# K-Theory of C*-algebras

Recall that two projections $p,q$ in a C*-algebra $A$ are equivalent, denoted $p\sim q,$ if there is a partial isometry $u$ such that $p=u^*u$ and $q=uu^*.$  Let $V_0(A)$ be the set of equivalence classes of projections and $V(A)=V_0(M_\infty(A)).$  We can define addition on $V(A)$ by

$[p]+[q]=[p'+q']$

where $p'\in [p]$, $q'\in [q]$, and $p'\perp q'$ (see [1] for why this is possible).  This gives an abelian monoid structure on $V(A)$ with identity $[0]$.  If $A$ is unital, then we define $K_0(A)=G(V(A))$ where $G(V(A))$ is the Grothendieck group of $V(A).$

To proceed with the nonunital case, first consider that $V:C^*algebras\to Mon$ is a covariant functor sending $A\mapsto V(A),$ since if $\varphi:A\to B$ is a *-homomorphism, then we can define $V(\varphi):V(A)\to V(B)$ by

$V(\varphi)([p])=V(\varphi\circ V^{-1}[p]).$

Recall the unitization of $A$ where $A^\dagger=A\oplus\mathbb{C}$ with funky structure.  We have a *-homomorphism $\mu:A^\dagger\to\mathbb{C}$ such that $\ker\mu=0.$  Since $V$ is a covariant functor, and by composition induces a covariant functor $K_0:C^*algebras\to Grp,$ we have a map

$\mu_*=K_0(\mu):K_0(A^\dagger)\to K_0(\mathbb{C})=\mathbb{Z}.$

In the nonunital case we then define $K_0(A)=\ker\mu_*.$  Now define

$U_n(A)=\{x\in U(M_n(A^\dagger)):x=1_n\mbox{~mod~}M_n(A)\}.$

If $A$ is unital, then the condition is always satisfied, so we have $U_n(A)=U(M_n(A)).$  Define a norm on $M_n(A)$ by $\|x\|=\max_{ij}\|x_{ij}\|$ and let $U_n(A)_0$ be the connected component (in the sense of the norm topology) containing $1_n.$  Then we define

$K_1(A)=\lim U_n(A)/U_n(A)_0.$

Definition 1.  Let $A$ be a C*-algebra.  We define its suspension, denoted $SA,$ as $C_0((0,1),A)=C_0(\mathbb{R},A).$

Hence these are continuous $A$-valued functions that vanish at infinity (so the the suspension is similar to the topological notion).

Theorem 2.  There is an isomorphism $\varphi:K_1(A)\to K_0(SA)$ defined by $\varphi([v])=[p]-[q_n]$ where $[p]=[p_t:(0,1)\to A]$ defined by $p_t=w_tq_nw_t^*$ and $w_t$ is a path in $U_{2n}(A)$ between $1_{2n}$ and $diag(v,v^*).$

$[q_n]$ in the image refers to the constant map $q_n:(0,1)\to\{q_n\}.$  We can then define by induction the higher K-groups: $K_{n+1}(A)=K_n(SA).$  All of these are clearly covariant functors since the definition can be reduced to $K_0$ of the $(n+1)$-th suspension of $A.$

Corollary 3.  Let

$0\longrightarrow J\stackrel{i}{\longrightarrow} A\stackrel{\pi}{\longrightarrow}A/J\to 0$

be a short exact sequence.  Then the induced sequence

$K_1(J)\stackrel{i_*}{\longrightarrow} K_1(A)\stackrel{\pi_*}{\longrightarrow}K_1(A/J)$

is exact in the middle.

