Properties of projections in a C*-algebra

A *projection* in a C*-algebra is an element $p$ satisfying $$p=p^*=p^2.$$
(The above equation is equivalent to $p=p^*p$, noting that $p^*=(p^*p)^*=p^*p=p$ and thus $p=p^*p=p^2$.)

**Property 1**. Let $A$ be a C*-algebra, $p$ and $q$ are two projections in $A$. Then $$p\leq q\Leftrightarrow pq=p\Leftrightarrow qp=p.$$

*Proof*. $p \leq q\Rightarrow pq=p$:

Suppose $a=q-p\geq 0$, then \begin{align*}
pq-p=pa,
\end{align*}
hence \begin{align*}
p(1-q)p=-pap,
\end{align*}
Thus $p(1-q)p=0$, $\|p-pq\|^2=\|p(1-q)p\|=0$ and $p=pq$.

**Property 2**.
$$pq=0\Leftrightarrow p\circ q=0$$ where $p\circ q=\frac{pq+qp}{2}$.

*Proof*. If $p\circ q=0$, then $$pq=-qp,$$ hence \begin{align*}pqp=p(pq)p=p(-qp)p=-pqp.\end{align*} So, $pqp=0$, $\|pq\|^2=\|pqp\|=0$, $pq=0$.

Ajunction from normed spaces to dual spaces

$\def \AA{A^{**}}\newcommand{\cC}{\mathbf{C\mbox{-}Vct}}
\newcommand{\cW}{\mathbf{W\mbox{-}Vct}}
\newcommand{\BB}{B^{**}}\newcommand{\ob}[1]{\operatorname{Ob}{\mathbf{#1}}}\newcommand{\ad}{\operatorname{Ad^2}}
\newcommand{\id}{\operatorname{Id}}$



Lemma 1.
Suppose $\pi :V\to W$ is a continuous linear map between two normed spaces, then
the adjoint of $\pi $ defined by \begin{align}
\pi^*:&W^*\to V^*\notag\\
& f\mapsto f\circ \pi\label{adjoint}
\end{align}

is a continous linear map respect to norms of two spaces or waek* topology of two
spaces. Moreover, the following equation always holds: \begin{align}
\label{adimbedding}
\pi^{**}\circ i=i\circ \pi,
\end{align}

i.e., the following diagram is always commutative: \begin{equation}
\label{icommutes}
\xymatrix{
V\ar[r]^\pi\ar[d]_i&W\ar[d]^i\\
V^{**}\ar[r]^{\pi^{**}}&W^{**}.
}
\end{equation}

Proof. Since \begin{equation}
\label{normdecrease}
\|f\circ \pi\|\leq\|f\|\|\pi\|,
\end{equation}

Suppose $(f_i)$ is a net converges to $f$ in $W^*$ with respect to the weak* topology,
i.e.,
$$f_i(w)\to f(w) \forall w\in W.$$
For any $v\in V$, $\pi (v)\in W$, so
$$f_i(\pi (v))\to f(\pi (v)),$$
so $\pi ^*(f_i)$ weak*-converges to $\pi ^*(f)$.

For any $v\in V$, \begin{align*}
&(\pi^{**}\circ i)(v)(w’)=\pi^{**}(i(v))(w’)=(i(v)\circ \pi^*)(w’)\\
=&i(v)(\pi^*(w’))=i(v)(w’\circ \pi)=(w’\circ \pi)(v)\\
=&w’(\pi(v))=i(\pi(v))(w’)=(i\circ \pi)(v)(w’)~(w’\in W^*),
\end{align*}

hence, $\pi ^{**}\circ i=i\circ \pi $. __

A normed space $M$ is said to be W-space if it admits a predual $^*M$, that is, $^*M$ is a normed
space and $M$ is isomorphic to $(^*M)^*$ as a normed space. Now Let $\cC $ and $\cW $ be two categories as
follows:

  • $\cC $: objects, a collection of normed spaces; arrows, all bouned linear maps
    between them.
  • $\cW $: objects, a collection of objects of form $V^*$ for some normed space $V$; arrows,
    all bouned linear maps between them that are also weak*-continuous.

