We show that a directed graph $E$ is a finite graph with no sinks if and only if, for each commutative unital ring $R$, the Leavitt path algebra $L_{R}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if the $C^{\ast }$-algebra $C^{\ast }(E)$ is unital and $\text{rank}(K_{0}(C^{\ast }(E)))=\text{rank}(K_{1}(C^{\ast }(E)))$. Let $k$ be a field and $k^{\times }$ be the group of units of $k$. When $\text{rank}(k^{\times })<\infty$, we show that the Leavitt path algebra $L_{k}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if $L_{k}(E)$ is unital and $\text{rank}(K_{1}(L_{k}(E)))=(\text{rank}(k^{\times })+1)\text{rank}(K_{0}(L_{k}(E)))$. We also show that any unital $k$-algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz–Krieger algebra, is isomorphic to an algebraic Cuntz–Krieger algebra. As a consequence, corners of algebraic Cuntz–Krieger algebras are algebraic Cuntz–Krieger algebras.

For a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.

We study a Morita-equivalent version of the Zariski cancellation problem.

We prove that the two-variable $\text{BMW}$ algebra embeds into an algebra constructed from the $\text{HOMFLY-PT}$ polynomial. We also prove that the $\mathfrak{s}{{\mathfrak{O}}_{2N}}-\text{BMW}$ algebra embeds in the $q$-Schur algebra of type $A$. We use these results to suggest a schema providing categorifications of the $\mathfrak{s}{{\mathfrak{D}}_{2N}}-\text{BMW}$ algebra.

A semiprime segment of a ring R is a pair P2 ⊂ P1 of semiprime ideals of R such that ∩ In ⊆ P2 for all ideals I of R with P2 ⊂ I ⊂ P1. In this paper semiprime segments with P1 a comparizer ideal are classified as either simple, exceptional, or archimedean, extending to several classes of rings a classification known for right chain rings. These three types of semiprime segments are also characterized in terms of the pseudo-radical.