Banach spaces webinar on Friday Sep 11 by Bence Horváth (Czech Academy of Sciences)
Dear all, The next Banach spaces webinar is on Friday September 11 9AM CDT (e.g., Dallas, TX time). Please join us at https://unt.zoom.us/j/512907580 Speaker: Bence Horváth, Czech Academy of Sciences Title: When are surjective algebra homomorphisms of $\mathcal{B}(X)$ automatically injective? Abstract. A classical result of Eidelheit asserts that if $X$ and $Y$ are Banach spaces then they are isomorphic if and only if their algebras of operators $\mathcal{B}(X)$ and $\mathcal{B}(Y)$ are isomorphic as Banach algebras, in the sense that there is a continuous bijective algebra homomorphism $\psi: \, \mathcal{B}(X) \rightarrow \mathcal{B}(Y)$. It is natural to ask whether for some class of Banach spaces $X$ this theorem can be strengthened in the following sense: If $Y$ is a non-zero Banach space and $\psi: \mathcal{B}(X) \rightarrow \mathcal{B}(Y)$ is a surjective algebra homomorphism, is $\psi$ automatically injective? It is easy to see that for a ``very nice'' class Banach spaces, such as $c_0$ and $\ell_p$, where $1 \leq p < \infty$, the answer is positive. Further examples include $\ell_{\infty}$ and $( \oplus_{n=1}^{\infty} \ell_2^n )_{c_0}$ and its dual space $\left( \oplus_{n=1}^{\infty} \ell_2^n \right)_{\ell_1}$, and the arbitrarily distortable Banach space $\mathbf{S}$ constructed by Schlumprecht. In recent joint work with Tomasz Kania it was shown that ``long'' sequence spaces of the form $c_0(\lambda)$, $\ell_{\infty}^c(\lambda)$ and $\ell_p(\lambda)$ (where $1 \leq p < \infty$) also enjoy this property. In the other direction, with the aid of a result of Kania--Koszmider--Laustsen we will show that for any separable, reflexive Banach space $X$ there is a Banach space $Y_X$ and a surjective algebra homomorphism $ \psi: \, \mathcal{B}(Y_X) \rightarrow \mathcal{B}(X)$ which is not injective. * For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Upcoming schedule September 18: Chris Phillips, University of Oregon Thank you, and best regards, Bunyamin Sari
participants (1)
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Sari, Bunyamin