Clearly, $\cW $ is a subcategory of $\cC $. Let $\ad $ assign to each bounded linear map $\pi :A\to B$ between two
normed space the double adjoint $\pi ^{**}: \AA \to \BB $, and $\id $ assign each bounded and weak*-continuous
map $\pi :M\to N$ between two W-spaces itself, then $\ad $ and $\id $ are two functors between $\cC $ and $\cW $.
Denotes the set of all arrows in $\cC $ from object $A$ to $B$ by $\cC (A,B)$, we have



Lemma 2.
for each object $A\in \cC $ and each obeject $M\in \cW $, there is a bijection of sets
\begin{align}
\label{n}
n=n_{A,M}:&\cW(\ad(A),M)\to\cC(A,\id(M))\notag\\
&(\xymatrix{\ad(A)\ar[r]^\pi&M})\mapsto(\xymatrix{A\ar[r]^{\pi\circ i}&\id(M)}),
\end{align}

which is natural between two hom-functors

$$\hom _\cW (\ad (*),*), \hom _\cC (*,\id (*)): \cC ^{op}\times \cW \to \mathbf {Set}$$
as indicated by the following commutative diagram for each $\xymatrix {B\ar [r]^f & A}$ and $\xymatrix {M\ar [r]^g & N}$, \begin{align}
\label{naturalhom}\xymatrix{
\cW(\ad A,M)\ar[d]_{((\ad f)^*,g_*)}\ar[r]^{n_{A,M}}&\cC(A,\id M)\ar[d]^{(f^*,(\id g)_*)}\\
\cW(\ad B,N)\ar[r]^{n_{B,N}}&\cC(B,\id N),
}
\end{align}

where $f^*$ is the operation turning an arrow $\pi :A\to B$ in $\cC $ into $f\circ \pi $, adn $g_*$ turning $\pi :M\to N$ in $\cW $ into $\pi \circ g$. In one
sentence, the triple $\langle \ad ,\id , n\rangle $ is an ajunction from $\cC $ to $\cW $.


Proof.
By lemma 1, the following map is well-defined: \begin{align}
\label{m}
m=m_{A,M}:&\cC(A,\id M)\to\cW(\ad A,M)\notag\\
&(\xymatrix{A\ar[r]^{\pi}&M})\mapsto(\xymatrix{\AA\ar[r]^{(\pi^*\circ i)^*}&M})
\end{align}

For each $A\in \cC $ and each $M\in \cW $, we prove that $m_{A,M}$ is the inverse of $n_{A,M}$.

Pick an arrow $\xymatrix {\AA \ar [r]^{\pi ”} & M}$ in $\cW $. For any $a\in A$, \begin{align*}
&((\pi’’\circ i)^*\circ i)^*(i(a))(’m)=i(a)(((\pi’’\circ i)^*(i(’m)))=i(a)(i(’m)\circ (\pi’’\circ i))\\
=&(i(’m)\circ (\pi’’\circ i))(a)=i(’m)(\pi’’(i(a)))=\pi’’(i(a))(’m)~(’m\in^*M),
\end{align*}

i.e.,

$$((\pi ”\circ i)^*\circ i)^*(i(a))=\pi ”(i(a)).$$
By Goldstine theorem (proposition V.4.1, [1]), $i(A)$ is weak* dense in $A^{**}$.
$$(\pi ”\circ i)^*\circ i)^* = \pi ”$$
since the maps of both sides are weak*-continuous.

Pick an arrow $\xymatrix {A\ar [r]^{\pi } & M}$ in $\cC $. For any $a\in A$, \begin{align*}
&((\pi^*\circ i)^*\circ i)(a)=(\pi^*\circ i)^*(i(a))=i(a)\circ (\pi^*\circ i)\\
=&(\pi^*\circ i)(a)=\pi^*(i(a))=i(a)\circ \pi=\pi(a).
\end{align*}

So,
$$(\pi ^*\circ i)^*\circ i=\pi .$$

Suppose $\pi \in \cW (\ad A, M)$. For each $\xymatrix {B\ar [r]^f & A}$ and $\xymatrix {M\ar [r]^g & N}$, \begin{align*}
\id(g)\circ n_{A,M}(\pi)\circ f=g\circ \pi\circ i\circ f,\\
g\circ n_{B,N}(\pi)\circ \ad(f)=g\circ \pi\circ f^{**}\circ i.
\end{align*}

By \eqref{ad imbedding}, $\id (g)\circ n_{A,M}(\pi )\circ f=g\circ n_{B,N}(\pi )\circ \ad (f)$, hence the diagram \eqref{natural hom} is
commutative. __

