–a notion of amenability for topological semigroups is introduced. a topological semigroup s iscalled johnson amenable if for every banach s -bimodule e , every bounded crossed homomorphism froms to e* is principal. in this paper it is shown that a discrete semigroup s is johnson amenable if and only if1(s) is an amenable banach algebra. also, we show that if a topological semigroup s is johns...

We extend certain homomorphisms defined on the higher Torelli subgroups of the mapping class group to crossed homomorphisms defined on the entire mapping class group. In particular, for every k ≥ 2, we construct a crossed homomorphism ǫk which extends Morita’s homomorphism τ̃k to the entire mapping class group. From this crossed homomorphism we also obtain a crossed homomorphism extending the kt...

On a closed symplectic surface Σ of genus two or more, we give a new construction of an extended flux map (a crossed homomorphism from the symplectomorphism group Symp(Σ) to the cohomology group H(Σ;R) that extends the flux homomorphism). This construction uses the topology of the Jacobian of the surface and a correction factor related to the Johnson homomorphism. For surfaces of genus three or...

We compute a twisted first cohomology group of the automorphism group of a free group with coefficients in the abelianization V of the IA-automorphism group of a free group. In particular, we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively. As a corollary, we see that the first Joh...

Given a crossed module of groupoids [Formula: see text], we construct (1) natural homomorphism from the product groupoid text] to and (2) transgression map singular cohomology nerve text]. The latter turns out be identical obtained by Tu–Xu in their study equivariant text]-theory.

Let G be a finite group and R a strongly G-graded ring. The question of when R is semisimple (meaning in this paper semisimple artinian) has been studied by several authors. The most classical result is Maschke’s Theorem for group rings. For crossed products over fields there is a satisfactory answer given by Aljadeff and Robinson [3]. Another partial answer for skew group rings was given by Al...

Suppose a locally compact groupG acts freely and properly on a locally compact Hausdorff space X , and let γ be the induced action on C0(X). We consider a category in which the objects are C∗-dynamical systems (A,G, α) for which there is an equivariant homomorphism of (C0(X), γ) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a gene...

Suppose a locally compact groupG acts freely and properly on a locally compact Hausdorff space X , and let γ be the induced action on C0(X). We consider a category in which the objects are C∗-dynamical systems (A,G, α) for which there is an equivariant homomorphism of (C0(X), γ) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a gene...

Suppose a locally compact groupG acts freely and properly on a locally compact Hausdorff space X , and let γ be the induced action on C0(X). We consider a category in which the objects are C∗-dynamical systems (A,G, α) for which there is an equivariant homomorphism of (C0(X), γ) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a gene...