Category Archives: Operator Algebras

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


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


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 xth 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


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


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


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


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


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


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.