A Jordan-isomorphism $J:A^{**}_a\to B^{**}_a$ that preserves q-open projections maps $A$ onto $B$

Suppose $A=C(X,\mathcal M_m)$ and $B=C(Y, \mathcal M_n)$ are two unital C*-algebras, $A_a''$ and $B_a''$ are the atomic parts of $A''$ and $B''$ respectively, $z_A$ is a central projection of $A_a''$ and $z_B$ is a central projection of $B_a''$. Let $J_1:z_AA_a''\to z_B B_a''$ be a *-isomorphism, $J_2: (1-z_A)A_a'' \to (1-z_B)B_a''$ be a *-antiisomorphism, and $J=J_1\oplus J_2:A_a''\to B_a''$ be a Jordan isomorphism such that $\newcommand{\qOP}{\operatorname{OP}}$ $J(\qOP(A))=\qOP(B) $, where $\qOP(A)$ denotes the set of all q-open projections of $A$.Then $J(A)=B$. Since there is a bijection between the set of q-open spectral projections and the set of open spectral projections, by [1, Theorem 2.2], a hermitian element in $A_a''$ belongs to $A$ if and only if each open set in its spectrum corresponds to a q-open spectral projection. Suppose $a$ is a hermitian element in $A$. For each open set $U$ in $\mathbb{R}$, $J_1(\chi_U(z_Aa))=\chi_U(J_1(z_Aa))$, and $TJ_2(\chi_U((1-z_A)a))=\chi_U(TJ_2((1-z_A)a))=T(\chi_U(J_2((1-z_A)a)))$, where $Tf(y)$ is the transpose of $f(y)$ $( f\in C(Y,\mathcal M_n), y\in Y)$. Hence, \begin{align*}J(\chi_U(a)) = &J(\chi_U(z_Aa+ (1-z_A)(a)))\\ = & J(\chi_U(z_Aa)+ \chi_U((1-z_A)a))\\ = &J_1(\chi_U(z_Aa))+ J_2(\chi_U((1-z_A)a)\\ = & \chi_U (J_1(z_Aa))+\chi_U(J_2(1-z_A)a)\\ = & \chi_U(J_1(z_Aa)+J_2((1-z_A)a)=\chi_U(J(a)).\end{align*}Therefore, $\chi_U(J(a))$ is a q-open spectral projection of $B$ and thus $J(a)\in B$. [1] Akemann, C. A., Pedersen, G. K., & Tomiyama, J. (1973). Multipliers of C∗-algebras. Journal of Functional Analysis, 13(3), 277-301.

a projection orthoisomorphism of closed projections of $C(X,\mathcal M_n)$ can be implemented by a Jordan *-isomorphism

Suppose $\theta$ is a orthoisomorpihism between closed projection of $A=C(X, \mathcal M_m)=\mathcal M_m(C(X))$ and $B=C(Y,\mathcal M_n)=\mathcal M_n(C(Y))$.$\newcommand{\P}{\operatorname{Proj}}$ We have known that [$\theta$ can be extended to a projection orthoisomorphism $\tilde\theta$ between $\ell^\infty(X,\P(A))$ and $\ell^\infty(Y,\P(B))$][1]. By Dye's theorem, $\tilde\theta$ is implemented by a Jordan *-isomorphism between $\ell^\infty(X,\mathcal M_m)$ and $\ell^\infty(Y,\mathcal M_n)$. Hence $\tilde\theta$ also preserves q-open projections in $\ell^\infty(X, \mathcal M_m)$. Since a projection $p$ is q-open projection and q-closed iff $p$ is a projection in $C(X,\mathcal M_m)$, by restricting $\tilde\theta$ to the $C(X,\P(\mathcal M_m))$, we get a projection orthoisomorphism between $C(X,\P(\mathcal M_m))$ and $C(Y,\P(\mathcal M_n))$. Denote the characteristic matrix in $A=\mathcal M_m(C(X))$ of $a\in C(X)$ by $$P_{ij}(a)=\left(\begin{matrix}(1+a^*a)^{-1} & (1+a^*a)^{-1}a^*\\ a(1+a^*a)^{-1} & a(1+a^*a)^{-1}a^*\end{matrix}\right)$$ and write $P_i=P_{ij}(0)$ for simplicity. The following can be varified: - $\tilde\theta$ preserves the projection of form $P_i$; - the lattice complement of $P_{j}$ in $P_i+P_j$ is of form $P_{ij}(a)$ if $X$ is connected; Therefore, if $X$ and $Y$ is connected, projections of form $P_{ij}(a)$ correspond under $\tilde\theta$ as $\tilde\theta$ preserves lattice order and addition. Hence the restriction of $\tilde\theta$ to $C(X,\P(\mathcal M_m))$ can be implemented by a Jordan *-isomorphism betwwen $A$ and $B$($\color{red}{unique?}$), by [[lemma 6, 1](#dyelemma6)], when $m=n\geq 3$. \[1\] Dye, H. (1955). On the Geometry of Projections in Certain Operator Algebras. Annals of Mathematics, 61(1), second series, 73-89. doi:10.2307/1969620 [1]: https://math.liveadvances.com/c-ding/a-projection-orthoisomorphism-of-q-closed-projections-can-be-extended-to-a-projection-orthoisomorphism-of-all-projections/

