Universal Property of the Multiplier Algerba

We call an injective $*$-isomorphism $i:A\to B$ between two C*-algebra an *ideal homomorphism* if $i(A)$ is an closed ideal of $B$. The universal property of the multiplier algebra $M(I)$ of a C*-algebra $I$ is: for any C*-algebra $A$ and an ideal homomorphism $i:I\to A$ there is a unique $*$-homomorphism $f$ from $A$ to $M(I)$ such that the following diagram is commutative:
\begin{xy}
\xymatrix{
A\ar[r]^f & M(I)\\
I\ar[u]^i\ar[ru]_c & ,
}
\end{xy}
where $c$ is the canonical ideal homomorphism.

*An application*. [If $I$ is a unital C*-algebra, then it cannot be an essential ideal of any other C*-algebra.][1]

[1]: https://math.stackexchange.com/q/3046342

What is “Free”

**Update**.Suppose $\mathbf{B}$ is a category, $\mathbf{A}$ is a subcategory of $\mathbf{B}$, and $i:X\to G$ is an injective morphism. If for each object $Y$ of $\mathbf{A}$ and each morphism $f:X\to Y$ there is a unique morphism $\tilde{f}:G\to Y$ such that the following diagram is commutative:
\begin{xy}
\xymatrix{
X\ar[r]^f \ar[d]_i&Y\\
G\ar[ur]_{\tilde{f}} & ,
}
\end{xy}
then $G$ is said to be a *free* object with a *basis* $X$.

——

Suppose $\mathbf{A}$ is a category, $G$ is an object of $\mathbf{A}$. If there is a parent category $\mathbf{B}$ of $\mathbf{A}$ with an object $X$ and an injective morphism $i:X\to G$ satisfying that for any object $Y$ of $\mathbf{A}$ and each morphism $f:X\to Y$ there is a unique morphism $\tilde{f}:G\to Y$ such that the following diagram is commutative:
\begin{xy}
\xymatrix{
X\ar[r]^f \ar[d]_i&Y\\
G\ar[ur]_{\tilde{f}} & ,
}
\end{xy}
then $G$ is said to be a *free* object.
(discussed with S.Q.Huang)

The isomorphism between $K_0$ groups under the unital case implies that under the non-unital case

Suppose $A$ is a C*-algebra.
$\newcommand{\diag}{\operatorname{diag}}\newcommand{\id}{\operatorname{id}}$The map $i_*:K_0(A)\to K_0(M_n(A))$ induced by $i: A\to M_n(A), a\mapsto\diag(a,0)$ is an isomorphism.

This is a supplementary explanation for lemma 6.2.10, [[1](#wegge-olsen)]. The author states that to prove the map $a\mapsto \diag(a,0)$ from $A$ to $M_n(A)$ induces an isomorphism between $K_0(A)$ and $K_0(M_n(A))$, it suffices to prove that the map induces a isomorphism between $V(A)$ and $V(M_n(A))$. It is true for the unital case in which $K_0(A)$ is just the Grothendieck group of $V(A)$. But for the non-unital case? It can be proved by the unital case!(known from Dadarlat and Chung)

Consider the following diagram:
$\newcommand{\C}{\mathbb{C}}$
\begin{xy}
\xymatrix{
0\ar[r]\ar[d] & A\ar[r]^i\ar[d]^\alpha &A^+\ar@/^/[r]^{\pi}\ar[d]^{\alpha^+} &\C\ar[r]\ar[d]^{\alpha_{\C}}\ar@/^/ [l]^j &0\ar[d] \\
0\ar[r] & M_n(A)\ar[r]^{i_n} &M_n(A^+)\ar@/^/[r]^{\pi_{ n}}&M_n(\C)\ar[r]\ar@/^/[l]^{j_n} &0
}
\end{xy}
(1)It is commutative. For example, $
\alpha_\C\circ\pi(a,\lambda)=\alpha_\C(\lambda)=\diag(\lambda,0)$ and
$\pi_n\circ\alpha^+(a,\lambda)=\pi_n((a,\lambda),0)=(\lambda,0)$ for each $(a,\lambda)\in A^+$, thus $\alpha_\C\circ\pi=\pi_n\circ\alpha^+$.

(2)Its rows are split exact sequences of C*-algebras. It is easy to verify that $\operatorname{im}i_n=\ker \pi_n$ and $j_n\circ\pi_n=\operatorname{id}$.

