Proof. Denote by 1 the unit element of B. Let a ∈ B×. We show that 1 and a can be connected by a path in B×. Let A be the subalgebra of B generated by 1 and a. That is A = {P (a) : P (t) ∈ C[t]}. Because A× ⊂ B×, it suffices to find a path in A× that connects 1 with a. First, we determine all multiplicative functionals of A. Since A is a linear subspace of B, it is finite dimensional. Then ther...

We show that if T is an isometry (as metric spaces) from an open subgroup of the invertible group A of a unital Banach algebra A onto an open subgroup of the invertible group B of a unital Banach algebra B, then T is extended to a real-linear isometry up to translation between these Banach algebras. We consider multiplicativity or unti-multiplicativity of the isometry. Note that a unital linear...

Let R be a linearly ordered ring with 1/2, Gn(R) (n ≥ 3) be the subsemigroup of GLn(R) consisting of all matrices with nonnegative elements. In [1], there is a description of all automorphisms of the semigroup Gn(R) in the case where R is a skewfield and n ≥ 2. In [2], there is a description of all automorphisms of the semigroup Gn(R) for the case where R is an arbitrary linearly ordered ring w...

We study the structure of invertible substitutions on three-letter alphabet. We show that there exists a finite set S of invertible substitutions such that any invertible substitution can be written as Iw ◦ σ1 ◦ σ2 ◦ · · · ◦ σk, where Iw is the inner automorphism associated with w, and σj ∈ S for 1 ≤ j ≤ k. As a consequence, M is the matrix of an invertible substitution if and only if it is a f...

The Graded Classification Conjecture states that the pointed $K_{0}^{\operatorname {gr}}$ -group is a complete invariant of Leavitt path algebras finite graphs when these are considered with their natural grading by $\mathbb {Z}$ . strong version this conjecture functor full and faithful on category graded homomorphisms modulo conjugations invertible elements zero components. We show for unital...

When constructing practical zero-knowledge proofs based on the hardness of the RingLWE or the Ring-SIS problems over polynomial rings Zp[X]/(X + 1), it is often necessary that the challenges come from a set C that satisfies three properties: the set should be large (around 2), the elements in it should have small norms, and all the non-zero elements in the difference set C −C should be invertib...

We extend the Larson–Sweedler theorem for weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We establish the autonomous monoidal category of the modules of a weak Hopf algebra A and show the semisimplicity of the unit and the invertible modules of A. We also reveal the connection of these modules to lef...

We show that if T is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then T is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, (T (eA)) −1 T is extended to a surjective real algebra isomorphism; if T is a surjective isometry...