We show that if T is an isometry (as metric spaces) from an open subgroup of the group of the invertible elements in a unital semisimple commutative Banach algebra onto an open subgroup of the group of the invertible elements in a unital Banach algebra, then T (1)T is an isometrical group isomorphism. In particular, T (1)T is extended to an isometrical real algebra isomorphism from A onto B.

Let A be a unital C -algebra with involution represented in a Hilbert space H, G the group of invertible elements of A, U the unitary group of A, G the set of invertible selfadjoint elements of A, Q = f" 2 G : " = 1g the space of re ections and P = Q \ U . For any positive a 2 G consider the a-unitary group Ua = fg 2 G : a g a = g g, i.e., the elements which are unitary with respect to the scal...

We call a ring R generalized uniquely clean (or GUC for short) if every not invertible element in is clean. Let be ring. It shown that and only it local or Thus the generalization of Some basic properties rings are proved.

All known finite sharply 4-transitive permutation sets containing the identity are groups, namely S4, S5, A6 and the Mathieu group of degree 11. We prove that a sharply 4-transitive permutation set on 11 elements containing the identity must necessarily be the Mathieu group of degree 11. The proof uses direct counting arguments. It is based on a combinatorial property of the involutions in the ...

Let A be a complex unital Banach algebra with unit 1. The sets of all invertible and quasinilpotent elements (σ(a) = {0}) of A will be denoted by A and A, respectively. The group inverse of a ∈ A is the unique element a ∈ A which satisfies aaa = a, aaa = a, aa = aa. If the group inverse of a exists, a is group invertible. Denote by A the set of all group invertible elements of A. The generalize...