Let $R$ be an associative algebra over a field $K$ generated by a vector subspace $V$. The polynomial $f(x_1,...,x_n)$ of the free associative algebra $K\langle x_1,x_2,...\rangle$ is a weak polynomial identity for the pair $(R,V)$ if it vanishes in $R$ when evaluated on $V$. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three....

Let ${ℛ}_{\alpha }$ be the Riesz distribution on a simple Euclidean Jordan algebra, parametrized by $\alpha \in ℂ$. I give an elementary proof of the necessary and sufficient condition for ${ℛ}_{\alpha }$ to be a locally finite complex measure (= complex Radon measure).

If the space $𝒬$ of quadratic forms in ${ℝ}^n$ is splitted in a direct sum ${𝒬}_1\oplus ...\oplus {𝒬}_k$ and if $X$ and $Y$ are independent random variables of ${ℝ}^n$, assume that there exist a real number $a$ such that $E\left(X\right|X+Y)=a(X+Y)$ and real distinct numbers $b_1,...,b_k$ such that $E\left(q\left(X\right)\right|X+Y)=b_{i}q(X+Y)$ for any $q$ in ${𝒬}_i.$ We prove that this happens only when $k=2$, when ${ℝ}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.