A net version of dominated convergence theorem

Lemma 1. Suppose $E$ is a Riesz space, $F$ is an order ideal of $E$ and $(f_i)_{i\in I}$ is a net in $F$. Then $f_i$ converges (in order) to an element $f$ in $F$ iff and only if $f_i$ converges to $f$ in $E$ and $|f_i|\leq \bar f$ for some $\bar f\in F$ when $i$ is sufficiently large.

Proof. Suppose $f$ is the order limit of $(f_i)$ in $F$, then we can find a decreasing net $(g_j)_{j\in J}$ in $F$ with $\inf_{j\in J} g_j=0$ such that for any $j\in J$ there exists some $i_0\in I$ satisfying $|f_i-f|\leq g_j$ for all $i\geq i_0.$ Pick $g\in E$ that $g\leq g_j$ for all $j\in J.$ Since $F$ is an ideal, $g$ is an element of $F$ and thus less than $0$. Therefore, the infimum of $(g_j)_{j\in J}$ in $E$ is also $0$. As a result, $f_i$ converges to $f$ in $E$ too. The proof for the only if part ends by taking $\bar f=|f|+g_{i_0}$.

Conversely, we suppose $f_i$ converges to $f$ in $E$ and $|f_i|\leq \bar f$ for some $\bar f\in F$ when $i$ is sufficiently large. Pick a decreasing net $(g_j)_{j\in J}$ with in $E$ with $\inf_{j\in J} g_j=0$ such that for any $j\in J$ there exists some $i_0\in I$ satisfying $|f_i-f|\leq g_j$ for all $i\geq i_0.$ We may assume $g_j\leq |f|+\bar f,$ otherwise, replace $g_j$ with $g_j\wedge(|f|+\bar f)$. It follows from $F$ is an ideal of $E$ that $g_j\in F$. Since $F$ is a subset of $E$, any $g\in F$ satisfying $g\leq g_j$ for all $j\in J$ satisfies $g\leq 0$. Consequently, the infimum of $(g_j)_{j\in J}$ in $F$ is also $0$ and hence $f_i$ converges to $f$ in $F$. ◻

The following theorem is an abstract statement of dominated convergence theorem in Banach lattice.

Theorem 2. Suppose $E$ is a Riesz space, $F$ is an order ideal of $E$ equipped with an order continuous norm. Let $(f_i)$ be a net order convergent to some $f$ in $E$ and $|f_i|\leq \bar f$ for some $\bar f\in F$. Then $f_i$ is norm convergent to $f$ in $F$.

Proof. By the above lemma, $f_i$ is order convergent to $f$ in $F$. The order continuity of the norm on $F$ yields that $f_i$ is norm convergent to $f$. ◻

Now, we can give a net version of the canonical Dominated Convergence Theorem.

Corollary 3. Let $(\Omega,\mathcal{A},\mu)$ be a measure space, $(f_i)$ a net of ${\mathcal {A}}$-measurable functions. Assume the net $(f_i)$ converges $\mu$-almost everywhere to an $\mathcal {A}$-measurable function $f$, and is dominated by a $g \in L^1(\Omega,\mathcal{A},\mu)$ i.e., for each $i$ we have: $|f_i| \leq g$, $\mu$-almost everywhere. Then f is in $L^1(\Omega,\mathcal{A},\mu)$ and $$\lim_{i} \int_\Omega |f_i-f|\,d\mu = 0$$ which also implies $$\lim_i\int_\Omega f_i\,d\mu = \int_\Omega f\,d\mu.$$

Proof. Let $E$ be the linear space of all $\mathcal{A}$-measurable functions. It is an Riesz space, with respect to the order defined by $f\leq g$ if $f(x)\leq g(x)$ $\mu$-a.e., in which a net $(f_i)$ is order convergent if and only if $(f_i(x))$ is convergent $\mu$-a.e.. By Theorem 2.4.2, (Meyer-Nieberg, 2012), the canonical norm on $L^1(\Omega,\mathcal{A},\mu)$ is order continuous. Applying Theorem 2 with $F=L^1(\Omega,\mathcal{A},\mu)$, we finishes the proof. ◻

Meyer-Nieberg, P. (2012). Banach lattices. Springer Science &Business Media.