Let us define a map $\partial_1:K_1(A/J)\to K_0(J)$ by

$\partial_1([u])=[wq_nw^*]-[q_n]$

for some $q_n\in J$ where $u\in U_n(A/J)$ and $w\in U_{2n}(A)$ be a lift of $diag(u,u^{-1}).$  Then $\partial_1([u])\in K_0(J)$ since $diag(u,u^{-1})$ commutes with $q_n$ since $u\notin J.$

Proposition 4$\partial_1$ makes the slightly longer sequence exact at $K_1(A/J)$ and $K_0(J).$

Corollary 5.  By induction we obtain maps $\partial=\{\partial_n\}$ and a long exact sequence

$\cdots K_n(J)\stackrel{i_*}{\to}K_n(A)\stackrel{\pi_*}{\to}K_n(A/J)\stackrel{\partial}{\to}K_{n-1}(J)\stackrel{i_*}{\to}\cdots\stackrel{\pi_*}{\to}K_0(A/J)\to 0.$

[1]  Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Vol. 122.  Springer-Verlag.  2006.

# Direct Integral Decomposition

Let $\{H_x\}_{x\in X}$ be a collection of Hilbert spaces such that $(X,\Sigma,\mu)$ is a measure space.  Now define

$\displaystyle H=\left\{s\in\bigoplus_{x\in X} H_x:\int_X |s(x)|^2\,d\mu<\infty\right\}$

where $s(x)=s_x$ (the $x$th component of $s$).  Then

$\displaystyle\langle s,t\rangle=\int_X\langle s_x,t_x\rangle_x\,d\mu$

defines a pre inner product on $H$.  Now let $H_0=span\{s-t:s=t\,a.e.\}$.  Then $H/H_0$ is an inner product space, and its completion is a Hilbert space called the direct integral of $\{H_x\}_{x\in X}$.  We denote the direct integral by

$\displaystyle H_X^\oplus=\int_X^\oplus H_x\,d\mu.$

Now suppose $\{T_x\}_{x\in X}$ is a collection of linear operators where $T_x\in L(H_x)$ such that $\{\|T_x\|\}$ is uniformly bounded.  Then there is an operator $T\in L(H_X^\oplus)$ where $T(s)_x=T_x(s_x)$ and where we can define

$\displaystyle \|T\|={\mbox{ess}\sup}_{x\in X}\{\|T_x\|\}.$

This gives a representation $\rho:L^\infty(X,\mu)\to L(H_X^\oplus)$ defined by

$\left(\rho(f)(s)\right)_x=f(x)s_x,$

which is essentially just a component-wise left action of $f$ on $s$$\rho(L^\infty(X,\mu))$ is called the algebra of diagonalizable operators of $H_X^\oplus$, which we will denote $D(H_X^\oplus)$.

Theorem 1.  Let $\rho:A\to L(H)$ be a representation of von Neumann algebra $A$ on a separable Hilbert space $H$ such that $B$ is a von Neumann subalgebra of $A'$.  Then there exists a measure space $(X,\Sigma,\mu)$, a collection of Hilbert spaces $\{H_x\}_{x\in X}$, and a  unitary map $U:H\to H_X^\oplus$ such that $U\left(\rho(B)(s)\right)=D(H_X^\oplus)(s')$ for all $s\in H$ and corresponding $s'\in H_X^\oplus$ and

$\displaystyle UTU^*=\int_X^\oplus T_x\,d\mu\in L(H_X^\oplus)$

for all $T\in\rho(B')$.

Thus if we let $B=A'$ from above, then $L(H_x)$ is a factor and we also write

$\displaystyle\rho(A')=\int_X^\oplus\rho(A')_x\,d\mu=\int_X^\oplus\rho(A_x)'\,d\mu.$

This is called the central decomposition of $A$.  This also gives a representation of $A$ on $H_X^\oplus$ defined by $a\mapsto U\rho(a)U^*$.

[1]  Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Springer-Verlag.  2006.

# von Neumann Algebras

Let $M$ be a magma and $S\subseteq M$.  The commutant of $S$ is defined as:

$S'=\{x\in M: sx=xs\}$

for all $s\in S$.  We have that $S'=S^{(2n+1)}$ and $S''=S^{(2n)}$ for $1\leq n\in\mathbb{N}$ and that $M=M'$ iff $M$ is abelian.  We can also refer to the commutant of the whole structure $M$ as the center of $M$ (i.e. $M'=Z(M)$).

