Let $0<\unicode[STIX]{x1D6FC}<n,1\leq p<q<\infty$ with $1/p-1/q=\unicode[STIX]{x1D6FC}/n$, $\unicode[STIX]{x1D714}\in A_{p,q}$, $\unicode[STIX]{x1D708}\in A_{\infty }$ and let $f$ be a locally integrable function. In this paper, it is proved that $f$ is in bounded mean oscillation $\mathit{BMO}$ space if and only if

$$\begin{eqnarray}\sup _{B}\frac{|B|^{\unicode[STIX]{x1D6FC}/n}}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q}<\infty ,\end{eqnarray}$$

$\unicode[STIX]{x1D714}^{p}(B)=\int _{B}\unicode[STIX]{x1D714}(x)^{p}\,dx$

$f_{\unicode[STIX]{x1D708},B}=(1/\unicode[STIX]{x1D708}(B))\int _{B}f(y)\unicode[STIX]{x1D708}(y)\,dy$

$f$

$Lip_{\unicode[STIX]{x1D6FC}}$

$$\begin{eqnarray}\sup _{B}\frac{1}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q}<\infty .\end{eqnarray}$$

We investigate the global versions of the Kleinecke–Shirokov theorem for skew derivations in Banach algebras. Centralizing skew derivations on Banach algebras are also studied.

Let 𝒟 be a strongly double triangle subspace lattice on a nonzero complex reflexive Banach space X and let δ:Alg 𝒟→Alg 𝒟 be a linear mapping. We show that δ is Jordan derivable at zero, that is, δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A) whenever AB+BA=0 if and only if δ has the form δ(A)=τ(A)+λA for some derivation τ and some scalar λ. We also show that if the dimension of X is greater than 2, then δ satisfies δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A) whenever AB=0 if and only if δ is a derivation.

The ‘polaroid’ property transfers from Banach algebra elements to their tensor product, and hence also to their induced multiplications on ‘ultraprime’ Banach bimodules.

Let A be a C*-algebra, and let X be a Banach A-bimodule. Johnson [B. E. Johnson, ‘Local derivations on C*-algebras are derivations’, Trans. Amer. Math. Soc. 353 (2000), 313–325] showed that local derivations from A into X are derivations. We extend this concept of locality to the higher cohomology of a C*-algebra and show that, for every , bounded local n-cocycles from A(n) into X are n-cocycles.