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Here's a collection of important and often used results. \vspace{.5cm} \begin{center} {\Large\it Order on positive elements and the original definition} \end{center} \vspace{.5cm} For positive elements $a,b \in A_+$, we write $a \precsim b$ (i.e., $a$ is \emph{Cuntz subequivalent} to $b$) if there exists a sequence $\{ c_n \} \subset A$ such that $\| c_n b c_n - a \| \to 0$. And we write $a \sim b$ (i.e., $a$ and $b$ are \emph{Cuntz equivalent}) if $a \precsim b$ and $b \precsim a$. Classes will be denoted $\langle a \rangle$. We let $W(A) = \M_\infty (A)_+/\sim$, where $\M_\infty(A)$ is the (algebraic) inductive limit of matrices over $A$. $W(A)$ is an ordered semigroup with $\langle a \rangle + \langle b \rangle = \langle a \oplus b \rangle$ and $\langle a \rangle \leq \langle b \rangle$ if and only if $a \precsim b$. The following functional calculus elements play a distinguished role in the theory: If $a \in A_+$ and $\e > 0$, then $(a - \e)_+ := f(a)$ where $f(t) = \max \{t - \e, 0\}$. Also, for $\e > 0$, let $f_\e \in C_0(0,\infty)$ be the continuous function that is zero on $(0,\e)$, identically one on $(2\e, \infty)$, and linear in between. \vspace{.5cm} \begin{enumerate} \item $\sim$ is an equivalence relation. \item For every $b \in A_+$, the set $\{ a \in A_+ | a \precsim b\}$ is norm closed. \item For all $a, b \in A_+$, the following are equivalent: \begin{enumerate} \item $a \precsim b$ \item There exist $r_n, s_n \in A$ such that $r_n b s_n \to a$. \item For every $\e > 0$, $(a - \e)_+ \precsim b$. \item For every $\e > 0$, there exists $\delta > 0$ and $x \in A$, such that $(a - \e)_+ = x^* (b - \delta)_+ x$. \item For every $\e > 0$ there exists $x \in A$ such that $x^* x = (a - \e)_+$ and $xx^* \in \mathrm{Her}(b)$ (the hereditary subalgebra generated by $b$). \item For every $\e > 0$ there exists $\delta > 0$ and $r \in A$ such that $f_\e (a) = rf_{\delta}(b) r^*$. \item For every $\e > 0$ there exists $r$ such that $f_{\e} (a) \leq r b r^*$. \end{enumerate} \item If $A$ has stable rank one, the following are equivalent: \begin{enumerate} \item $a \precsim b$ \item The exists $x \in A$ such that $x^*x = a$ and $xx^* \in \mathrm{Her}(b)$. \item For every $\e > 0$, there exists a unitary such that $u(a - \e)_+ u^* \in \mathrm{Her}(b)$. \item For every $\e > 0$, there exists a unitary such that $uf_\e (a) u^* \in \mathrm{Her}(b)$. \end{enumerate} \item (Robert-Santiago) $a, b \in (A \otimes \mathcal{K})_+$ and $a \precsim b$, then for every $\e > 0$ there exists $b'$ such that $b \sim b'$ and $\| a - b'\| < \e$. \item (Pedersen) If $a \leq b$ and $0 \leq \alpha < 1$ there exists $u \in A$ such that $u^* b^{\alpha} u = a$. \item (Kirchberg-R{\o}rdam) If $a = x^* (b - \delta)_+ x$, then there exists $y \in A$ such that $\| y \| \leq d^{1/2} \|a \|^{1/2}$ and $y^* b y = a$. \item (Kirchberg-R{\o}rdam) If $a \leq b$ and $\e >0$, there exists $d \in A$ such that $\| d \| \leq 1$ and $d^* b d = (a - \e)_+$. \item If $\| a - b\| < \e$, then $(a - t - \e)_+ \precsim (b - t)_+$ for all $t > 0$. \item There is a universal constant $C$ such that if $\| a - x^*x \| < \e$, then there exists $y$ such that $(a - \e)_+ = y^*y, yy^* \leq xx^*$ and $\| x - y \| \leq C\sqrt{\e}$. \item