Since $K_0$ is a covariant functor that preserves split exactness (corollary 8.2.2, [[1](#wegge-olsen)]), the following diagram is also communicative and the rows are split exact too.
\begin{xy}
\xymatrix{
0\ar[r]\ar[d] & K_0(A)\ar[r]^{i_*}\ar[d]^{\alpha_*} & K_0(A^+)\ar[r]^{\pi_*}\ar[d]^{\alpha^+_*} &K_0(\C)\ar[r]\ar[d]^{\alpha_{\C *}} &0\ar[d] \\
0\ar[r] & K_0(M_n(A))\ar[r]^{i_{n*}} &K_0(M_n(A^+))\ar[r]^{\pi_{ n *}}&K_0(M_n(\C))\ar[r] &0
}
\end{xy}

From the unital case, we know that $\alpha^+_*:K_0(A^+)\to K_0(M_n(A^+))$ and $\alpha_{\C *}:K_0(\C)\to K_0(M_n(\C))$ are isomorphisms. Proof ends by applying the five-lemma to
\begin{xy}
\xymatrix{
0\ar[r]\ar[d] & 0\ar[r]\ar[d] & K_0(A)\ar[r]^{i_*}\ar[d]^{\alpha_*} & K_0(A^+)\ar[r]^{\pi_*}\ar[d]^{\alpha^+_*} &K_0(\C)\ar[d]^{\alpha_{\C *}} \\
0\ar[r] & 0\ar[r] & K_0(M_n(A))\ar[r]^{i_{n*}} &K_0(M_n(A^+))\ar[r]^{\pi_{ n *}}&K_0(M_n(\C))
}
\end{xy}

\[1\] Wegge-Olsen, N. E. (1993). K-theory and C*-algebras.

a Positive Bilinear Map is Necessarily Bounded

Suppose $A$ and $B$ are Banach spaces, $C$ is a normed space, and $\sigma:A\times B\to C$ is a bilinear form. For each $a\in A$, denote the map $b\mapsto \sigma(a,b)$ by $\hat{a}:B\to C$; and for each $b\in B$ denote the map $a\mapsto \sigma(a,b)$ by $\hat{b}:A\to C$.

**Lemma**. The bilinear form $\sigma$ is bounded is equivalent to $\hat{a}$ and $\hat{b}$ are both bounded for all $a\in A$ and $b\in B$.

*Proof*. Since $\hat{b}$ is a bounded map on $A$ for all elment $b$ in the closed unit ball of $B$, $\sup_{a\in A_1}\|\hat{b}(a)\|\leq \|\hat{b}\|$, i.e., $\sup_{a\in A_1}\|\hat{a}(b)\|\leq \|\hat{b}\|$. By PUB, the family $\{\hat{a}|a\in A_1\}$ of bounded maps is bounded uniformly, that is, $\sup_{a\in A_1}\|\hat a\|\leq M$ for some $M>0$ . Hence, $\sup_{a\in A_1, b\in B_1}\|\sigma(a, b)\|=\sup_{a\in A_1, b\in B_1}\|\hat{a}(b)\|\leq M$.

**Application**. Let $A$, $B$ and $C$ be C*-algebras.$\newcommand{\tensor}{\otimes}$
By a *positive* bilinear form $\sigma$ from $A\times B$ to $C$, we mean $\sigma(a,b)\geq 0$ for all positive elements $a$ in $A$ and $b$ in $B$, or equivalently, $\hat{a}$ and $\hat{b}$ are positive linear maps on $B$ and $A$ respectively.
Since every positive linear map between two C*-algebra is bounded, by the above lemma,
>a positive bilinear form in C*-algebras must be bounded.

If $A$ and $B$ are C*-algebras, $\gamma$ is a C*-norm on $A\tensor B$, then the map $A\times B\to A\tensor_\gamma B$ is a positive bilinear form and thus is bounded, that is, $\gamma(a,b)\leq M\|a\|\|b\|$ for some positive number $M$.

