When are maps preserving semi-inner products linear?

We observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space $$\ell _2$$ ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

The wigner property for CL-spaces and finite-dimensional polyhedral Banach spaces

We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

Similarity isometries of point packings

A linear isometry R of {\bb R}^{d} is called a similarity isometry of a lattice \Gamma\subseteq{\bb R}^{d} if there exists a positive real number β such that βRΓ is a sublattice of (finite index in) Γ. The set βRΓ is referred to as a similar sublattice of Γ. A (crystallographic) point packing generated by a lattice Γ is a union of Γ with finitely many shifted copies of Γ. In this study, the notion of similarity isometries is extended to point packings. A characterization for the similarity isometries of point packings is provided and the corresponding similar subpackings are identified. Planar examples are discussed, namely the 1 × 2 rectangular lattice and the hexagonal packing (or honeycomb lattice). Finally, similarity isometries of point packings about points different from the origin are considered by studying similarity isometries of shifted point packings. In particular, similarity isometries of a certain shifted hexagonal packing are computed and compared with those of the hexagonal packing.

2-локальные изометрии некоммутативных пространств Лоренца

Let mathcal M be a von Neumann algebra equipped with a faithful normal finite trace tau, and let Sleft( mathcalM, tauright) be an ast -algebra of all tau -measurable operators affiliated with mathcal M . For x in Sleft( mathcalM, tauright) the generalized singular value function mu(x):trightarrow mu(tx), t0, is defined by the equality mu(tx)infxp_mathcalM:, p2pp in mathcalM, , tau(mathbf1-p)leq t. Let psi be an increasing concave continuous function on 0, infty) with psi(0) 0, psi(infty)infty, and let Lambda_psi(mathcal M,tau) left x in Sleft( mathcalM, tauright): x _psi int_0inftymu(tx)dpsi(t) infty right be the non-commutative Lorentz space. A surjective (not necessarily linear) mapping V:, Lambda_psi(mathcal M,tau) to Lambda_psi(mathcal M,tau) is called a surjective 2-local isometry, if for any x, y in Lambda_psi(mathcal M,tau) there exists a surjective linear isometry V_x, y:, Lambda_psi(mathcal M,tau) to Lambda_psi(mathcal M,tau) such that V(x) V_x, y(x) and V(y) V_x, y(y). It is proved that in the case when mathcalM is a factor, every surjective 2-local isometry V:Lambda_psi(mathcal M,tau) to Lambda_psi(mathcal M,tau) is a linear isometry.

Изометрии действительных подпространств самосопряженных операторов в банаховых симметричных идеалах

Let (mathcal C_E, cdot_mathcal C_E) be a Banach symmetric ideal of compact operators, acting in a complex separable infinite-dimensional Hilbert space mathcal H. Let mathcal C_Ehxin mathcal C_E : xx be the real Banach subspace of self-adjoint operators in (mathcal C_E, cdot_mathcal C_E). We show that in the case when (mathcal C_E, cdot_mathcal C_E) is a separable or perfect Banach symmetric ideal (mathcal C_E eq mathcal C_2) any skew-Hermitian operator H: mathcal C_Ehto mathcal C_Eh has the following form H(x)i(xa - ax) for same aain mathcal B(mathcal H) and for all xin mathcal C_Eh. Using this description of skew-Hermitian operators, we obtain the following general form of surjective linear isometries V:mathcal C_Eh to mathcal C_Eh. Let (mathcal C_E, cdot_mathcal C_E) be a separable or a perfect Banach symmetric ideal with not uniform norm, that is p_mathcal C_E 1 for any finite dimensional projection p inmathcal C_E with dim p(mathcal H)1, let mathcal C_E eq mathcal C_2, and let V: mathcal C_Eh to mathcal C_Eh be a surjective linear isometry. Then there exists unitary or anti-unitary operator u on mathcal H such that V(x)uxu orV(x)-uxu for all x in mathcal C_Eh.

ON THE EXTENSION OF ISOMETRIES BETWEEN THE UNIT SPHERES OF A -TRIPLE AND A BANACH SPACE

We prove that if $M$ is a $\text{JBW}^{\ast }$ -triple and not a Cartan factor of rank two, then $M$ satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another real Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$ .

Extreme non-Arens regularity of the group algebra

The Banach algebras of Harmonic Analysis are usually far from being Arens regular and often turn out to be as irregular as possible. This utmost irregularity has been studied by means of two notions: strong Arens irregularity, in the sense of Dales and Lau, and extreme non-Arens regularity, in the sense of Granirer. Lau and Losert proved in 1988 that the convolution algebra {L^{1}(G)} is strongly Arens irregular for any infinite locally compact group. In the present paper, we prove that {L^{1}(G)} is extremely non-Arens regular for any infinite locally compact group. We actually prove the stronger result that for any non-discrete locally compact group G, there is a linear isometry from {L^{\infty}(G)} into the quotient space {L^{\infty}(G)/\mathcal{F}(G)}, with {\mathcal{F}(G)} being any closed subspace of {L^{\infty}(G)} made of continuous bounded functions. This, together with the known fact that {\ell^{\infty}(G)/\mathscr{W\!A\!P}(G)} always contains a linearly isometric copy of {\ell^{\infty}(G)}, proves that {L^{1}(G)} is extremely non-Arens regular for every infinite locally compact group.

Dynamics of Biholomorphic Self-Maps on Bounded Symmetric Domains

Let $g$ be a fixed-point free biholomorphic self-map of a bounded symmetric domain $B$. It is known that the sequence of iterates $(g^n)$ may not always converge locally uniformly on $B$ even, for example, if $B$ is an infinite dimensional Hilbert ball. However, $g=g_a\circ T$, for a linear isometry $T$, $a=g(0)$ and a transvection $g_a$, and we show that it is possible to determine the dynamics of $g_a$. We prove that the sequence of iterates $(g_a^n)$ converges locally uniformly on $B$ if, and only if, $a$ is regular, in which case, the limit is a holomorphic map of $B$ onto a boundary component (surprisingly though, generally not the boundary component of $\frac{a}{\|a\|}$). We prove $(g_a^n)$ converges to a constant for all non-zero $a$ if, and only if, $B$ is a complex Hilbert ball. The results are new even in finite dimensions where every element is regular.

ON ISOMETRIC REPRESENTATION SUBSETS OF BANACH SPACES

Let $X,Y$ be two Banach spaces and $B_{X}$ the closed unit ball of $X$. We prove that if there is an isometry $f:B_{X}\rightarrow Y$ with $f(0)=0$, then there exists an isometry $F:X\rightarrow Y^{\ast \ast }$. If, in addition, $Y$ is weakly nearly strictly convex, then there is an isometry $F:X\rightarrow Y$. Making use of these results, we show that if $Y$ is weakly nearly strictly convex and there is an isometry $f:B_{X}\rightarrow Y$ with $f(0)=0$, then there exists a linear isometry $S:X\rightarrow Y$.