Homomorphisms on normed algebras
... continuous. If B is primitive with minimal one-sided ideals, T(Bj) is dense in B and intersects at least one minimal ideal of B then T is closed. Other examples are given. In these results we can obtain the conclusion for ring homomorphism as well as algebra homomorphism if we assume that p(T(x))^p( ...
... continuous. If B is primitive with minimal one-sided ideals, T(Bj) is dense in B and intersects at least one minimal ideal of B then T is closed. Other examples are given. In these results we can obtain the conclusion for ring homomorphism as well as algebra homomorphism if we assume that p(T(x))^p( ...
On Idempotent Measures of Small Norm
... for compact K ⊆ G, ε > 0, and their translates, form an open neighbourhood base for G. Proof. This is simply the topology of compact convergence on Γ, hence is the topology of G by the Pontryagin duality theorem. The above statement is proved as Proposition 1.2.6. in [9], without reference to the Po ...
... for compact K ⊆ G, ε > 0, and their translates, form an open neighbourhood base for G. Proof. This is simply the topology of compact convergence on Γ, hence is the topology of G by the Pontryagin duality theorem. The above statement is proved as Proposition 1.2.6. in [9], without reference to the Po ...
skew-primitive elements of quantum groups and braided lie algebras
... where H is a braided graded Hopf algebra over a commutative group G [L, Ma, R, S]. In this situation the primitive elements of H are skew-primitive elements of L. So the structure of a braided Lie algebra on the set of primitive elements in H induces a similar structure on a subset of the skew-primi ...
... where H is a braided graded Hopf algebra over a commutative group G [L, Ma, R, S]. In this situation the primitive elements of H are skew-primitive elements of L. So the structure of a braided Lie algebra on the set of primitive elements in H induces a similar structure on a subset of the skew-primi ...
THE ENDOMORPHISM SEMIRING OF A SEMILATTICE 1
... so that x ≤ g(x); similarly, a = ρa,b,x (x) = ρa,b,x (g(x)), so that g(x) ≤ x; we get g = idM also in this case. It follows that it suffices to prove the equivalence of the conditions (1) and (4). Let FM = EM . Then every element of EM can be expressed as the sum of finitely many elements ρa,b,c wit ...
... so that x ≤ g(x); similarly, a = ρa,b,x (x) = ρa,b,x (g(x)), so that g(x) ≤ x; we get g = idM also in this case. It follows that it suffices to prove the equivalence of the conditions (1) and (4). Let FM = EM . Then every element of EM can be expressed as the sum of finitely many elements ρa,b,c wit ...
Rings with no Maximal Ideals
... x(A0 x) ⊆ (x2 ). Therefore, N is not a maximal ideal of M . Next, suppose that (x2 ) 6⊆ N . If (x2 ) + N ⊂ M , then N is not a maximal ideal of M . Suppose that (x2 ) + N = M . Write x = x2 f + xg with xg ∈ N ; we can write any element of N in this form since N ⊆ M . Then 1 = xf + g, so g = 1 − xf ...
... x(A0 x) ⊆ (x2 ). Therefore, N is not a maximal ideal of M . Next, suppose that (x2 ) 6⊆ N . If (x2 ) + N ⊂ M , then N is not a maximal ideal of M . Suppose that (x2 ) + N = M . Write x = x2 f + xg with xg ∈ N ; we can write any element of N in this form since N ⊆ M . Then 1 = xf + g, so g = 1 − xf ...
An introduction to random walks on groups
... g ∈ H, which should be proved. Amenable groups can be characterized as groups which admit fixed points in convex invariant subsets for certain actions on Banach spaces. Given a locally compact group G and a Banach space E, by an isometric continuous action of G on E, we mean a linear action of G by ...
... g ∈ H, which should be proved. Amenable groups can be characterized as groups which admit fixed points in convex invariant subsets for certain actions on Banach spaces. Given a locally compact group G and a Banach space E, by an isometric continuous action of G on E, we mean a linear action of G by ...
Diagonalisation
... is not invertible.) However, we can solve this fairly directly just by looking at the equations. We have to solve x1 + x2 = 0, 2x1 + 2x2 = 0. Clearly both equations are equivalent. From either one, we obtain x1 = −x2 . We can choose x2 to be any number we like. Let’s take x2 = 1; then we need x1 = − ...
... is not invertible.) However, we can solve this fairly directly just by looking at the equations. We have to solve x1 + x2 = 0, 2x1 + 2x2 = 0. Clearly both equations are equivalent. From either one, we obtain x1 = −x2 . We can choose x2 to be any number we like. Let’s take x2 = 1; then we need x1 = − ...
Algebraic Groups
... Lemma 1.5. Let G be an algebraic group and let X ⊆ G be a constructible dense subset. Then G = X · X −1 = X · X. Proof. X contains a subset U which is open and dense in G, and the same holds for V := U ∩ U −1 . This implies that gV ∩ V #= ∅ for any g ∈ G, and so ...
... Lemma 1.5. Let G be an algebraic group and let X ⊆ G be a constructible dense subset. Then G = X · X −1 = X · X. Proof. X contains a subset U which is open and dense in G, and the same holds for V := U ∩ U −1 . This implies that gV ∩ V #= ∅ for any g ∈ G, and so ...
weakly almost periodic functions and almost convergent functions
... group,f E UC(G). Then f • FL(G) C FL(G) if and only if f E FL0(G) © C (The constant functions on G are identified with C) The discrete version of the above theorem is contained in [4]. Another natural question one may ask is the following: If the condition "amenable" is skipped from (2) above can on ...
... group,f E UC(G). Then f • FL(G) C FL(G) if and only if f E FL0(G) © C (The constant functions on G are identified with C) The discrete version of the above theorem is contained in [4]. Another natural question one may ask is the following: If the condition "amenable" is skipped from (2) above can on ...
GENERIC SUBGROUPS OF LIE GROUPS 1. introduction In this
... Theorem 2: For S non compact the statement follows from cor. 6 and cor. 7 combined with cor. 1. If S is compact and positive dimensional then k (G) is a set of measure zero for all k 2 by prop. 11. Theorem 1 combined with prop. 6 implies that either 1 (G) or G \ 1(G) is a set of measure zero for S c ...
... Theorem 2: For S non compact the statement follows from cor. 6 and cor. 7 combined with cor. 1. If S is compact and positive dimensional then k (G) is a set of measure zero for all k 2 by prop. 11. Theorem 1 combined with prop. 6 implies that either 1 (G) or G \ 1(G) is a set of measure zero for S c ...