Continuous Group of Transformations

Let $B$ be Banach algebra and $M$ be topological space. If there exists homeomorphism $$f:M\to N$$ of topological space $M$ into convex set $N$ of the space $B^n$, then homeomorphism $f$ is called chart of the set $M$. The set $M$ is called simple $B$-manifold of class $C^k$ if for any two charts $$f_1:M\to N_1\subseteq B^n$$ $$f_2:M\to N_2\subseteq B^n$$ there exists diffeomorphism $$f:B^n\to B^n$$ of class $C^k$ such that $$f_1\circ f=f_2$$ Topological space $M$ is called differential $B$-manifold of class $C^k$ if topological space $M$ is a union of simple $B$-manifolds $M_i$, $i\in I$, and intersection $M_i\cap M_j$ of simple $B$-manifolds $M_i$, $M_j$ is also simple $B$-manifold. Differential $B$-manifold $G$ equipped with group structure such that map $$(f,g)\to f{g}^{-1}$$ is differentiable is called Lie group. Module $T_{e}G$ equipped with product $$[v,w{]}^c=R_{Ljm}^c\circ (v^m,w^j)-R_{Lmj}^c\circ (w^j,v^m)\in T_{e}G$$ is Lie algebra $g_L$ of Lie group $G$.