For open projections $p,q \in A^{**}$, we write $p \sim q$ if there exists a $u \in A^{**}$ such that $u^*u=p, uu^* = q$ and $u^* a u \in A \cap (q A^{**} q)$ for all $a \in A \cap (p A^{**} p)$. Let $[ p ]$ denote the corresponding equivalence class. If $A$ is simple and has stable rank one, then the map $\langle a \rangle \mapsto [p_a]$, where $p_a$ is the range projection of $a$, is an order isomorphism. \item If $p \in A$ is a projection and $p \precsim a$, then $a \sim (p \oplus b)$ for some $b$. \item If $A$ is simple and has stable rank one, suprema of bounded increasing sequences in $W(A)$ exist. \item If $a = x^*x$ and $b = xx^*$, then for all $f \in C_0(0,\infty)$ we have $f(a) = y^*y$ and $f(b) = yy^*$ for some $y \in A$. \item If $B \subset A$ is hereditary and $a,b \in B_+$, then $a \precsim b$ in $B$ if and only if $a \precsim b$ in $A$. \item If $\| a - b \| < \e$, then $(a - \e)_+ = x^* b x$ for some contraction $x \in A$. (This fundamental lemma of Kirchberg and R{\o}rdam is crucial to the Hilbert module picture of the Cuntz semigroup. See \cite[Lemma 2.2]{KR2}.) \item If $x \in \mathrm{Her}(y)$, then $x \precsim y$. \item If $a,b,x,y \in A_+, xy = 0, a \precsim x,$ and $b \precsim y$, then $a + b \precsim x + y$. \item If $a \in A_+$ and $0$ is an isolated point in the spectrum $\sigma(a)$, then $a \sim p$ for some projection $p \in A$. \item If $X$ is a compact space, and $f, g \in C(X)_+$, then $f \precsim g$ if and only $\{ x | f(x) > 0 \} \subset \{ x | g(x) > 0 \}$. (There's no nice characterization of $\precsim$ for $\M_n(C(X))_+$.) \item For all $a \in A_+$, $a \sim a^n$ for all $n \in \N$. The case $n = 2$ can be used to show $x^* x \sim xx^*$ for all $x \in A$. More generally, if $f \in C_0(0,\infty)$ is never zero, then $f(a) \sim a$ for all $a \in A_+$. \item If $a \leq b$, then $a \precsim b$. \item If $T, S \in \M_n(\C)_+$, then $T \precsim S$ if and only if $\mathrm{rank}(T) \leq \mathrm{rank}(S)$. Hence, $W(\M_n(\C)) = \N$ (with canonical ordered semigroup structure). \item If $\K = \K(\ell^2)$, then $W(\K) = \N \cup \infty$ (with canonical ordered semigroup structure). \item If $A$ is unital, simple and purely infinite, then $W(A) = \{ 0, \infty \}$. (If you'd like a huge hint, recall Cuntz's result that for all $a \in A_+$ there exists $x \in A$ such that $x^* a x = 1$. If not, you shouldn't have read this far.) \item If $p,q \in A$ are projections, then $p \precsim q$ if and only if $p$ is Murray-von Neumann subequivalent to $q$ (i.e., $v^* v = p$ and $vv^* \leq q$). \item It is not true that $p \sim q$ if and only if $p$ and $q$ are Murray-von Neumann equivalent. \item If $A$ is finite (i.e., contains no infinite projections), then $p \sim q$ if and only if $p$ and $q$ are Murray-von Neumann equivalent. Hence if $A$ is stably finite, there is a natural inclusion $V(A) \subset W(A)$, where $V(A)$ denotes the Murray-von Neumann semigroup of projections. \item If $A \cong A \otimes \K$ with strictly positive\footnote{Recall that $e$ is strictly positive if and only if $e^{1/n}$ is an approximate unit.