The author is grateful for support from a PhD grant from the Engineering and Physical Sciences Research council of the UK and post-doctoral fellowship number SFRH/BPD/40762/2007 from the Fundação para a Ciéncia e a Tec-nologia of Portugal. This paper contains some work included in the authors PhD thesis ([5]). Abstract We introduce notions of compactness and weak compactness for multi-linear ma...

We introduce notions of compactness and weak compactness for multilinear maps from a product of normed spaces to a normed space, and prove some general results about these notions. We then consider linear maps T : A → B between Banach algebras that are “close to multiplicative” in the following senses: the failure of multiplicativity, defined by ST (a, b) = T (a)T (b) − T (ab) (a, b ∈ A), is co...

Abstract. In this paper, we first derive specific results concerning the continuity and upper semi-continuity of the spectral radius and spectrum functions on fundamental locally multiplicative topological algebras. We continue our investigation by further determining the automatic continuity of linear mappings and homomorphisms in these algebras.

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 $a_1, a_2, ldots, a_nin A$. Th...

We introduce and study “norm-multiplicative” homomorphisms φ:L1(F)→Mr(G) between group measure algebras, φ:L1(ωF)→M(ωG) Beurling where F G are locally compact groups with continuous weights ωF ωG. Through a unified approach we recover, sometimes strengthen, many of the main known results concerning isomorphisms these (Beurling) algebras. provide first description all positive φ:L1(F)→Mr(G). sta...

‎We extend the notion of approximately multiplicative to approximately n-multiplicative maps between locally multiplicatively convex algebras and study some properties of these maps‎. ‎‎W‎e prove that every approximately n-multiplicative linear functional on a functionally continuous locally multiplicatively convex algebra is continuous‎. ‎We also study the relationship between approximately mu...

In addition, define certain lattices and subsets: L∨ = X∗(T ) = group of algebraic homomorphisms Gm → T L = Hom(X∗(T ),Z), the dual of L ∨ X(T ) = group of algebraic homomorphisms T → Gm, the weights of T . There is a canonical map from L toX(T )—to λ corresponds the multiplicative character e defined by the formula e(μ(x)) = x 〉 , which makes sense because all algebraic homomorphisms Gm → Gm a...