Exercises on von Neumann Algebra

$\textbf{4.4}$ Let $A$ be a von Neumann algebra on a Hilbert space $H$, and suppose that $\tau$ is a bounded linear functional on $A$. We say that $\tau$ is normal if, whenever an increasing net $(u_\lambda)_{\lambda\in \Lambda}$ in $A_{sa}$ converges strongly to an operator $u\in A_{sa}$, we have $\lim_{\lambda}\tau(u_\lambda)=\tau(u)$. Show that every $\sigma$-weakly continuous functional $\tau\in A^*$ is normal.
$\newcommand{\vN}{\text{von Neumann algebra}}\renewcommand{\l}[1]{\lVert #1\rVert}$

$\textbf{Lemma}.$ Suppose $u,v$ are two bounded linear operators on some Hilbert space, then

\begin{equation*}
\lvert uv\rvert\leq \lVert u\rVert\lvert v\rvert.
\end{equation*}

Indded,Let $v=w\lvert u\rvert$ and $uv=w’\lvert uv\rvert$ be the Polar Decomposition, then
$\lvert uv\rvert^2=w’^*uw\lvert v\rvert$, thus \begin{align*}
\lvert uv\rvert^2 & =\lvert v\rvert w^* u^* w’ w’^*uw\lvert v\rvert\\
&\leq \lVert w’^*uw\rVert^2\lvert v\rvert^2\leq\l{u}^2\lvert v\rvert^2,
\end{align*}
Hence $ \lvert uv\rvert\leq \lVert u\rVert\lvert v\rvert.$

$\textbf{Theorem}.$ Let $A$ be a $\vN$ on Hilbert space containing $\operatorname{id}_H$, then for any $\sigma$-continuous linear $\tau$ on $A$, there exists $u\in L^1(H)$ such that $$\tau(v)=\operatorname{tr}(uv)$$ for all $v\in A$.

Proof.
Suppose $\{e_\alpha\}_{\alpha\in\Gamma}$ is a basis for $H$, $u\in L^1(H)$ such taht $\tau(v)=\operatorname{tr}(uv)$ for all $v\in A$. and a bounded net $(v_\lambda)_{\lambda\in \Lambda}$ in $A_{sa}$ converges strongly to an operator $v\in A_{sa}$. For every $\varepsilon>0$, there exists a finite subset $\Gamma_0$ of $\Gamma$ such that $$\sum_{\alpha\notin \Gamma_0}\langle \lvert u\rvert e_\alpha,e_\alpha\rangle\leq \varepsilon,$$
and there exists $\lambda_0\in \Lambda$ such that $\forall\lambda\geq\lambda_0$
$$\l{(v_\lambda-v) u e_\alpha}\leq \varepsilon, \alpha\in\Gamma_0.$$
Therefore, \begin{align*}\l{(v_\lambda-v) u}_1 &=\sum_{\alpha\in \Gamma}\langle \lvert (v_\lambda-v) u\rvert e_\alpha,e_\alpha\rangle\\
& \leq \sum_{\alpha\notin \Gamma_0}\langle \lvert u\rvert e_\alpha,e_\alpha\rangle\l{v_\lambda-v} + \sum_{\alpha\in \Gamma_0}\l{(v_\lambda-v) u e_\alpha}\\
&\leq M\varepsilon
\end{align*}
for some constant $M$. Hence $\lvert \tau(v_\lambda)-\tau(v)\rvert\leq\l{(v_\lambda-v) u}_1\to 0$.

$\newcommand{\vphi}{\varphi} \newcommand{\rank}{\operatorname{rank}} \def\ni{^{-1}}$
$\textbf{4.7}$ Let $A$ be a $C^*$-algebra.

(1)Show that if $p, q$ are equivalent projections in $A$, and $r$ is a projection orthogonal to both (that is, $rp = rq = 0$), then the projections $r + p$ and $r + q$ are equivalent.

(2)If $H$ is a separable Hilbert space and $p$ is a projection not of finite rank, set $\operatorname{rank}(p) =\infty$. If $p$ has finite rank, set $\operatorname{rank}(p) = \operatorname{dim}p(H)$. Show that $r \sim q$ in $B(H)$ if and only if $ \operatorname{rank}(p) = \operatorname{rank}(q).$
Thus, the equivalence class of a projection in a C*-algebra can be
thought of as its “generalised rank.”

(3)We say a projection $p$ in a $C^*$-algebra $A$ is finite if for any projection $q$ such that $q\sim p$ and $q \leq p$ we necessarily have $q = p.$ Otherwise, the projection is said to be infinite. Show that if $p, q$ are projections such that $q\leq p$ and $p$ is finite, then $q$ is finite.

