Group C * algebra

Group C * algebras are examined in the mathematical sub-areas of harmonic analysis and functional analysis. A locally compact group is naturally assigned a C * -algebra , so that it contains the representation theory of the group.

Unitary representations of locally compact groups

definition

For a Hilbert space denote the C * -algebra of the bounded linear operators on and denote the multiplicative group of the unitary operators . ${\displaystyle H}$ ${\displaystyle B(H)}$ ${\displaystyle H}$ ${\displaystyle U(H)\subset B(H)}$

Let it be a locally compact group . A unitary representation of on a Hilbert space is a homomorphism that is continuous with respect to the weak operator topology . ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle H}$ ${\displaystyle u:G\rightarrow U(H)}$

The left-regular representation

In order to build a successful theory of unitary representations, there must be enough of such representations to be able to represent the group faithfully, that is, injectively . This is done by the left-regular display . As is well known, there is a left hair measure for a locally compact group . Therefore one can construct the Hilbert space , which one writes briefly as , omitting the Haar measure . For each is now by defining where and are. ${\displaystyle G}$ ${\displaystyle \mu }$ ${\displaystyle L^{2}(G,\mu )}$ ${\displaystyle L^{2}(G)}$ ${\displaystyle s\in G}$ ${\displaystyle U_{s}:L^{2}(G)\rightarrow L^{2}(G)}$ ${\displaystyle U_{s}f(t)\,=\,f(s^{-1}t)}$ ${\displaystyle f\in L^{2}(G)}$ ${\displaystyle t\in G}$

From the left invariance of the Haar measure it follows that they are unitary operators. It is shown that is a unitary representation; this is the so-called left-regular representation. ${\displaystyle U_{s}}$ ${\displaystyle \lambda :G\rightarrow U(L^{2}(G)),\lambda (s):=U_{s}}$

Remark: If you were to replace the one on the right in the formula with , you would still get unitary operators, but if there were no homomorphism, you would be in "wrong order" . Using in the above formula sets the order in order. ${\displaystyle U_{s}f(t)=f(s^{-1}t)}$ ${\displaystyle s^{-1}}$ ${\displaystyle s}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda (s_{1}s_{2})=\lambda (s_{2})\lambda (s_{1})}$ ${\displaystyle s^{-1}}$

The group algebra

As in algebraic representation theory , the group representations are expanded to include representations of related algebras because representations of algebras are easier to use.

The - Banach space is considered for the locally compact group with left-Haar's measure . For one defines and by the formulas ${\displaystyle G}$ ${\displaystyle \mu }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle L^{1}(G)=L^{1}(G,\mu )}$ ${\displaystyle f,g\in L^{1}(G)}$ ${\displaystyle f\star g}$ ${\displaystyle f^{*}}$

• ${\displaystyle (f\star g)(t)\,=\,\int _{G}f(s)g(s^{-1}t)\,\mathrm {d} \mu (s)}$,
• ${\displaystyle f^{*}(t)\,=\,\Delta (t)^{-1}{\overline {f(t^{-1})}}}$,

where the slash stands for the complex conjugation and is the modular function of . One shows that , the so-called convolution from and , is defined almost everywhere , and that with the convolution as the product and the involution, a Banach - * - algebra with approximation of one is. ${\displaystyle \Delta }$ ${\displaystyle G}$ ${\displaystyle f\star g}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle L^{1}(G)}$ ${\displaystyle f\mapsto f^{*}}$

For each unitary representation of the group one constructs a representation , whereby the following formula is defined: ${\displaystyle u:G\rightarrow U(H)}$ ${\displaystyle \pi \colon L^{1}(G)\rightarrow B(H)}$ ${\displaystyle \pi (f),\,f\in L^{1}(G)}$

${\displaystyle \langle \pi (f)\xi ,\eta \rangle :=\int _{G}\langle u(s)\xi |\eta \rangle f(s)\mathrm {d} \mu (s),\quad \quad \xi ,\eta \in H}$.

One can show that the representation defined in this way is a non-degenerate Hilbert space representation that is also compatible with the involution, that is, it applies to all -functions , where the * on the right is the involution in the C * - Algebra is. ${\displaystyle \pi }$ ${\displaystyle \pi (f^{*})=\pi (f)^{*}}$ ${\displaystyle L^{1}}$ ${\displaystyle f}$ ${\displaystyle B(H)}$

Conversely , if there is a non-degenerate * representation, then there is exactly one unitary representation , so that, according to the above construction, results from . Hence the representation theory of is equivalent to that of . ${\displaystyle \pi \colon L^{1}(G)\rightarrow B(H)}$ ${\displaystyle u\colon G\rightarrow U(H)}$ ${\displaystyle \pi }$ ${\displaystyle u}$ ${\displaystyle G}$ ${\displaystyle L^{1}(G)}$

