a homomorphism if the bilinear map \(\mathop{T}\limits^{ \circ } (a,b) = T(a,b) - T(a)T(b)\) is small. This occurs if T is near a genuine homomorphism and we consider whether every map which is almost a homomorphism is near a homomorphism. The main result is that if A is an amenable algebra and B is the dual of a suitable Banach space then the pair (A, B) has this property.

In order to summarize the contents of this paper it is necessary to use some of the definitions in [5]. These definitions are repeated below at the beginning of Section 1. Although this paper adds to the material in [5] we have attempted to reduce the number of references to [o] to a minimum. An approximate diagonal in a Banach algebra 31 is a bounded net {ma} in 31 a for all a in 31 in the norm topologies where 31? 31 is the usual projective tensor product [2] and -k is the continuous linear map 31?31->31 defined by ir(a?b) =ab. The net {m 31) ** with aM = Ma and ?r** (M) a = a for all a ? 31. In the algebraic cohomology of algebras the existence of an element m of A A with ma = am and ir(m) the identity of A is equivalent to the statement that H1(A, X) =0 for all 31 modules X almost never holds?in fact, in the commutative case for all A bimodules X. Among Banach algebras the statement jEP^S^X) =0 (see [3; Theorem 1] and [5; Proposition 8.1]) we can always find a Banach 31 module ? with S^1(3I,36)=t^O unless 31 zz?n. The existence of an approxi? mate diagonal is not unusual and is equivalent to the statement !M* (31,36*) = 0 for all Banach 31 modules X; thatis 31 is an emenable Banach algebra. The element M plays the same role in the theory of amenable tlgebras that the invariant mean does for amenable groups. The definitions used from [5] and the theorem connecting amenability with approximate diagonals are given in ? 1. In ? 2 we show that if 31 has a virtual diagonal with Ma? 31? St for all a in 3T then 913 (31, 3E) ? .0 for all Banach 31 modules 3?. Algebras satisfying this condition are grdup algebras of compact topological groups and the algebra c0 of convergent sequences of complex numbers. Although we do not prove it the result also applies to any annihilator B* algebra with only finite dimensional minimal ideals. In ? 3 we show that if 31 is a semi simple commutative amenable Banach algebra then there is a Banach 31 module X with 3/2(3I,3E) ^0, unless 31 is finite

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)) .

We suggest to model the electrical activity of an isolated neuron by using piecewise linear stochastic and piecewise continuous dynamic maps based on phenomenological notions about neuron dynamics. It is demonstrated that the proposed models provide for time series qualitatively similar to those observed in experiments with biological neurons. The possibility of using these models for constructing coupled neuron systems is discussed.

A linear mapping T from a subspace E of a Banach algebra into another Banach algebra is called spectrally bounded if there is a constant M ≥ 0 such that r(T x) ≤ Mr(x) for all x ∈ E, where r (·) denotes the spectral radius. We establish the equivalence of the following properties of a unital linear mapping T from a unital C* -algebra A into its centre:(a) T is spectrally bounded;(b) T is a spectrally bounded trace;(c) T is a bounded trace.