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Let A and B be two non-unital reduced Banach *-algebras and φ: A → B be a vector space isomorphism. The two following statement holds: If φ is a *-isomorphism, then φ is isometric (with respect to the C*-norms), bipositive and φ maps some approximate identity of A onto an approximate identity of B. Conversely, any two of the later three properties imply that φ is a *-isomorphism. Finally, we show that a unital and self-adjoint spectral isometry between semi-simple Hermitian Banach algebras is an...
In this paper, we establish some new sufficient conditions for uniform global asymptotic stability for certain classes of nonlinear systems. Lyapunov approach is applied to study exponential stability and stabilization of time-varying systems. Sufficient conditions are obtained based on new nonlinear differential inequalities. Moreover, some examples are treated and an application to control systems is given.
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