By theorem IV.1.1, [2],
$$\eta _A:=n_{A,A}(\operatorname {1}_{\ad A})=i:A\to A^{**}$$
together with $\ad A$ is a universal arrow from $A\in \cC $ to the functor $\id :\cW \to \cC $ in the sense for any $M\in \cW $ and
any $\xymatrix { A\ar [r]^\pi & \id M}$ in $\cC $ there is an arrow $\xymatrix @C=1.75cm{\ad M\ar [r]^{\pi ”=n_{A,M}^{-1}(\pi )} & N}$ in $\cW $ filled in the commutative digram
$$\xymatrix { A\ar [r]^i\ar [rd]_\pi & \id \ad A\ar @{–>}[d]^{\id \pi ”} & \ad A\ar @{–>}[d]^{\pi ”=n_{A,M}^{-1}(\pi )} \\ & \id M. & M } $$
In other words,


Theorem 3. Each linear map $\pi :A\to M$ from a normed space $A$ and a W-space $M$ can be
extended to a unique linear map $(\pi ^*\circ i)^*:A^{**}\to M$ that are both bouded and weak*-continuous.

References


[1]   
John B Conway. A course in functional analysis, volume 96. Springer
Science; Business Media, 2013.


[2]   
Saunders MacLane. Categories for the Working Mathematician.
Springer-Verlag, New York, 1971. Graduate Texts in Mathematics, Vol. 5.

$z(M)M$ is atomic

Suppose $M$ is a W*-algebra, $z$ is the supremum of all minimal projections in $M$. It is well-known that $z$ is a central porjection in $M$. Let $M_0=zM$, then $M_0$ is also a W*-algebra. Denote the collection of all minimal projections in a W*-algebra $M$ by $\mathscr P_m^M$.

**Lemma**. $\mathscr P_m^M=\mathscr P_m^{M_0}$.

*Proof*. Since $m=zm$ for any minimal projection $m$ in $M$, $\mathscr P_m^M\subset\mathscr P_m^{M_0}$. Conversely, suppose $m_0$ is a minimal projection in $M_0$, $p\in M$ and $p\leq m_0$, then $zp\leq zm=m$, hence $zp=0$ or $zp=m_0$. Therefore, $m_0p=m_0zp$ equals to $0$ or $m_0$, i.e., $m_0$ is a minimal projection in $M$.

Now, we can denote $\mathscr P_m^M$ and $\mathscr P_m^{M_0}$ by the same notation $\mathscr P_m$.

**Theorem**. $M_0$ is atomic, that is, each projection in $M_0$ dominates a minimal projection.

*Proof*. Suppose $p$ is a non-zero projection in $M_0$, then $p\leq z$, so, there is a minimal projection $m$ such that $pm\neq 0$( otherwise, $z=\sup\{m\in \mathscr P_m\}\leq 1-p$, $pz=0$ ), which is equivalent to $pmp\neq 0$. Since $$pmpMpmp\subset pmMmp=p(\mathbb{C}m) p=\mathbb{C}pmp,$$ $(pmp)^2$ is a scalar multiple of $pmp$. Let $m_p=pmp/\|pmp\|$, then $m_p$ is a projection and $m_pMm_p= \mathbb C m_p$, that is, $m_p$ is a minimal projection. What’s more, $m_p\leq p$ for $m_pp=pm_p=m_p$.

**Theorem**. For any $q$ in $M_0$, we have $q=\sup\{m\in \mathscr P_m|m\leq q\}$.

*Proof*. Denote $\sup\{m\in \mathscr P_m|m\leq q\}$ by $q_0$, then $q_0\leq q$ and $q_0\in M_0$. If $q_0\neq q$, then $q-q_0$ dominates a minimal projection, say $m$. So, $m\leq q$, and thus $m\leq q_0$ by the definition of $q_0$. Therefore, $m(q-q_0)=mq-mq_0=m-m=0$, a contradiction to $m\leq q-q_0$.

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] Ng, Chi-Keung, and Ngai-Ching Wong. “Comparisons of equivalence relations on open projections.” J. Operator Theory 74.1 (2015): 101-123.–>

[1]: https://math.liveadvances.com/c-ding/a-projection-orthoisomorphism-of-q-closed-projections-can-be-extended-to-a-projection-orthoisomorphism-of-all-projections/