Definition 1.  A von Neumann algebra is a C*-algebra $A$ such that $A=A''$.

A projection in a *-algebra is an element $p$ such that $p^2=p=p^*$.  A partial isometry is an element $u$ such that $u^*u$ is a projection. Recall an element $x$ is positive, denoted $x\geq 0$, if $x^*=x$ and $\sigma_A(x)\subseteq [0,\infty)$, and that $x\leq y$ if $y-x\geq 0$.  We will also say that two elements $x,y$ are orthogonal, denoted $x\perp y$, if $xy=0$.

Definition 2.  Two projections $p,q$ are Murray-von Neumann equivalent, denoted $p\sim q$, if there is a partial isometry $u$ such that $p=u^*u$ and $q=uu^*$.  We say $p$ is subordinate to $q$, denoted $p\preceq q$, if there is a projection $q'$ such that $p\sim q'$ and $q'\leq q$.

Proposition 3.  Let $A$ be a *-algebra and $\{(p_i,q_i)\}$ be a sequence of pairs of projections such that $p_i\perp p_j$ and $q_i\perp q_j$ for $i\neq j$ and $p_i\sim q_i$ for all $i$.  Then $\sum p_i\sim\sum q_i.$  Also, if $p_i\preceq q_i$ for all $i,$ then $\sum p_i\preceq\sum q_i.$

Proof.  If $p_i\sim q_i$, then $p_i=u_i^*u_i$ and $q_i=u_iu_i^*.$  Hence $\sum p_i=\sum u_i^*u_i$ and $\sum q_i=u_i^*u_i$.  Now define $u=\sum u_i$ and $u^*=\sum u_i^*.$  Then we have

$\displaystyle u^*u=\left(\sum u_i^*\right)\left(\sum u_i\right)=\sum u_i^*u_j=\sum u_i^*u_i$

where the last equality follows from orthogonality.  Hence $\sum p_i\sim\sum q_i.$

Now suppose $p_i\preceq q_i.$  Then there are $q_i'$ such that $p_i\sim q_i'$ and $q_i'\leq q_i.$  Then since $\sum p_i\sim\sum q_i'$ by the previous claim, it remains to show that $\sum q_i'\leq\sum q_i.$  But $\sum q_i-\sum q_i'=\sum (q_i-q_i')\geq 0$ since $q_i'\leq q_i$ for all $i.$  So we have the result.

Proposition 4 (Schroder-Bernstein).  Let $p$ and $q$ be projections in a *-algebra such that $p\preceq q$ and $q\preceq p.$  Then $p\sim q$.

It turns out that if a unital von Neumann algebra $A$ is a factor ($Z(M)=1$), then $\preceq$ is a total order on the projections.

Definition 5.  A projection $p\in A$ is

1. abelian if $pAp$ is commutative;
2. finite if $p\sim p'$ where $p'\leq p$ implies that $p=p'$;
3. infinite if it is not finite;
4. properly infinite if $p\sim p_1$ and $p\sim p_2$ where $p_1,p_2\leq p$ and $p_1\perp p_2.$

Lemma 6.  If $p$ is an infinite projection in a von Neumann algebra $A$, then there is a projection $z\in Z(A)$ such that $pz$ is nonzero and properly infinite.

This allows for a somewhat complicated decomposition (see [1] for details) of $A$

$A=Az_1\oplus Az_{2_1}\oplus Az_{2_\infty}\oplus Az_3$

where $z_1$ is a discrete central projection, $z_{2_1}$ is the largest finite continuous central projection, $z_{2_\infty}$ is the largest properly infinite semifinite continuous projection, and $z_3$ is a purely infinite projection.  The algebra $A$ is said to be of pure type $\alpha$ if $z_\beta=0$ for all $\beta\neq\alpha.$

[1]  Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Springer-Verlag.  2006.