Half exactness of $K_0$ functor

Theorem. Let $\require{amscd}$ $\begin{CD} 0 @> >> A @>i>> [email protected]>\pi>>[email protected]>>>0 \end{CD}$ be an exact sequence of $\mathbb{C}$-algebras, then $\begin{CD}K_0(A)@>{i_*}>>K_0(B)@>{\pi_*}>>K_0(C)\end{CD}$ is exact in the middle.

Key point. For any invertible element $u$ in $M_m(B)$, $$\begin{pmatrix}u & \\ & -u^{-1}\end{pmatrix}=\begin{pmatrix}& 1 \\ 1& \end{pmatrix}\begin{pmatrix}1 & \\u & 1\end{pmatrix}\begin{pmatrix}1 & -u^{-1}\\ & 1\end{pmatrix}\begin{pmatrix}1 & \\u & 1\end{pmatrix}.$$ Since each factor on the right hand side lifts to a invertible element in $M_{2m}(B^+)$, $\begin{pmatrix}u & \\ & -u^{-1}\end{pmatrix}$ lifts to an invertible element in $M_{2m}(B^+)$.

Proof. Let $[e']-[e'_0]$ be an element in the kernel of $\pi_*$, where $e'\in M_{n'}(B^+)$ and $e'_0\in M_{n'}(\mathbb{C})$ satisfies $e'- e'_0\in M_{n'}(B)$. Then $[\pi_* e']=[\pi_* e'_0]$ and thus there is a unitary $w$ in $M_{n}(C)$ such that $w (\pi_* e) w^{-1}=\pi_* e_0=e_0$, where $n=2n', e= \begin{pmatrix}e & \\ & 0\end{pmatrix}, e_0=\begin{pmatrix}e_0 & \\ & 0\end{pmatrix}.$

Pick a lift $v\in M_{2n}(B^+)$ of $\begin{pmatrix}w& \\ & -w^{-1}\end{pmatrix}$, we have $$\pi_*(v\begin{pmatrix}e & \\ & 0\end{pmatrix}v^{-1}) =\begin{pmatrix}w& \\ & -w^{-1}\end{pmatrix} \begin{pmatrix}\pi_*e& \\ & 0\end{pmatrix} \begin{pmatrix}w^{-1}& \\ & -w\end{pmatrix} =\begin{pmatrix}e_0& \\ & 0\end{pmatrix}.$$ Since $\begin{CD} A @>i>> [email protected]>\pi>>C \end{CD}$ is exact, there is some $d\in M_{2n}(A^+)$ such that $i_* d=v\begin{pmatrix}e & \\ & 0\end{pmatrix}v^{-1}.$ Moreover, $d=d^2$ due to that $i$ is an injection. A direct computation $$i_*([d]-[e_0])=[v\begin{pmatrix}e & \\ & 0\end{pmatrix}v^{-1}]-[e_0]=[e]-[e_0]$$ yields that $\begin{CD}K_0(A)@>{i_*}>>K_0(B)@>{\pi_*}>>K_0(C)\end{CD}$ is exact in the middle.

elements of $K_0$ group

Let $e$ be an idempotent in a unital algebra $A$ and $u=\begin{pmatrix}1-e & e\\ e & 1-e\end{pmatrix}$, then $u$ is invertible and $u\begin{pmatrix}1-e & 0\\ 0 & e\end{pmatrix}=\begin{pmatrix}1 & 0\\ 0 & 0\end{pmatrix}u.$ So, $\begin{pmatrix}e & 0\\ 0 & 1-e\end{pmatrix}$ is Murrary-von Neumann equivalent to $\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}$ in $M_2(A)$.