properties of a projection orthoisomorphism on the set of closed projections

By a *projection orthoisomorphism* we mean a one-one mapping $\theta$ that sends the set of projections of a C*-algebra to that of another C*-algebra which preserves orthogonality, in the sense that $pq=0$ if and only if $\theta(p)\theta(q)=0$. $\newcommand{\P}{\operatorname{Proj}}$ Let $X$ is a compact Hausdorff space and $A=C(X,\mathcal M_m)$. The atomic part of $A$ is of form $\ell^\infty(X, \mathcal M_m)$, so the q-closed projections of $A$ is of form $\ell^\infty(X, \P(\mathcal M_m))$, where $\P(\mathcal M_m)$ denotes the set of all projections of $\mathcal M_m$. A rank-one projection $m$ of $\ell^\infty(X,M_m)$ is such a projection that $m(x_0)$ is a rank-one projection in $\mathcal M_m$ for some point $x_0\in X$ and $m(x)=0$ for others. It is q-closed. $\newcommand{\qCP}{\operatorname{CP}}$ **Proposition 1.** Suppose $A=C(X, \mathcal M_m)$, $B=C(Y, \mathcal M_n)$, and $\theta$ is a projection orthoisomorphism between the set $\qCP(A)$ and $\qCP(B)$ of q-closed projections of $A$ and $B$ respectively. Some properties are as follows: (a) $\theta(0)=0$;
(b) $\theta$ preserves order in the sense that $p\leq q$ implies that $\theta(p)\leq \theta(q)$;
(c) $\theta$ sends a rank-one projection to a rank-one projection. *Proof*. (a) $0$ can be characterized as the only q-closed projection orthogonal to itself; (b) If $p\leq q$ but $\theta(p)\nleq\theta(q)$, then there is a rank-one projection $n$ in $\ell^\infty(Y,\mathcal M_n)$ such that $n$ is not orthogonal to $\theta(p)$ but $n\perp \theta(q)$, thus $\theta^{-1}(n)$ is not orthogonal to $p$ but $\theta^{-1}(n)$ is orthogonal to $q$, a contradiction. (c) A rank-one projection can be characterized as follows: itself is the only one non-zero q-closed projection less than it. By (a) and (b), $(c)$ is obtained.