# Covariant Systems

Recall a topological group $G$ is a group and a topological space such that the maps $(x,y)\mapsto xy$ and $x\mapsto x^{-1}$ are continuous.  Let $\Sigma$ be the $\sigma$-algebra generated by the compact subsets of $G$.  A measure $\mu$ on $\Sigma$ is left-invariant if $\mu(tU)=\mu(U)$ for all $t\in G$ and $U\in\Sigma$.  A left Haar measure on $G$ is a left-invariant Radon measure on $\Sigma$.

Theorem 1.  Let $G$ be a locally compact group.  Then there exists a unique left and unique right Haar measure on $G$ (up to multiplication by a constant).

See [2] for its construction.  Let $\mu$ be a left Haar measure on $G$.  We can define right translate Haar measures by $\mu_t(U)=\mu(Ut)$.  Clearly these are left invariant as they simply changes the input of the original left invariant Haar measure.

Proposition 2.  If $\mu$ is a left Haar measure, then there is a unique function $\Delta_G:G\to\mathbb{R},$ called the modular function of $G$, such that $\mu_t(U)=\Delta_G(t)\mu(U)$.

Let $\rho:G\to U(L(H))$ be a strongly continuous unitary representation of $G$ on a Hilbert space $H$.  That is, the group homomorphism is continuous with respect to the norm topology of $U(L(H))$, and $U(L(H))$ is the subset of $L(H)$ consisting of unitary linear operators on $H$.  Let $\mu$ be a left Haar measure on $G$ and $f\in L^1(G)$, then the operator

$\displaystyle\rho^*(f)=\int_G f(t)\rho(t)\,d\mu(t)$

in $L(H)$ is bounded and in fact defines a nondegenerate representation $\rho^*:L^1(G)\to L(H)$ as a Banach *-homomorphism with $f\mapsto\rho^*(f)$.  The product on $L^1(G)$ is convolution:

$\displaystyle (f*g)(t)=\int_G f(s)g(s^{-1}t)\,d\mu(s)$

and the involution is defined by

$f^*(t)=\Delta_G(t^{-1})\bar{f}(t^{-1}).$

Recall in the GNS construction we started with a positive functional $f$ on a C*-algebra $A$ and induced a representation $\rho_f:A\to L(H_f)$.  Here we start with one representation $\rho$ on $G$ and induce another, $\rho^*$, on $L^1(G)$.  We call $\rho^*$ the integrated form of $\rho$.  We can impose another norm (other than the default sup norm $\|f\|=\sup_{t\in G}|f(t)|$) on $L^1(G)$ defined by

$\|f\|=\sup_{\rho}\|\rho(f)\|$

where $\rho$ is a representation of $L^1(G)$.  The completion of $L^1(G)$ with respect to this norm is a C*-algebra called the group C*-algebra of $G$, which we denote by $C^*(G)$.  So every strongly continuous unitary representation of $G$ induces a nondegenerate representation of $L^1(G)$, and in fact, the converse is also true (that every nondegenerate representation of $L^1(G)$ is induced by a scu representation of $G$).  Moreover there is a bijection between the irreducible ones in each case.

Definition 3.  A covariant system is a triple $(G,A,\alpha)$ where $G$ is a locally compact group, $A$ is a C*-algebra, and $\alpha:G\to Aut(A)$ is a continuous representation (where $Aut(A)$ possesses the point-norm topology).  A covariant representation on a covariant system $(G,A,\alpha)$ is a pair $(\rho,\psi)$ of representations of $G$ and $A$ respectively on a Hilbert space $H$ such that $\rho$ is strongly continuous unitary, $\psi$ is nondegenerate, and

$\rho(t)\psi(a)\rho(t)^*=\psi(\alpha(t)(a))$

for all $t\in G$ and $a\in A$.

