Let G be a smooth affine group over a field k, and λ : Gm → G a k-homomorphism (possibly trivial, though that case is not interesting). One often calls λ a 1-parameter k-subgroup of G, even when kerλ 6= 1. Such a homomorphism defines a left action of Gm on G via the functorial procedure t.g = λ(t)gλ(t)−1 for g ∈ G(R) and t ∈ R× for any k-algebra R. In lecture we introduced the following associa...

We establish a necessary condition for a commutative Banach algebra A so that there exists a homomorphism θ from A into another Banach algebra such that the prime radical of the continuity ideal of θ is not a finite intersection of prime ideals in A. We prove that the prime radical of the continuity ideal of an epimorphism from A onto another Banach algebra (or of a derivation from A into a Ban...

Although every computable data type has an initial algebra speci-cation with hidden functions, it may happen that some of the homo-morphic images of the data type are not models of the speciication. The latter are reducts of algebras that would be models of the specii-cation if all its functions were visible, whereas the homomorphic images of the data type are independent of the speciication an...

In 1939, Gelfand and Kolmogorov published a paper [4], where they showed that for a (compact Hausdorff) topological space X, homomorphisms of the algebra of continuous functions C(X) to numbers are in a one-to-one correspondence with points of X. Paper [4] is famous for giving birth to the theory of normed rings. However, it is worth stressing that the Gelfand–Kolmogorov theorem may be viewed a...

For the generality of fuzzy ideals in TM-algebra, a cubic ideal this algebra has been studied, such as and T-ideals. Some properties these are investigated. Also, we show that T-ideal is ideal, but converse not generally valid. In addition, sub-algebra defined, new relations between level subset discussed. After that, T-ideals under homomorphism image (pre-image) Finally, Cartesian product TM-a...

In this paper we define $varphi$-Connes module amenability of a dual Banach algebra $mathcal{A}$ where $varphi$ is a bounded $w_{k^*}$-module homomorphism from $mathcal{A}$ to $mathcal{A}$. We are mainly concerned with the study of $varphi$-module normal virtual diagonals. We show that if $S$ is a weakly cancellative inverse semigroup with subsemigroup $E$ of idemp...

The loop homology of a closed orientable manifoldM of dimensionm is the commutative algebra H∗(LM) = H∗+m(LM) equipped with the Chas-Sullivan loop product. Here LM denotes the free loop on M . There is also a homomorphism of graded algebras I from H∗(LM) to the Pontryagin algebra H∗(ΩM). The primary purpose of this paper is to study the loop homology and the homomorphism I, when M is a fibered ...

Let A be a unital AH-algebra and let α ∈ Aut(A) be an automorphism. A necessary condition for A ⋊α Z being embedded into a unital simple AF-algebra is the existence of a faithful tracial state. If in addition, there is an automorphism κ with κ∗1 = −idK1(A) such that α ◦ κ and κ ◦ α are asymptotically unitarily equivalent, then A⋊α Z can be embedded into a unital simple AF-algebra. Consequently,...

This paper is concerned with complemented modular lattices containing the elements O and 7. The first part treats of homomorphisms of the lattice L, their existence, determination and invariant properties. The second considers norms (i.e., sharply positive or, alternatively, strictly monotone modular functionals) and quasi-norms (i.e., positive or monotone modular functionals) on L, their inter...