We study conditions on an infinite dimensional separable Banach space $X$ implying that $X$ is the only non-trivial invariant subspace of $X^{**}$ under the action of the algebra $𝔸\left(X\right)$ of biconjugates of bounded operators on $X$: $𝔸\left(X\right)=\{T^{**}T\in ℬ\left(X\right)\}$. Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of $c_0$, and show in particular that any space which does not contain ${\ell }_1$ and has property (u) of Pelczynski is simple.

We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [12]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of examples for which our conditions are satisfied. Included are known examples of PI spaces, such as Laakso spaces,...

Suppose that a real nonatomic function space on $[0,1]$ is equipped with two rearrangement-invariant norms $\parallel ·\parallel$ and $\left|\right||·|\left|\right|$. We study the question whether or not the fact that $(X,\parallel ·\parallel )$ is isometric to $(X,|\left|\right|·\left|\right|\left|\right)$ implies that $\parallel f\parallel =\left|\right|\left|f\right|\left|\right|$ for all $f$ in $X$. We show that in strictly monotone Orlicz and Lorentz spaces this is equivalent to asking whether or not the norms are defined by equal Orlicz functions, respĿorentz weights. We show that the above implication holds true in most rearrangement-invariant spaces, but we also identify a class...

Isometric Sobolev spaces on finite graphs are characterized. The characterization implies that the following analogue of the Banach-Stone theorem is valid: if two Sobolev spaces on 3-connected graphs, with the exponent which is not an even integer, are isometric, then the corresponding graphs are isomorphic. As a corollary it is shown that for each finite group and each p which is not an even integer, there exists n ∈ ℕ and a subspace $L\subset \ell {ⁿ}_p$ whose group of isometries is the direct product × ℤ₂.

We investigate isometric composition operators on the weighted Dirichlet space ${𝒟}_{\alpha }$ with standard weights ${(1-|z|}^2{)}^{\alpha }$, $\alpha >-1$. The main technique used comes from Martín and Vukotić who completely characterized the isometric composition operators on the classical Dirichlet space $𝒟$. We solve some of these but not in general. We also investigate the situation when ${𝒟}_{\alpha }$ is equipped with another equivalent norm.

We prove that some Banach spaces X have the property that every Banach space that can be isometrically embedded in X can be isometrically and linearly embedded in X. We do not know if this is a general property of Banach spaces. As a consequence we characterize for which ordinal numbers α, β there exists an isometric embedding between $C_0(\alpha +1)$ and $C_0(\beta +1)$.

Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z)\ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of ${\ell }_{\infty }$.

It is shown that $l_2^n$ imbeds isometrically into $l_4^{n^2+1}$ provided that n is a prime power plus one, in the complex case. This and similar imbeddings are constructed using elementary techniques from number theory, combinatorics and coding theory. The imbeddings are related to existence of certain cubature formulas in numerical analysis.