Rank, trace and determinant in Banach algebras: generalized Frobenius and Sylvester theorems
Rank, trace and determinant in Banach algebras: Generalized Frobenius and Sylvester Theorems G. Braatvedt University of Johannesburg As a follow-up to a paper of Aupetit and Mouton, we consider the spectral definitions of rank, trace and determinant applied to elements in a general Banach algebra. We prove a generalization of Sylvester’s Determinant Theorem to Banach algebras and thereafter a generalization of the Frobenius Inequality. This is a joint work with R. Brits and F. Schulz, University of Johannesburg. Truncation and Spectral Variation in Banach Algebras R. Brits University of Johannesburg Let a and b be elements of a semisimple, complex and unital Banach algebra A. Using subharmonic methods, we show that if the spectral containment σ(ax) ⊆ σ(bx) holds for all x ∈ A, then ax belongs to the bicommutant of bx for all x ∈ A. Given the aforementioned spectral containment, the strong commutation property then allows one to derive, for a variety of scenarios, a precise connection between a and b. The current paper gives another perspective on the implications of the above spectral containment which was also studied, not long ago, by J. Alaminos, M. Brešar et. al. This is a joint work with C. Touré and F. Schulz, University of Johannesburg. On Fréchet algebras with the dominating norm property Tomasz Ciaś Adam Mickiewicz University A Fréchet space E with a fundamental sequence (|| · ||n)n∈N of seminorms has the property (DN) if there is a continuous norm || · || on E such that for all k ∈ N there is n ∈ N and a constant C > 0 such that ||x||k ≤ C||x|| ||x||n for all x ∈ E. Every norm ||·|| with this property is called a dominating norm on E. The property (DN) plays a key role in the structure theory of nuclear Fréchet spaces. In 1977 D. Vogt proved that a nuclear Fréchet space is isomorphic to a closed subspace of the space s of rapidly decreasing sequences if and only if it admits a dominating norm. In the talk we consider Fréchet ∗algebras E with unit – called DN-algebras – admitting a dominating Hilbert norm || · || = √ (·, ·) such that, in particular, (xy, z) = (y, x∗z) for all x, y, z ∈ E. This class seems to be interesting at least for two reasons. Firstly, many natural Fréchet algebras are DN-algebras, e.g., the algebras C∞(M) of smooth functions on smooth compact manifolds and the algebras C∞(Ω) of smooth functions with uniformly continuous partial derivatives on open, bounded subsets of Rn with Lipschitz boundary. Secondly, it appears