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.

The relationships between the JB*-triple structure of a complex spin factor S and the structure of the Hilbert space H associated to S are discussed. Every surjective linear isometry L of S can be uniquely represented in the form L(x) = mu.U(x) for some conjugation commuting unitary operator U on H and some mu belonging to C, |mu|=1. Automorphisms of S are characterized as those linear maps (continuity not assumed) that preserve minimal tripotents in S and the orthogonality relations among them.

We show that if T is an isometry (as metric spaces) from an open subgroup of the group of invertible elements in a unital semisimple commutative Banach algebra A onto a open subgroup of the group of invertible elements in a unital Banach algebra B, then $T{\left(1\right)}^{-1}T$ is an isometrical group isomorphism. In particular, $T{\left(1\right)}^{-1}T$ extends to an isometrical real algebra isomorphism from A onto B.

A characterization of isometries of complex Musielak-Orlicz spaces $L_{\Phi }$ is given. If $L_{\Phi }$ is not a Hilbert space and $U:L_{\Phi }\to L_{\Phi }$ is a surjective isometry, then there exist a regular set isomorphism τ from (T,Σ,μ) onto itself and a measurable function w such that U(f) = w ·(f ∘ τ) for all $f\in L_{\Phi }$. Isometries of real Nakano spaces, a particular case of Musielak-Orlicz spaces, are also studied.

We give a complete description of the structure of surjective isometries between the unitary groups of unital C*-algebras. While any surjective isometry between the unitary groups of von Neumann algebras can be extended to a real-linear Jordan *-isomorphism between the relevant von Neumann algebras, this is not the case for general unital C*-algebras. We show that the unitary groups of two C*-algebras are isomorphic as metric groups if and only if the C*-algebras are isomorphic in the sense that...

Let $X$ and $E$ be a Banach space and a real Banach lattice, respectively, and let $\Gamma$ denote an infinite set. We give concise proofs of the following results: (1) The dual space $X^{*}$ contains an isometric copy of $c_0$ iff $X^{*}$ contains an isometric copy of ${\ell }_{\infty }$, and (2) $E^{*}$ contains a lattice-isometric copy of $c_0\left(\Gamma \right)$ iff $E^{*}$ contains a lattice-isometric copy of ${\ell }_{\infty }\left(\Gamma \right)$.

We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use ℓψ-Hilbertian and ∞-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition to the case of Banach spaces and introduce the class of Banach spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type,...

Analogues of the classical Banach-Stone theorem for spaces of continuous functions are studied in the context of the spaces of absolutely continuous functions introduced by Ashton and Doust. We show that if AC(σ₁) is algebra isomorphic to AC(σ₂) then σ₁ is homeomorphic to σ₂. The converse however is false. In a positive direction we show that the converse implication does hold if the sets σ₁ and σ₂ are confined to a restricted collection of compact sets, such as the set of all simple polygons.

Suppose ℒ₁ and ℒ₂ are subspace lattices on complex separable Banach spaces X and Y, respectively. We prove that under certain lattice-theoretic conditions every isomorphism from algℒ₁ to algℒ₂ is quasi-spatial; in particular, if a subspace lattice ℒ of a complex separable Banach space X contains a sequence $E_i$ such that $\left(E_i\right)₋\ne X$, $E_i\subseteq E_{i+1}$, and ${\bigvee }_{i=1}^{\infty }{E}_i=X$ then every automorphism of algℒ is quasi-spatial.