An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dual space of a JB*-triple, and up to complete isometry, of one-sided ideals in C*-algebras.
We prove that every JBW$^*$-triple $M$ with rank one or rank bigger than or equal to three satisfies the Mazur--Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$. We also show that the same conclusion holds if $M$ is not a JBW$^*$-triple factor, or more generally, if the atomic part of $M^{**}$ is not a rank two Cartan factor.
Linear isometries between real JB*-triples and C*-algebras
设$T: A\to B$是两个实JB $^*$ -三元组之间的线性等距（不一定是满射）.则对每个$a\in A$，在$B$的双对偶$B'',$中存在一个三幂$u_a$，使得\begin{enumerate}[$(a)$] \item $\{u_a,T(\{f,g,h\}),u_a\}=\{u_a,\{T(f),T(g),T(h)\},u_a\}$, for all $f,g,h$ in the real JB$^*$-subtriple, $A_a,$ generated by $a$; \item The mapping $\{u_a,T(\cdot),u_a\} :A_a\rightarrow B''$ is a linear isometry. \end{enumerate}.此外，当$B$是实C-$^*$-代数时，投影$p=p_a= u_a^* u_a$满足$T(\cdot)p :A_a\rightarrow B''$是等距三同态.当$A$和$B$是实C-$^*$-代数，且$A$是实型交换时，存在部分等距$u\in B''$使得映射$T(\cdot)u^*u :A\rightarrow B''$是等距三同态.这些结果推广到实际环境中，由于C.- H.朱和N.- C. Wong和C.- H. Chu和M. 2004年和2005年的麦基。我们给出了一个例子，一个非满射的实线性等…
We examine the structure of two possible candidates of isometry groups for the spectral triples on $AF$-algebras introduced by Christensen and Ivan. In particular, we completely determine the isometry group introduced by Park, and observe that these groups coincide in the case of the Cantor set. We also show that the construction of spectral triples on crossed products given by Hawkins, Skalski, White and Zacharias, is suitable for the purpose of lifting isometries.
We prove that every surjective isometry from the unit sphere of the space $K(H),$ of all compact operators on an arbitrary complex Hilbert space $H$, onto the unit sphere of an arbitrary real Banach space $Y$ can be extended to a surjective real linear isometry from $K(H)$ onto $Y$. This is probably the first example of an infinite dimensional non-commutative C$^*$-algebra containing no unitaries and satisfying the Mazur--Ulam property. We also prove that all compact C$^*$-algebras and all weakly compact JB$^*$-triples satisfy the Mazur--Ulam property.
Tingley's problem asks whether every surjective isometry between the unit spheres of two Banach spaces admits an extension to a real linear surjective isometry between the whole spaces. In this paper, we give an affirmative answer to Tingley's problem when both spaces are preduals of von Neumann algebras, the spaces of self-adjoint operators in von Neumann algebras or the spaces of self-adjoint normal functionals on von Neumann algebras. We also show that every surjective isometry between the unit spheres of unital C$^*$-algebras restricts to a bijection between their unitary groups. In additi…
We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed probability measure. In the former case it also follows from recent results of the second author that the quantum isometry group is classical, i.e. the commutative $C^*$-algebra of continuous functions on the Riemannian isometry group.
Let $M$ and $N$ be two unital JB$^*$-algebras and let $\mathcal{U} (M)$ and $\mathcal{U} (N)$ denote the sets of all unitaries in $M$ and $N$, respectively. We prove that the following statements are equivalent: $(a)$ $M$ and $N$ are isometrically isomorphic as (complex) Banach spaces; $(b)$ $M$ and $N$ are isometrically isomorphic as real Banach spaces; $(c)$ There exists a surjective isometry $Δ: \mathcal{U}(M)\to \mathcal{U}(N).$ We actually establish a more general statement asserting that, under some mild extra conditions, for each surjective isometry $Δ:\mathcal{U} (M) \to \mathcal{U} (N…
Let $(Ω,μ)$ be a $σ$-finite measure space. Given a Banach space $X$, let the symbol $S(X)$ stand for the unit sphere of $X$. We prove that the space $L^{\infty} (Ω,μ)$ of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm, satisfies the Mazur-Ulam property, that is, if $X$ is any complex Banach space, every surjective isometry $Δ: S(L^{\infty} (Ω,μ))\to S(X)$ admits an extension to a surjective real linear isometry $T: L^{\infty} (Ω,μ)\to X$. This conclusion is derived from a more general statement which assures that every surjective isometry …
Existence and examples of quantum isometry group for a class of compact metric spaces
给出了紧致度量空间上紧致量子群（CQG）的等距作用的定义，推广了Banica在有限度量空间上的定义。对于可以等距嵌入到某个欧氏空间中的度量空间$(X,d)$，证明了在等距作用于$(X,d)$上的紧量子群范畴中存在泛对象.事实上，我们的存在性定理适用于更大的一类，即对于任何紧致度量空间$(X,d)$，它允许一个一对一连续映射$f : X \raro \IR^n$对于某些$n$，使得$d_0(f(x),f(y))=ϕ(d(x,y))$（其中$d_0$是欧几里得度量）对于$\IR^+$的某些同胚$ϕ$。作为具体的例子，我们得到了Wang的量子置换群$\cls_n^+$和$\IZ_2$与$\cls_n^+$的自由圈积作为在\cite{huang1}中取区间拓扑并构造的紧致连通度量空间的量子等距群.
