We prove that for every group $G$ and any two sets $I,J$, the Brandt semigroup algebras $ell(B(I,G))$ and $ell(B(J,G))$ are Morita equivalent with respect to the Morita theory of self-induced Banach algebras introduced by Gronbaek. As applications, w
e show that if $G$ is an amenable group, then for a wide class of Banach $ell(B(I,G))$-bimodules $E$, and every $n>0$, the bounded Hochschild cohomology groups $H^n(ell(B(I,G)),E^*)$ are trivial, and also, the notion of approximate amenability is not Morita invariant.
In this short note, we give some new results on continuous bounded cohomology groups of topological semigroups with values in complex field. We show that the second continuous bounded cohomology group of a compact metrizable semigroup, is a Banach sp
ace. Also, we study cohomology groups of amenable topological semigroups, and we show that cohomology groups of rank greater than one of a compact left or right amenable semigroup, are trivial. Also, we give some examples and applications about topological lattices.
We show that any quantum family of maps from a non commutative space to a compact quantum metric space has a canonical quantum semi metric structure.
We show that the quantum family of all maps from a finite space to a finite dimensional compact quantum semigroup has a canonical quantum semigroup structure.
Let $Top_c$ be the category of compact spaces and continuous maps and $Top_fsubset Top_c$ be the full subcategory of finite spaces. Consider the covariant functor $Mor:Top_f^{op}times Top_cto Top_c$ that associates any pair $(X,Y)$ with the space of
all morphisms from $X$ to $Y$. In this paper, we describe a non commutative version of $Mor$. More pricelessly, we define a functor $mathfrak{M}mathfrak{o}mathfrak{r}$, that takes any pair $(B,C)$ of a finitely generated unital C*-algebra $B$ and a finite dimensional C*-algebra $C$ to the quantum family of all morphism from $B$ to $C$.
