Duality of multiplicative functionals

Almost multiplicative linear functionals and approximate spectrum

We define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital Banach algebra A and show that the δ-approximate spectrum σ_δ (a) of a is compact. The relation between the δ-approximate spectrum and the usual spectrum is investigated. Also an analogue of the classical Gleason-Kahane-Zelazko theorem is established: For each ε>0, there is δ>0 such that if ϕ is...

Superstability of Generalized Multiplicative Functionals

LetX be a set with a binary operation ◦ such that, for each x, y, z ∈ X, either x ◦y ◦z x ◦z ◦y, or z◦ x◦y x◦ z◦y . We show the superstability of the functional equation g x◦y g x g y . More explicitly, if ε ≥ 0 and f : X → C satisfies |f x ◦ y − f x f y | ≤ ε for each x, y ∈ X, then f x ◦ y f x f y for all x, y ∈ X, or |f x | ≤ 1 √1 4ε /2 for all x ∈ X. In the latter case, the constant 1 √ 1 4...

Multiplicative properties of Atiyah duality

Let Mn be a closed, connected n-manifold. Let M−τ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that M−τ is homotopy equivalent to the Spanier-Whitehead dual of M with a disjoint basepoint, M+. This dual can be viewed as the function spectrum, F (M,S), where S is the sphere spectrum. F (M,S) has the structure of a commutative, symmetric ring spectru...

Multiplicative Operator Functionals of a Markov Process

Multiplicative Functionals on Semigroups of Continuous Functions

Let X be a compact Hausdorff space. We denote by C(X) the multiplicative semigroup of all continuous real-valued functions on X. Milgram [2] has shown that C(X) as a semigroup determines X. In this paper we investigate the set %(X) of all continuous positive nontrivial2 multiplicative functionals on C(X), where C(X) has the topology of uniform convergence. If multiplication is defined pointwise...