Structure and Hochschild cohomology group of a class of Kadison-Singer algebras

Abstract

In 2018, Dong et al. reviewed the backgroundof Kadison-Singer algebras (KS-algebras, for short),introduced the definitions and basic properties,and listed 16 open problems about KS-algebras.In this paper, we mainly study the problems 2 and 8 on matrix algebras $M_{n}(\mathbb{C})$.Suppose that $\mathcal~L$ is a KS-lattice on a Hilbert space $\mathcal~H$.Problem 2: Is every KS-algebra $\mathrm{Alg}\mathcal~L$ non-selfadjoint if $\mathcal~L$ is non-trivial?Problem 8: Is every $n$-th Hochschild cohomology group ${H^n}(\mathrm{Alg}\mathcal~L,~\mathcal~B(\mathcal~H))$ trivial?We prove that if $\mathcal~A\subseteq~M_{3}(\mathbb{C})$ is a KS-algebra, thenevery $n$-th ($n\geqslant1$) Hochschild cohomology group ${H^n}(\mathcal~A,~M_{3}(\mathbb{C}))$ is trivial;and we also prove that if $\mathcal~L$ is a non-trivial KS-lattice of $M_{n}(\mathbb{C})$ ($n\geqslant2$),then the corresponding KS-algebra $\mathrm{Alg}\mathcal~L$ is a non-selfadjoint operator algebra.