Dear all,

The next Banach spaces webinar is on Friday July 10 9AM CDT (e.g., Dallas, TX time). Please join us at

https://unt.zoom.us/j/512907580

Speaker: Niels Laustsen (Lancaster University)

Title: A C(K) space with few operators and few decompositions

Abstract. I shall report on joint work with Piotr Koszmider (IMPAN) concerning the closed subspace of $\ell_\infty$ generated by $c_0$ and the characteristic functions of elements of an uncountable, almost disjoint family $A$ of infinite subsets of the natural numbers. This Banach space has the form $C_0(K_A)$ for a locally compact Hausdorff space $K_A$ that is known under many names, including $\Psi$-space and Isbell--Mr'ow-ka space.

We construct an uncountable, almost disjoint family $A$ such that the algebra of all bounded linear operators on $C_0(K_A)$ is as small as possible in the precise sense that every bounded linear operator on $C_0(K_A)$ is the sum of a scalar multiple of the identity and an operator that factors through $c_0$ (which in this case is equivalent to having separable range). This implies that $C_0(K_A)$ has the fewest possible decompositions: whenever $C_0(K_A)$ is written as the direct sum of two infinite-dimensional Banach spaces $X$ and $Y$, either $X$ is isomorphic to $C_0(K_A)$ and $Y$ to $c_0$, or vice versa. These results improve previous work of Koszmider in which an extra set-theoretic hypothesis was required.

* 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

July 10: Alejandro Chávez-Domínguez (University of Oklahoma)

Thank you, and best regards,

Bunyamin Sari