(4)A projection $ p$ in a von Neumann algebra $A$ is abelian if the algebra $pAp$ is abelian. Show that abelian projections are finite.

(5)A von Neumann algebra is said to be finite or infinite according as its unit is a finite or infinite projection. If $H$ is a Hilbert space, show that the von Neumann algebra $B(H)$ is finite or infinite according as $H$ is finite- or infinite- dimensional.

Proof.
(1) Suppose $p=u^*u $ and $q=uu^*$, then $ru^*u r=rpr=0$ and $ ruu^*r=rqr=0$, so $ur=ru^*=ru=u^*r=0$.Hence
\begin{align*}
& r+p=r^2+u^*u+u^*r+ru=(r+u^*)(r+u);\\
& r+q=r^2+uu^*+ur+ru^*=(r+u)(r+u^*),
\end{align*}
that is, $r+p\sim r+q.$

(2)If $\operatorname{ran}(p)$ and $\operatorname{ran}(q)$ have a same dimension less than $\aleph_0$, choose orthonormal bases $\{e_n\}_{n=1}^N$ and
$\{f_n\}_{n=1}^\infty$ for $ p(H)$ and $q(H)$, respectively($N$ may be infty). Let $$ u:\operatorname{ran}(p)\to \operatorname{ran}(q), u(e_n)=f_n, n=1,2,\cdots,N.$$
and let $$v(x)=\begin{cases}
u(x), x\in \operatorname{ran}(p)=(\ker(p))^\perp;\\
0, x\in\ker(p),
\end{cases} $$
then $v^* v=p$(because they coincide on $\ker(p)$ and $(\ker(p))^\perp$) and $v v^*=q$.
Conversely, if $p\sim q$, then $p=u^*u $ and $q=uu^*$ for some $u\in B(H)$, so \begin{align*}
\rank(p)=\rank(u^*qu )\leq\rank(q);\\
\rank(q)=\rank(upu^*)\leq\rank(p).
\end{align*}

(3)Since $(p-q)r=0, (p-q)q=0$, $r+(p-q)\sim q+(p-q)=p$ by (1). Because $r+(p-q)\leq p$, $r+(p-q)=p$, i.e. $r=q.$
(4)Since $pAp$ is an abelian $C^*$-algebra with the identity $p$, there is a $*$-isomorphism $\vphi: pAp\to C(\Omega)$, where $\Omega$ is the compact set of all non-zero algebra homomorphism from $pAp$ to $\mathbb{C}$.
Suppose $q\leq p$, $p=u^*u$ and $q=uu^*$, then
\begin{align*}
u & =qu=pqu\\
&=pqup\in pAp,
\end{align*}
thus \begin{align*}
p=\vphi\ni(\vphi(p))=\vphi\ni(\vphi(u)^*\vphi(u))=\vphi\ni(\lvert \vphi(u)\rvert^2);\\
q=\vphi\ni(\vphi(q))=\vphi\ni(\vphi(u)\vphi(u)^*)=\vphi\ni(\lvert \vphi(u)\rvert^2),\\
\end{align*}
so $p=q$.

(5)If $H$ is finite-demensional, then $\rank(1)<\infty$, so $1$ is finite, i.e., $B(H)$ is finite. If $H$ is infinite-demensional, suppose $\{e_n\}_{n=0}^\infty \sqcup \{e_\lambda\}_{\lambda\in\Lambda}$ is an orthonormal basis for $H$, and set $$u(e_n)=e_{n+1}(n=0,1,\cdots), u(e_\lambda)=e_\lambda(\lambda\in\Lambda).$$ Then $$u^*(e_n)=e_{n-1}(n=1,2,\cdots),$$ $$ u^*(e_0)=0, u^*(e_\lambda)=e_\lambda.$$ Thus $$u^*u(e_n)=e_n(n=0,1,\cdots),$$ $$ uu^*(e_n)=e_n(n=1,2,\cdots), uu^*(e_0)=0.$$ Let $p=uu^*$, then $p\sim u^*u=1,$ and $p\leq 1$, but $p\neq 1$, that is $1$ is infinite, so is the Hilbert space. 1. https://math.stackexchange.com/questions/2355169/non-trivial-commutant-implies-proper-projection/2358469#2358469

2.https://math.stackexchange.com/questions/2362496/what-generates-ell-infty