The group C * algebra

definition

Let it be the universal representation of . The group C * algebra of a locally compact group is defined as the normative closure of in . So if there is any non-degenerate * -representation, there is a surjective homomorphism according to construction , where the dash stands for the norm closure in . ${\displaystyle \pi _{u}:L^{1}(G)\rightarrow B(H_{u})}$ ${\displaystyle L^{1}(G)}$ ${\displaystyle C^{*}(G)}$ ${\displaystyle G}$ ${\displaystyle \pi _{u}(L^{1}(G))}$ ${\displaystyle B(H_{u})}$ ${\displaystyle \pi :L^{1}(G)\rightarrow B(H)}$ ${\displaystyle \psi :C^{*}(G)\rightarrow {\overline {\pi (L^{1}(G))}}}$ ${\displaystyle B(H)}$

The commutative case

If, for example, is commutative and the dual group , then each defines a homomorphism via Pontryagin's duality . The multiplication operator on defined by is unitary, since it only accepts values ​​of the absolute value 1. A unitary representation is therefore obtained , which leads to a non-degenerate * representation , the norm closure of which is isomorphic to the C * algebra of continuous functions that vanish at infinity . According to the above construction one thus obtains a surjective homomorphism , of which one can show that it is even an isomorphism; so you have the formula . ${\displaystyle G}$ ${\displaystyle {\hat {G}}}$ ${\displaystyle s\in G}$ ${\displaystyle {\hat {s}}:{\hat {G}}\rightarrow \mathbb {C} ,\,\chi \mapsto \chi (s)}$ ${\displaystyle {\hat {s}}}$ ${\displaystyle M_{\hat {s}}}$ ${\displaystyle L^{2}({\hat {G}})}$ ${\displaystyle {\hat {s}}}$ ${\displaystyle s\mapsto M_{\hat {s}}}$ ${\displaystyle \pi :L^{1}(G)\rightarrow B(L^{2}({\hat {G}}))}$ ${\displaystyle C_{0}({\hat {G}})}$ ${\displaystyle {\hat {G}}\rightarrow \mathbb {C} }$ ${\displaystyle C^{*}(G)\rightarrow C_{0}({\hat {G}})}$ ${\displaystyle C^{*}(G)\cong C_{0}({\hat {G}})}$

In general the relationships are not so simple, which is also due to the fact that the Hilbert space is inaccessible to universal representation.

The reduced group C * algebra

In order to avoid the difficulties associated with the universal representation, it makes sense to consider the left-regular representation , because then one only has to do with the Hilbert space . The corresponding representation is nothing more than the convolution:, where and . The standard closure of in is called the reduced group C * -algebra and denotes this with . ${\displaystyle \lambda :G\rightarrow U(L^{2}(G))}$ ${\displaystyle L^{2}(G)}$ ${\displaystyle \pi :L^{1}(G)\rightarrow B(L^{2}(G))}$ ${\displaystyle \pi (f)g=f\star g}$ ${\displaystyle f\in L^{1}(G)}$ ${\displaystyle g\in L^{2}(G)}$ ${\displaystyle \pi (L^{1}(G))}$ ${\displaystyle B(L^{2}(G))}$ ${\displaystyle C_{r}^{*}(G)}$

According to the construction presented above, the left-regular representation continues to a surjective homomorphism . This is generally not injective, although the left-regular representation of it is. One can show that this is an isomorphism if and only if the group is indirect . ${\displaystyle C^{*}(G)\rightarrow C_{r}^{*}(G)}$ ${\displaystyle L^{1}(G)}$

The reduced group C * -algebra does not contain the full representation theory of the group, unless it is indirect, as the example of the group freely generated by two elements shows. One can prove that many have finite-dimensional representations, whereas is simple and therefore cannot have finite-dimensional representations. ${\displaystyle \mathbb {F} _{2}}$ ${\displaystyle C^{*}(\mathbb {F} _{2})}$ ${\displaystyle C_{r}^{*}(\mathbb {F} _{2})}$

Individual evidence

1. ↑ Jacques Dixmier : C * -Algebras. North-Holland Publishing Company, 1977, ISBN 0-7204-0762-1 , chapter 13.2.
2. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups. Academic Press Inc., 1979, ISBN 0-12-549450-5 , sentence 7.1.4.
3. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups. Academic Press Inc., 1979, ISBN 0-12-549450-5 , sentence 7.1.6.
4. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups. Academic Press Inc., 1979, ISBN 0-12-549450-5 , sentence 7.2.1.
5. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups. Academic Press Inc., 1979, ISBN 0-12-549450-5 , Theorem 7.3.9.
6. ^ KR Davidson: C * -Algebras by Example. American Mathematical Society, 1996, ISBN 0-8218-0599-1 , sentence VII.6.1.
7. ^ KR Davidson: C * -Algebras by Example. American Mathematical Society, 1996, ISBN 0-8218-0599-1 , Corollary VII, 7.5