Tag Archives: Haar measure

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.