Definition 4.  We define the covariance algebra $L^1(G,A)$ of the covariant system $(G,A,\alpha)$ as the completion of $C_C(G,A)$ under the norm

$\displaystyle\|f\|_1=\int_G \|f(t)\|\,d\mu(t)$

where we define

$\displaystyle (f*g)(t)=\int_G f(s)\cdot\alpha(s)(g(s^{-1}t))\,d\mu(s)$

$\displaystyle f^*(t)=\Delta_G(t^{-1})\cdot\alpha(t)(f(t^{-1})^*).$

It is a Banach *-algebra.

Definition 5.  If $(\rho,\psi)$ is a covariant representation of $(G,A,\alpha)$, then there is a nondegenerate representation $\rho\times\psi:L^1(G,A)\to L(H)$ defined by

$\displaystyle(\rho\times\psi)(f)=\int_G \rho(f(t))\psi(t)\,d\mu(t)$

called the integrated form of $(\rho,\psi)$$L^1(G,A)$ together with a new norm

$\displaystyle\|f\|=\sup_{(\rho,\psi)}\|(\rho\times\psi)(f)\|$

is a C*-algebra called the crossed product of $(G,A,\alpha)$, which is denoted $C^*(G,A,\alpha)$.

[1]  Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Vol. 122.  Springer-Verlag.  2006.

[2]  Folland, Gerald.  Real Analysis: Modern Techniques and Their Applications.  2nd Edition.  John Wiley and Sons.  1999.

# Tensor Products of C*-algebras

Let $A$ and $B$ be C*-algebras.  We can define their *-algebra tensor product as the standard tensor product of algebras $A\otimes B$ with product $(a\otimes b)(a'\otimes b')=aa'\otimes bb'$ and involution $(a\otimes b)^*=a^*\otimes b^*$.  There are a variety of norms one can impose on this tensor product to make $A\otimes B$ a Banach *-algebra.  For example we may define

$\displaystyle\left\|\sum a_i\otimes b_i\right\|_\wedge=\sum \|a_i\|\|b_i\|$.

This seminorm becomes a norm on $A\otimes B$ modulo the appropriate subspace, and its completion is denoted $A\hat{\otimes}B$ and is called the projective tensor product of $A$ and $B$.  We also have

$\left\|\left(\sum a_i\otimes b_i\right)^*\right\|_\wedge=\left\|\sum a_i^*\otimes b_i^*\right\|_\wedge=\sum \|a_i^*\|\|b_i^*\|=\sum \|a_i\|\|b_i\|=\left\|\sum a_i\otimes b_i\right\|,$

so $A\hat{\otimes}B$ is a Banach *-algebra.  But it fails to satisfy the C*-axiom ($\|x^*x\|=\|x\|^2$):

$\begin{array}{lcl}\left\|\left(\sum a_i\otimes b_i\right)^*\left(\sum a_i\otimes b_i\right)\right\|&=&\left\|\left(\sum a_i^*\otimes b_i^*\right)\left(\sum a_i\otimes b_i\right)\right\|\\&=&\left\|\sum a_i^*a_j\otimes b_i^*b_j\right\|\\&=&\sum\|a_i^*a_j\|\|b_i^*b_j\|\\&\leq&\sum \|a_i\|\|a_j\|\|b_i\|\|b_j\|\\&=&\left(\sum \|a_i\|\|b_i\|\right)^2\\&=&\left\|\sum a_i\otimes b_i\right\|^2\end{array}.$

It turns out that representations on $A$ and $B$ allow us to define norms on $A\otimes B$ that make it a C*-algebra.

Definition 1.  Let $\rho_A:A\to L(H_1)$ and $\rho_B:B\to L(H_2)$ be representations on $A$ and $B$.  We define the product representation $\rho=\rho_A\otimes\rho_B$ on $H_1\otimes H_2$ as

$\rho(a\otimes b)=\rho_A(a)\otimes\rho_B(b)\in L(H_1)\otimes L(H_2)$.

