C. Ding wrote a new post, A net version of dominated convergence theorem (Wrong), on the site C. Ding's blog 1 year, 5 months ago

::: {.lemma}

**Lemma 1**. *Suppose $E$ is a Riesz space, $F$ is an order ideal of $E$

and $(f_i)_{iin I}$ is a net in $F$. Then $f_i$ converges (in order) to

an element $f$ in $F$ iff and only if $f_i$ […]C. Ding wrote a new post, Half exactness of $K_0$ functor, on the site C. Ding's blog 1 year, 9 months ago

**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 […]C. Ding wrote a new post, elements of $K_0$ group, on the site C. Ding's blog 1 year, 10 months ago

Let $e$ be an idempotent in a unital algebra $A$ and $u=begin{pmatrix}1-e & e e & 1-eend{pmatrix}$, then $u$ is invertible and $ubegin{pmatrix}1-e & 0 0 & eend{pmatrix}=begin{pmatrix}1 & 0 0 & […]

C. Ding wrote a new post, On Homotopy, on the site C. Ding's blog 1 year, 10 months ago

**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) […]

C. Ding wrote a new post, Examples of Idempotents, on the site C. Ding's blog 1 year, 11 months ago

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 […]C. Ding's profile was updated 1 year, 11 months ago