q-open projections of $C(X)$

Let $A$ be a C*-algebra. We call a projection $p$ in the atomic part of $A$ *q-open* if there is an increasing net of positive elements of $A$ converges to $p$ respect to the strong topology. $\newcommand{\Supp}{\operatorname{Supp}}$ **Lemma 1.** If $\{f_i\}$ is an increasing net of positive functions in the closed unit ball of $\ell^\infty(X)\subset B(\ell^2(X))$, then $f_i$ converges to some $f\in\ell^\infty(X)$ strongly is equivalent to $f_i$ converges to $f$ pointwise. *Proof.* \begin{align*} & f_i\xrightarrow{strongly}f\\ \Leftrightarrow & f_i\xrightarrow{weakly}f\\ \Leftrightarrow & \langle f_i h,h\rangle\to \langle fh,h\rangle, \forall h\in \ell^2(X)\\ \Leftrightarrow & \sum_{x\in X}f_i(x)|h(x)|^2\to \sum_{x\in X}f(x)|h(x)|^2, \forall h\in \ell^2(X)\\ \Leftrightarrow & \sum_{x\in X}(f_i(x)-f)g(x)\to 0, \forall g\in \ell^1(X) \end{align*} Suppose $f_i$ converges to $f$ pointwise. For all $g\in \ell^1(X)$ and $\varepsilon >0$, there is a finite set $F$ of $X$ such that $\sum_{x\notin F}|g(x)|<\varepsilon$ and there is a $i_0$ such that $|f_i(x)-f(x)|<\varepsilon $ for all $x\in F$ whenever $i>i_0$. Hence $|\sum_{x\in X}(f_i(x)-f)g(x)|\leq \varepsilon (\|g\|_1+ \#F)$. **Theorem 2.** The mapping $f\to \Supp f$ is a bejection between q-open projections of $C(X)$ and the open set of $X$, where $X$ is a compact Hausdorff space and $\Supp(f):=\{x\in X|f(x)\neq 0\}$ denotes the support of $f\in \ell^\infty(X)$. *Proof.* Suppose $f$ is a q-open projection of $C(X)$, then there is an increasing net $(f_i) $ of positive continuous functions on $X$ converges to $f$ respect to SOT. This is equivalent to $f_i$ increasingly converges to $f$ pointwise by lemma 1. Therefore, $\Supp f=\cup_i\Supp f_i$ is open as the support of each continuous function is open. Suppose $U$ is an open set of $X$, we will prove that the projection with supprot $U$ in the atomic part $\ell^\infty(X)$ of $C(X)$ is open. For every compact set $F$ contained in $U$, by Urysohn's lemma, there is a continuous function $f_F$ such that $f_F({K})=\{1\}$ and ${\Supp f}\subset U$. For each finite collection $\{F_1, F_2, \cdots, F_n\}$ of compact sets contained in $U$, let $f_{\{F_1,\cdots, F_n\}}=\max\{f_{F_1},\cdots, f_{F_n}\}$, then $f_{\{F_1,\cdots, F_n\}}$ forms a increasing net, respect to containment of collections, of positive continuous functions converges pointwise to the projection with support $U$.

the atomic part of $C(X)$

Let $X$ be a compact Hausdorff space and $A=C(X)$. $\newcommand{\id}{\operatorname{id}}\newcommand{\C}{\mathbb{C}}$ For each state $\tau$ of $A$, dnote the associated GNS representation by $(\pi_\tau, H_\tau, x_\tau)$(or $(\pi_\tau,H_\tau)$). If $\tau$ is pure, then $\pi_\tau(A)'=\mathbb{C}\id_\tau$. Since $\pi_\tau(A)\subset \pi_\tau(A)'$, $\pi_\tau(A)=\mathbb{C}\id_\tau$ and thus $\pi_\tau(A)'=B(H_\tau)$. Hence $B(H_\tau)=\C\id_\tau$ and $H_\tau\cong \C$. Suppose $\pi_\tau(a)=k\id_\tau$, then $k=\langle \pi_\tau(a)x_\tau,x_\tau\rangle=\tau(a)$, so $\pi_\tau(a)=\tau(a)\id_\tau$. If $\tau$ and $\rho$ are two pure states of $A$, $(\pi_\tau, H_\tau)$ is unitarily equivalent to $(\pi_\rho, H_\rho)$. Then $\tau(a)\id_\tau=\pi_\tau(a)=u^*\pi_\rho(a) u=\rho(a)u^*\id_\rho u=\rho(a)\id_\tau$,where $u$ is a unitary from $H_\tau$ to $H_\rho$. Therefore, $\tau=\rho$. That is, the atomic representation of $C(X)$ is $(\pi_{atomic}, H_{atomic})=(\oplus_{\tau\in PS(A)}\pi_\tau,\oplus_{\tau\in PS(A)} H_\tau)=(\oplus_{\tau\in \Sigma(A)}\tau(\cdot)\id_\tau, \oplus_{\tau\in \Sigma(A)}\C)=(\oplus_{\tau\in \Sigma(A)}\tau(\cdot)\id_\tau, \ell^2(\Sigma(A))),$ where $\Sigma(A)$ is the character space of $A$. Since $\oplus_{\tau\in PS(A)}\tau(a)\id_\tau=\widehat{a}:\tau\to \tau(a)\id_\tau$, $\pi_{atomic}(A)=C(\Sigma(A))$. As $\pi_{atomic}$ is non-degenerate, operators in $B(\oplus_{\tau\in PS(A)}H_\tau)$ communicated with $\oplus_{\tau\in PS(A)}\pi_\tau(a)$ are of form $\oplus_{\tau\in PS(A)}u_\tau$. By $\|\oplus_{\tau\in PS(A)}u_\tau\|=\sup_{\tau\in PS(A)}\|u_\tau\|< \infty$ we know that $\pi_{atomic}(A)'=\ell^\infty(\Sigma(A))$ and thus $\pi_{atomic}(A)''=\ell^\infty(\Sigma(A))$