Since we always have the trivial representations, the set of representations of $A$ on $H_1$ and $B$ on $H_2$ are never empty.  Let us define the minimal C*-norm on $A\otimes B$ by

$\begin{array}{lcl}\displaystyle\left\|\sum a_i\otimes b_i\right\|_{\mbox{min}}&=&\displaystyle\sup_{\rho_A,\rho_B}\left\|\rho\left(\sum a_i\otimes b_i\right)\right\|\\&=&\displaystyle\sup_{\rho_A,\rho_B}\left\|\sum \rho_A(a_i)\otimes\rho_B(b_i)\right\|\end{array}$

where the two norms on the right are operator norms.  This is clearly finite (hence a norm) and satisfies the C*-axiom.  The completion of $A\otimes B$ with this norm is a C*-algebra called the minimal (or spatial) tensor product of $A$ and $B$ with respect to $\rho_A$ and $\rho_B$, and is denoted $A\underline{\circledast} B$.

Definition 2.  Let $\rho_A:A\to L(H)$ be a representation and $N\leq H$ be the largest subspace of $H$ such that $\rho(a)(x)=0$ for all $a\in A$ and $x\in N$.  Then $N^\perp$ is called the essential subspace of $H$, and we will denote it $E(H)$.  If $E(H)=H$, then $\rho_A$ is said to be nondegenerate. $\rho_A$ is degenerate if it is not nondegenerate.

In other words, $\rho_A$ is nondegenerate if $N=0$.

Proposition 3.  If $\rho:A\otimes B\to L(H)$ is a nondegenerate representation, then there are unique nondegenerate representations $\rho_A:A\to L(H)$ and $\rho_B:B\to L(H)$ such that $\rho(a\otimes b)=\rho_A(a)\rho_B(b)=\rho_B(b)\rho_A(a)$.

But arbitrary representations of the tensor product of algebras cannot be broken into pieces.  This gives us the following.

Definition 4.  Let $H$ be a Hilbert space and $A,B$ be C*-algebras.  We define the maximal C*-norm on $A\otimes B$ as

$\displaystyle\left\|\sum a_i\otimes b_i\right\|_{\mbox{max}}=\sup_{\rho}\left\|\rho\left(\sum a_i\otimes b_i\right)\right\|$

where $\rho:A\otimes B\to L(H)$.  This is also a C*-norm, and the completion of $A\otimes B$ under this norm is a C*-algebra called the maximal tensor product of $A$ and $B$ and is denoted $A\overline{\circledast}B$.

We also have that $\|\cdot\|_{\mbox{min}}\leq\|\cdot\|_*\leq\|\cdot\|_{\mbox{max}}\leq\|\cdot\|_\wedge$ where $\|\cdot\|_*$ is any C*-norm.  It follows that $\|(a\otimes b)\|_*=\|a\|\|b\|$.

Definition 5.  A functional on $A\otimes B$ is positive if $f(x^*x)\geq 0$ for all $x\in A\otimes B$.  A state on $A\otimes B$ where $A$ and $B$ are unital is a positive linear functional $f$ on $A\otimes B$ such that $f(1\otimes 1)=1$.  We denote the set of states by $S(A\otimes B)$.

As in the previous post, there is a GNS construction that gives a representation $\rho_f:A\overline{\circledast}B\to L(H_f)$ for a positive linear functional $f$, although one must show the left action on $H_f$ is by bounded operators.

Definition 6.  A C*-algebra $A$ is nuclear if for every C*-algebra $B$, there is a unique C*-norm on $A\otimes B$.

Hence in such a case, we would have $A\underline{\circledast} B=A\overline{\circledast} B$, and thus denote the product C*-algebra by $A\circledast B$.  The class of nuclear C*-algebras includes all of the commutative ones, finite ones, and is itself closed under inductive products and quotients.  Non nuclear ones are exotic; $C^*(\mathbb{F}_2),$ the group C*-algebra of $\mathbb{F}_2$ (see next post), is an example.

[1]  Blackadar, Bruce.  Operator Algebras.  Encyclopedia of Mathematical Sciences.  Vol. 122.  Springer-Verlag. 2006.