} element $e$, then $\langle a \rangle + \langle e \rangle = \langle e \rangle$ for all $a \in A_+$. Thus $W(A)$ has an absorbing element, so taking the Groethendieck group is \emph{not} a good idea for stable algebras. \item If $a_1 \leq a_2 \leq \cdots$, then the set $\{ \langle a_n \rangle \}$ has a least upper bound in $W(A)$. What if you only know $a_1 \precsim a_2 \precsim \cdots$? \end{enumerate} \vspace{.5cm} \begin{center} {\Large\it States and Order Issues.} \end{center} \vspace{.5cm} Essentially all the results below can be generalized to quasitraces. But since Haagerup proved that on unital exact C$^*$-algebras these are the same thing as usual traces, we'll stick to usual traces. Some of the facts below are quite hard; see \cite{BH} for proofs. Let $S(W(A)) = \{ f: W(A) \to [0,\infty] | f \textrm{ is positive and order preserving}\}$ and given $u \in W(A)$, let $S(W(A))$ be the states sending $u$ to $1$. If $A$ is stable, we define a state $f$ to be \emph{lower semicontinuous} if $$f(x) = \sup_{x' << x} f(x').$$ Given $f \in S(W(A))$ we replace it with a lower semicontinuous state by defining $$\bar{f}(x) = \sup_{x' << x} f(x').$$ \begin{enumerate} \item If $\tau$ is a tracial state on $A$, then $$d_{\tau} ( a ) := \lim_n \tau(a^{1/n})$$ respects Cuntz subequivalence. Hence $d_{\tau}$ defines a \emph{state} on $W(A)$ (i.e., order preserving map $W(A) \to \R_+$).\footnote{Of course, if $A$ is unital then states should be normalized at $1$; otherwise at some order unit.} \item $d_{\tau}$ is lower semicontinuous. \item $d_{\tau}(\langle a \rangle) = \sup_{\e > 0} d_{\tau}(\langle (a-\e)_+ \rangle)$ for all $a \in A_+$. \item If $A$ is unital and exact, then every lower semicontinuous\footnote{In this context, lower semicontinuity means that if $\| a_n - a\| \to 0$, then $d(\langle a \rangle) \leq \liminf_n d(\langle a_n \rangle)$.} state $d$ on $W(A)$ is equal to $d_{\tau}$ for some $\tau$. \item If the tracial state space $\mathrm{T}(A)$ is endowed with the weak-$*$ topology, then for every $a \in A_+$ the map $\tau \mapsto d_{\tau} (a)$ is lower semicontinuous. Hence we can define an order-preserving semigroup map $W(A) \to \mathrm{LAff}(\mathrm{T}(A))$, where $\mathrm{LAff}(\mathrm{T}(A))$ denotes the lower semicontinuous affine maps on $\mathrm{T}(A)$. \item If $A$ is unital, simple and $d_{\tau}(a) < d_{\tau}(b)$ for all $\tau$, then for every $\e > 0$, there exists $\delta > 0$ such that $d_{\tau}((a-\e)_+) < d_{\tau}((b-\delta)_+)$ for all $\tau$. \item (Blackadar-R{\o}rdam) If $W_0 \subset W$ is a subsemigroup of an ordered semigroup $W$, then any state on $W_0$ extends to a state on $W$. \item Given $x, y \in W(A)$, write $x <_s y$ if there exists $k \in \N$ such that $(k+1)x \leq k y$. Then $x <_s y$ if and only if $f(x) < f(y)$ for all $f \in S(W(A), y)$ \item $W(A)$ is \emph{almost unperforated} if for all $x,y \in W(A)$, $x <_s y \Longrightarrow x \leq y$. (R{\o}rdam) $W(A\otimes \mathcal{Z})$ is almost unperforated. \item (R{\o}rdam) If $A$ is exact, unital, simple, stable rank one and $W(A)$ is almost uperforated, then $A$ has real rank zero if and only if the image of $K_0(A)$ in $\mathrm{Aff}(\mathrm{T}(A))$ is (norm) dense. \end{enumerate} \vspace{.5cm} \begin{center} {\Large\it Hilbert Modules} \end{center} \vspace{.5cm} In order to invoke Kasparov's stabilization theorem, and avoid other unpleasantries, we stick to countably generated Hilbert modules. If $E$ is a Hilbert $A$-module, then we'll denote the