Tag Archives: C*-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

[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.

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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.

States and Representations of C*-algebras

Definition 1.  Let A be a C*-algebra and x\in Ax is positive if it is self-adjoint and \sigma_A(x)\subseteq [0,\infty).  We denote the set of positive elements of A by A_+ and write x\geq 0 for all x\in A_+.  We also write x\leq y if y-x\geq 0.

It turns out that if B is a C*-subalgebra of A, then \sigma_B(x)=\sigma_A(x).  Hence B_+=B\cap A_+.

Definition 2.  A functional f:A\to\mathbb{C} is positive if x\geq 0\Rightarrow f(x)\geq 0 (hence f(x)\in\mathbb{R}).  A state is a positive linear functional such that \|f\|=\sup_{x\in A}|f(x)|=1.  The set of all states is called the state space and is denoted S(A).

If f is a positive linear functional, it defines a pre inner product on A:

\langle x,y\rangle_f=f(y^*x).

If H is a Hilbert space, we can endow a *-algebra structure on L(H), the space of linear operators on H where multiplication is composition, and if X\in L(H), then X^* is defined as the adjoint of X.

Definition 3.  Let A be a C*-algebra and H be a Hilbert space.  A representation is a *-homomorphism \rho:A\to L(H).  A subrepresentation is a representation \rho':A\to L(H') where H'\leq H is a closed subspace of H which is invariant under action from A.  A representation is irreducible if it has no nontrivial subrepresentations.  A representation \rho is faithful if \ker\rho=0.

We now present an important connection discovered by Gelfand, Naimark, and Segal.

Let A be a C*-algebra and f be a positive linear functional on A.  Define

N_f=\{x\in A:f(x^*x)=0\}.

N_f is a closed left ideal in A, and (A/N_f,\langle\cdot\rangle_f) is an inner product space.  Let H_f=L^2(A,f) be its Hilbert completion.  Now define \rho_f:A\to L(H_f) by action of x: \rho_f(x)(a+N_f)=xa+N_f\rho_f is clearly a representation, and is called the GNS representation of A associated with f.

If A is unital, let 1_f denote the image of 1 in the completion/quotient composition A\to A/N_f\to H_f.  Then f induces a positive linear functional on L(H_f) defined by

\phi_f(X)=\langle X(1_f),1_f\rangle_f

for X\in L(H_f).

Theorem 4.  Let f,g be positive linear functionals on a C*-algebra A such that g\leq f (meaning g(x)\leq f(x) for all x\in A_+).  Then there is a unique operator X\in\rho_f(A)\subseteq L(H_f) such that 0\leq X\leq 1 and g(x)=\phi_f(X\rho_f(x))=\langle (X\rho_f(x)(1_f),1_f\rangle_f for all x\in A.

This is a generalization of the Radon-Nikodym theorem (which is special case for A=L^\infty(X,\mu) where (X,\mu) is a finite measure space and \phi(f)=\int f\,d\mu).

Definition 5.  We define an extreme point in topological vector space V as a point that does not belong to any open line segment in V.  A pure state is an extreme point in S(A).  The set of pure states of A is denoted P(A).

It follows that S(A)=hull(P(A)).

Proposition 6.  Let f be a state on a C*-algebra A.  Then \rho_f is irreducible if and only if f is pure.

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

C*-Algebras

Definition 1.  A Banach algebra is a complex algebra which is a Banach space and satisfies subnormality:

\|xy\|\leq\|x\|\|y\|.

A *-algebra is a complex algebra with a unary operation * that satisfies (x+y)^*=x^*+y^*, (xy)^*=y^*x^*, x^{**}=x, and (\lambda x)^*=\overline{\lambda}x^*.  A Banach *-algebra is a Banach algebra and a *-algebra that satisfies \|x^*\|=\|x\|.  A C*-algebra is a Banach *-algebra that satisfies \|x^*x\|=\|x\|^2.  A map between two *-algebras is a *-homomorphism if f(x^*)=f(x)^*.

Proposition 2.  Any nonunital Banach algebra A has a unitization A^\dagger.