Suppose $g=[e]-[f]$ is an element of a $K_0$ group, then \begin{align*}g=&[e]+[1-f]-[1-f]-[f]=[\begin{pmatrix}e & 0\\ 0 & 1-f\end{pmatrix}]-[\begin{pmatrix}1-f & 0\\ 0 & f\end{pmatrix}]\\=&[\begin{pmatrix}e & 0\\ 0 & 1-f\end{pmatrix}]-[\begin{pmatrix}1 & 0\\ 0 & 0\end{pmatrix}].\end{align*}

On Homotopy

Lemma. Suppose $A$ is a unital Banach algebra and $\gamma$ is a continuous path in the idempotents of $A$. Then there is a continuous path $u$ in invertible elements of $A$ such that $u(t)\gamma(t) u^{-1}(t)=\gamma(0)$.

Proof. For any $s\in [0,1]$, let $$u_s(t)=\gamma(s)\gamma(t)+(1-\gamma(s))(1-\gamma(t)),$$ then $u_s(t)\gamma(t)=\gamma(s)u_s(t)$ and $\|u_s(t)-1\|=\|(2\gamma(s)-1)(\gamma(t)-\gamma(s))\|\leq \|2\gamma(s)-1\|\|\gamma(s)-\gamma(t)\|.$ Let $r_s>0$ be such that $\|\gamma(s)-\gamma(t)\|<\frac{1}{\|2\gamma(s)-1\|}$ for all $t\in (s-r_s, s+r_s)$, then $u_s$ is invertible on the open set $(s-r_s, s+r_s)$. Since $[0,1]$ is compact, we can find finite points $s_1< s_2<\cdots<s_n$ such that $\{(s_i-r_{s_i}, s_i+r_{s_i}): i=1,2,\cdots,n\}$ is a cover of $[0,1]$, $(s_i-r_{s_i}, s_i+r_{s_i}) \cap(s_{i+1}-r_{s_{i+1}}, s_{i+1}+r_{s_{i+1}})\neq\emptyset$, $0\in (s_1-r_1, s_1+r_1)$ and $1\in (s_n-r_{n}, s_n+r_{s_n})$.

Pick $s^i\in (s_i-r_{s_i}, s_i+r_{s_i})\cap (s_{i+1}-r_{s_{i+1}}, s_{i+1}+r_{s_{i+1}})$ such that $s_i<s^i<s_{i+1}$, $i=1,\cdots,n-1$ and s^0=0, s^n=1. For each $i\in \{1,2,\cdots, n\}$, define $$v_i(t)=\begin{cases}u_{s_i}(s^{i-1})^{-1}u_{s_i}(t), t\in [s^{i-1}, s^i]\\ 1, t\in [0,s^{i-1}]\\ u_{s_i}(s^{i-1})^{-1}u_{s_i}(s^i), t\in [s^i,1]\end{cases}$$ and let $u=v_1v_2\cdots v_n$, then $u$ is a continuous path in invertible elements of $A$ and $u(t) \gamma(t) u^{-1}(t)=\gamma(0)$.

Examples of Idempotents

If $e$ is an idempotent in an algebra, so is $ueu^{-1}$ whenever $u$ is invertible. So, if $e$ and $f$ are homotopic in $M_\infty(A)$ if they are homtopic in some $M_n(A)$ where $A$ is a C*-algebra. Note that $\begin{pmatrix} I & \\ & 0\end{pmatrix}e_t \begin{pmatrix} I & \\ & 0\end{pmatrix}$ is a homotopy if $e_t:[0,1]\to M_\infty(A)$ is.

Examples of idempotents in $M_\infty(\mathbb{C})$. (a)\begin{pmatrix} & & & d\\ & & & c\\ & \Huge 0 & & b \\ & & & a \\ & & & 1\end{pmatrix} (b)$\frac{1}{1-t^2}\begin{pmatrix} -t^2 & t\\ -t &1\end{pmatrix}(t\in(1,\infty))$