compact operators on $E$ by $\K(E)$ and the adjointable operators by $\B(E)$. As usual, we only consider \emph{right} $A$-modules and the inner product is linear in the second variable. Given an inclusion $E \subset F$ of Hilbert $A$-modules, we say $E$ is \emph{compactly contained} in $F$ if there exists $S = S^* \in \K(F)$ such that $S|_E = \id_E$; in this case we write $E \subset \subset F$. We say $E$ is \emph{CEI-subequivalent} to $F$, and write $E \precsim F$, if every compactly contained submodule of $E$ is isomorphic to a compactly contained submodule of $F$. Two Hilbert $A$-modules $E$ and $F$ are \emph{CEI-equivalent}, denoted $E \sim F$, if $E \precsim F$ and $F \precsim E$; that is, if a third Hilbert $A$-module $X$ is isomorphic to a compactly contained submodule of $E$ if and only if it's isomorphic to a compactly contained submodule of $F$. Finally, $Cu(A)$ denotes the set of (countably generated) Hilbert $A$-modules modulo $\sim$, with the classes denoted $[E]$. It's an ordered semigroup with $[E] + [F] = [E \oplus F]$ and $[E] \leq [F]$ if and only if $E \precsim F$. Essentially all the facts below are either folklore, or proved in \cite{CEI}. (In fact, items (2), (11), (13) and (14) below are main theorems of \cite{CEI}, and the rest are nontrivial lemmas.) \begin{enumerate} \item Addition in $Cu(A)$ is well-defined. \item $Cu(\cdot)$ is a functor (into a category that is rather technical to describe; see \cite{CEI}). \item $Cu(A) \cong Cu(A\otimes \K)$ for all $A$. \item If $E \subset F$ are Hilbert $A$-modules, there is a canonical inclusion $\K(E) \subset \K(F)$. \item If $E \subset F$, then $E \precsim F$. \item If $A$ is stable, $\ell^2(A) \cong A$ (as right $A$-modules). \item If $a \in A_+$, then the closed right ideal, denoted $H_a$, is a Hilbert $A$-module. \item For every $a \in A_+$ and $\e > 0$, $H_{(a - \e)_+} \subset \subset H_a$. \item Given $a,b \in A_+$, $H_a$ and $H_b$ are isomorphic if and only if there exists $s \in A$ such that $s^* s = a$ and $ss^*$ generates $H_b$. \item If $A$ is stable, every Hilbert $A$-module is isomorphic $aA$ for some $a \in A_+$. \item The map $W(A\otimes \K) \to Cu(A\otimes \K)$ defined by $\langle a \rangle \mapsto [\bar{a(A\otimes \K)}]$ is a bijection. Actually, it's an isomorphism of ordered semigroups. \item It turns out $W(A) \subset Cu(A)$ need not be a hereditary subset, meaning it can happen that $x \leq y \in W(A)$, yet $y \in Cu(A)\setminus W(A)$. (Examples due to Leonel Robert.) \item If $[F_1] \leq [F_2] \leq \cdots$ in $Cu(A)$, there is a least upper bound denoted $\sup_n [F_n]$. (This need not be true in $W(A)$, even for bounded increasing sequences, as Robert's examples show.) \item Write $[E] << [F]$ if for every increasing sequence $[F_1] \leq [F_2] \leq \cdots$ with $[F] \leq \sup_n [F_n]$, $[E] \leq [F_n]$ for some $n\in \N$. \begin{enumerate} \item If $E \subset \subset F$, then $[E] << [F]$. \item $E \precsim E'$ with $E' \subset \subset F$ if and only if $[E] << [F]$. \item If $x \in Cu(A)$ then there exists $x_1 << x_2 << \cdots$ such that $x = \sup_n x_n$. (Such a sequence is called \emph{rapidly increasing}.) \end{enumerate} \item Given an inductive system $A_1 \to A_2 \to \cdots$ with limit $A$ and $x \in Cu(A)$, there is an nondecreasing sequence $x_n \in Cu(A_n)$ such that $x = \sup_n \dot{x}_n$, where $\dot{x}_n$ denotes the image of $x_n$ in $Cu(A)$. (This is crucial to $Cu(\cdot)$ being a continuous functor.) \item If $A$ has stable rank one and $E$ and $F$ are given, then $[E] \leq [F]$ if and only if $E$ is isomorphic to a submodule of $F$. Also, $[E] = [F]$ if and only if $E \cong F$. \item If $x, x', y \in Cu(A)$ and $x' << x < y$, then there exists $z \in Cu(A)$ such that $x' + z \leq y \leq x + z$. \item If $A$ has stable rank one, $x,y,z,z' \in Cu(A)$ such that $z' << z$ and $x + z \leq y + z'$, then $x \leq y$. \item (L. Brown \cite{LBrown:closehered}): If $p,q$ are open projections and $\| p - q\| < 1$, then the associated Hilbert modules are isomorphic. Here, the associated Hilbert module to $p$ is $A \cap (pA^{**})$ (in fact a right ideal). \end{enumerate} \begin{thebibliography}{999} %\bibitem{BC} B. Blackadar and J. Cuntz, \emph{The structure of stable algebraically simple $C\sp{*}$-algebras} Amer. J. Math. \textbf{104} (1982), 813--822. \bibitem{BH} B. Blackadar and D. Handelman, \emph{Dimension functions and traces on $C\sp{*}$-algebras}, J. Funct. Anal. \textbf{45} (1982), 297--340. %\bibitem{BKR} B. Blackadar, A. Kumjian and M. R{\o}rdam, \emph{Approximately central matrix units and the structure of noncommutative tori}, $K$-Theory \textbf{6} (1992), 267--284. %\bibitem{LBrown} Brown, L. G.: \emph{Stable isomorphism of hereditary subalgebras of $C\sp*$-algebras}, Pacific J.\ Math.\ \textbf{71} (1977), 335--348. \bibitem{LBrown:closehered} Brown, L. G., \emph{Close hereditary $C^*$-subalgebras and the structure of quasi-multipliers}, preprint, (1985). \bibitem{CEI} Coward, K. T., Elliott, G. A.\ and Ivanescu, C.: \emph{The Cuntz semigroup as an invariant for C$^*$-algebras}, J. Reine Angew. Math., \textbf{623} (2008), 161--193.. \bibitem{Cuntz} J. Cuntz, \emph{Dimension functions on simple $C\sp*$-algebras}, Math. Ann. \textbf{233} (1978), 145--153. \bibitem{CP} J. Cuntz and G.K. Pedersen, \emph{Equivalence and traces on $C\sp{*}$-algebras}, J. Funct. Anal. \textbf{33} (1979), 135--164. %\bibitem{marius} M. Dadarlat, \emph{Nonnuclear subalgebras of AF algebras}, Amer. J. Math. \textbf{122} (2000), 581--597. %\bibitem{handelman} D. Handelman, \emph{Homomorphisms of $C\sp{*}$ algebras to finite $AW\sp{*}$\ algebras}, Michigan Math. J. \textbf{28} (1981), 229--240. %\bibitem{kirchberg} E. Kirchberg, \emph{On the existence of traces on exact stably projectionless simple $C\sp *$-algebras}, Operator algebras and their applications (Waterloo, ON, 1994/1995), 171--172, Fields Inst. Commun., 13, Amer. Math. Soc., Providence, RI, 1997. \bibitem{KR} E. Kirchberg and M. R{\o}rdam, \emph{Non-simple purely infinite $C\sp *$-algebras}, Amer. J. Math. \textbf{122} (2000), 637--666. \bibitem{KR2} E. Kirchberg and M. R{\o}rdam, \emph{Infinite non-simple $C\sp *$-algebras: absorbing the Cuntz algebras $\mathcal{O}_\infty$}, Adv. Math. \textbf{167} (2002), 195--264. %\bibitem{PT} F. Perera and A.S. Toms, \emph{Recasting the Elliott conjecture}, Math. Ann. \textbf{338} (2007), 669--702. \bibitem{R} M. R{\o}rdam, \emph{On the structure of simple $C\sp *$-algebras tensored with a UHF-algebra. II}, J. Funct. Anal. \textbf{107} (1992), 255--269. \end{thebibliography} \end{document}