We study II${}_1$ factors $M$ and $N$ associated with good generalized Bernoulli actions of groups having an infinite almost normal subgroup with the relative property (T). We prove the following rigidity result : every finite index $M$-$N$-bimodule (in particular, every isomorphism between $M$ and $N$) is described by a commensurability of the groups involved and a commensurability of their actions. The fusion algebra of finite index $M$-$M$-bimodules is identified with an extended Hecke fusion algebra, providing the...

We deal with projective limits of classes of functions and prove that: (a) the Chebyshev polynomials constitute an absolute Schauder basis of the nuclear Fréchet spaces ${}_{\left(\right)}\left({[-1,1]}^r\right)$; (b) there is no continuous linear extension map from ${\Lambda }_{\left(\right)}^{\left(r\right)}$ into ${}_{\left(\right)}\left({ℝ}^r\right)$; (c) under some additional assumption on , there is an explicit extension map from ${}_{\left(\right)}\left({[-1,1]}^r\right)$ into ${}_{\left(\right)}\left({[-2,2]}^r\right)$ by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].

We study imbeddings of the Sobolev space $W^{m,\varrho }\left(\Omega \right)$: = u: Ω → ℝ with $\varrho ({\partial }^{\alpha }u/\partial x^{\alpha })$ < ∞ when |α| ≤ m, in which Ω is a bounded Lipschitz domain in ℝⁿ, ϱ is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n - 1. For such a space we have shown there exist r.i. norms, ${\tau }_{\varrho }$ and ${\sigma }_{\varrho }$, that are optimal with respect to the inclusions $W^{m,\varrho }\left(\Omega \right)\subset W^{m,{\tau }_{\varrho }}\left(\Omega \right)\subset L_{{\sigma }_{\varrho }}\left(\Omega \right)$. General formulas for ${\tau }_{\varrho }$ and ${\sigma }_{\varrho }$ are obtained using the -method of interpolation. These lead to explicit expressions when ϱ is a Lorentz Gamma norm or an Orlicz norm.

We consider a compact linear map T acting between Banach spaces both of which are uniformly convex and uniformly smooth; it is supposed that T has trivial kernel and range dense in the target space. It is shown that if the Gelfand numbers of T decay sufficiently quickly, then the action of T is given by a series with calculable coefficients. This provides a Banach space version of the well-known Hilbert space result of E. Schmidt.

It is shown that a finite system T of matrices whose real linear combinations have real spectrum satisfies a bound of the form $\left|\right|e^{i⟨T,\zeta ⟩}\left|\right|\le C(1+{\left|\zeta \right|)}^s{e}^{r\left|\Im \zeta \right|}$. The proof appeals to the monogenic functional calculus.

It is shown that for each nonzero point x in the open unit disc D, there is a measure whose support is exactly ∂D ∪ {x} and that is also a weak*-exposed point in the set of representing measures for the origin on the disc algebra. This yields a negative answer to a question raised by John Ryff.