A note on compact groups

A Note on Locally Compact Groups

In this note we shall prove that every locally compact group can be embedded as a closed subgroup in a unimodular group. If the original group is locally Euclidean, the enlarged group will be also, hence the fifth problem of Hilbert is reduced to the unimodular case. We shall use certain results concerning Haar measure whose proof may be found in A. Weil, U integration dans les groupes topologi...

A Note on Semi-groups in a Locally Compact Group

A Note on Operator Biprojectivity of Compact Quantum Groups

Given a (reduced) locally compact quantum group A, we can consider the convolution algebra L(A) (which can be identified as the predual of the von Neumann algebra form of A). It is conjectured that L(A) is operator biprojective if and only if A is compact. The “only if” part always holds, and the “if” part holds for Kac algebras. We show that if the splitting morphism associated with L(A) being...

A note on quasi irresolute topological groups

In this study, we investigate the further properties of quasi irresolute topological groups defined in [20]. We show that if a group homomorphism f between quasi irresolute topological groups is irresolute at $e_G$, then $f$ is irresolute on $G$. Later we prove that in a semi-connected quasi irresolute topological group $(G,*,tau )$, if $V$ is any symmetric semi-open neighborhood of $e_G$, then...

On component extensions locally compact abelian groups

Let $pounds$ be the category of locally compact abelian groups and $A,Cin pounds$. In this paper, we define component extensions of $A$ by $C$ and show that the set of all component extensions of $A$ by $C$ forms a subgroup of $Ext(C,A)$ whenever $A$ is a connected group. We establish conditions under which the component extensions split and determine LCA groups which are component projective. ...