Proof.  Let A^\dagger=A\oplus\mathbb{C}.  Let a\in A, and send a\mapsto (a,0).  Define (a,\lambda)+(b,\mu)=(a+b,\lambda+\mu), (a,\lambda)(b,\mu)=(ab+\lambda b+\mu a,\lambda\mu), and \|(a,\lambda)\|=\|a\|+|\lambda|.  Clearly A^\dagger is a Banach space.  We also have

\begin{array}{lcl}\|(a,\lambda)(b,\mu)\|&=&\|(ab+\lambda b+\mu a,\lambda\mu)\|\\&=&\|ab+\lambda b+\mu a\|+|\lambda\mu|\\&\leq&\|ab\|+\|\lambda b\|+\|\mu a\|+|\lambda||\mu|\\&\leq&\|a\|\|b\|+|\mu|\|a\|+|\lambda|\|b\|+|\lambda||\mu|\\&=&\|(a,\lambda)\|\|(b,\mu)\|.\end{array}

The unit is (0,1): (a,\lambda)(0,1)=(a,\lambda)=(0,1)(a,\lambda).

In fact, if A is a nonunital Banach *-algebra, then there is a unitization of A which is also a Banach *-algebra.  We simply use A^\dagger and define (a,\lambda)^*=(a^*,\overline{\lambda}).  And if A is a C*-algebra, then we modify the norm on A^\dagger to \|(a,\lambda)\|=\sup\{\|ab+\lambda b\|:\|b\|=1\}.

Definition 3.  Let A be a Banach algebra and x\in A.  The spectrum of x is the set

\sigma_A(x)=\{\lambda:x-\lambda\cdot 1\mbox{~is not invertible in }A^\dagger\}.

Clearly for nonunital A, 0\in\sigma_A(x) for all x\in A.

Lemma 4.  Let A be a unital Banach algebra.  If \|1-x\|<1, then x^{-1} exists and is defined by

\displaystyle x^{-1}=1+\sum_{n=1}^\infty(1-x)^n.

Proof.  Denote x by 1-(1-x).  Then

\displaystyle\begin{array}{lcl}xx^{-1}&=&\left(1-(1-x)\right)\left(1+\sum_{n=1}^\infty(1-x)^n\right)\\&=&1+(1-x)+(1-x)^2+\cdots-(1-x)-(1-x)^2-\cdots.\end{array}

Since \|1-x\|<1, the series is absolutely convergent and hence convergent.  So the two sums cancel–leaving 1.

Corollary 5.  If |\lambda|>\|x\|, then \lambda\cdot 1-x is a unit.

Proposition 6.  Let A be a unital Banach algebra and x,y\in A.

  1. \sigma_A(x) is a nonempty compact subset of \mathbb{C}.
  2. \rho(x)=\max_{\lambda\in\sigma_A(x)}|\lambda|=\lim_{n\to\infty}\|x^n\|^{1/n}=\inf_{n\in\mathbb{N}} \|x^n\|^{1/n}.

Proof.  (1) By Heine-Borel, it suffices to show \sigma_A(x) is closed and bounded for all x\in A.  Let U denote the set of units of A and a scalar \lambda be called a regular point of x if \lambda\notin\sigma_A(x).  Let \lambda_0 be a regular point.  Then

x_{\lambda_0}=x-\lambda_0\cdot 1\in U.

Since U is open, there is a neighborhood B_\varepsilon(x_{\lambda_0})\subset U.  Moreover, since the function f:\mathbb{C}\to A defined by f(\lambda)=x-\lambda\cdot 1 is continuous, f^{-1}(\lambda_0) is open in \mathbb{C}.  Thus the set of regular points is open–implying that the spectrum is closed.

Now if |\lambda|>\|x\|, then \lambda is a regular point by the previous claim.  Hence diam(\sigma_A(x))\leq 2\|x\|.

Nonemptiness and (2) are omitted.

Corollary 7 (Gelfand-Mazur). If A is a Banach division algebra, then A=\mathbb{C}.

Proposition 8.  Let A be a C*-algebra and x be self-adjoint (x=x^*).  Then \rho(x)=\|x\|.

Proof.  From the C*-axiom we have \|x^*x\|=\|x^2\|=\|x\|^2.  Iterating this yields \|x^{2^n}\|=\|x\|^{2^n}.  Hence by (2) of Proposition 6 and the C*-axiom,

\displaystyle \rho(x)=\lim_{n\to\infty}\|x^{2^n}\|^{2^{-n}}=\|x\|.

Since x^*x is self-adjoint for all x\in A, this says that the norm of a C*-algebra is uniquely determined:  \|x\|=\sqrt{\rho(x^*x)}.

Theorem 9 (Gelfand-Naimark).  Let A be a commutative unital C*-algebra and \hat{A} be a closed subset of the unit ball of the dual space A^*.  Then the map \varphi:A\to C_0(\hat{A}) defined by \varphi(x)(f)=f(x), called the Gelfand transform, is an isometric *-isomorphism.

[1] Bachman, George and Lawrence Narici.  Functional Analysis.  Dover Publications. 2000.

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