Takai duality

Takai duality , named after Hiroshi Takai , is a concept from the mathematical branch of functional analysis . Is a C * -Dynamic system with an abelian , locally compact group , so that operates Dual group in such a way that the C * algebra until tensoring with the compact operators from can recover. ${\displaystyle (A,G,\alpha )}$ ${\displaystyle A\ltimes _{\alpha }\!G}$ ${\displaystyle A}$ ${\displaystyle A\ltimes _{\alpha }\!G}$

The dual operation

Let it be a C * -dynamic system with an Abelian, locally compact group . Then there is the dual group of continuous group homomorphisms , which with the topology of compact convergence is again an Abelian, locally compact group. Next, let the convolutional algebra of continuous functions with a compact carrier lie in close proximity . For be ${\displaystyle (A,G,\alpha )}$ ${\displaystyle G}$ ${\displaystyle {\hat {G}}}$ ${\displaystyle G\rightarrow \{z\in \mathbb {C} ;\,|z|=1\}}$ ${\displaystyle K(A,G,\alpha )}$ ${\displaystyle A\ltimes _{\alpha }\!G}$ ${\displaystyle G\rightarrow A}$ ${\displaystyle \chi \in {\hat {G}}}$

${\displaystyle {\hat {\alpha }}_{\chi }:K(A,G,\alpha )\rightarrow K(A,G,\alpha ),\,({\hat {\alpha }}_{\chi }(x))(t):=\chi (t)x(t)}$, where . ${\displaystyle x\in K(A,G,\alpha ),t\in G}$

Then be a like named automorphism to expand and is a group homomorphism from the dual group in the automorphism group of which a C * pharmacodynamic system makes, * called to the dual C toxicodynamic system. ${\displaystyle {\hat {\alpha }}_{\chi }}$ ${\displaystyle A\ltimes _{\alpha }\!G}$ ${\displaystyle \chi \mapsto {\hat {\alpha }}_{\chi }}$ ${\displaystyle {\hat {G}}}$ ${\displaystyle A\ltimes _{\alpha }\!G}$ ${\displaystyle (A\ltimes _{\alpha }\!G,{\hat {G}},{\hat {\alpha }})}$

Takai's duality theorem

Let it be a C * -dynamic system with an Abelian, locally compact group and be the dual C * -dynamic system. If the C * -algebra of the compact operators is over the Hilbert space of the functions that can be square-integrated with respect to the hair measure , then . ${\displaystyle (A,G,\alpha )}$ ${\displaystyle G}$ ${\displaystyle (A\ltimes _{\alpha }\!G,{\hat {G}},{\hat {\alpha }})}$ ${\displaystyle K(L^{2}(G))}$ ${\displaystyle L^{2}(G)}$ ${\displaystyle (A\ltimes _{\alpha }\!G)\ltimes _{\hat {\alpha }}{\hat {G}}\cong A\otimes K(L^{2}(G))}$

Remarks

This is an analogy to the duality for W * -dynamic systems, which goes back to Takesaki . Tensoring with the full operator algebra for Von Neumann algebras has been replaced in the Takai duality presented here by tensing with the C * algebra of compact operators.

If separable , for example if countable is infinite and discrete, then the compact operators are isomorphic to the C * -algebra over the sequence space . One calls two C * -algebras and stable-isomorphic if . The theorem about the Takei duality thus says that the cross product of the dual C * -dynamic system is stable-isomorphic to . ${\displaystyle L^{2}(G)}$ ${\displaystyle G}$ ${\displaystyle K(L^{2}(G))}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\otimes K(\ell ^{2})\cong B\otimes K(\ell ^{2})}$ ${\displaystyle (A,G,\alpha )}$ ${\displaystyle A}$

If order is a finite group , then is and therefore . In particular, except for isomorphism , this results in a handy implementation of the cross product as a sub-algebra of a matrix algebra. ${\displaystyle G}$ ${\displaystyle n}$ ${\displaystyle K(L^{2}(G))\cong K(\mathbb {C} ^{n})\cong M_{n}(\mathbb {C} )}$ ${\displaystyle (A\ltimes _{\alpha }\!G)\ltimes _{\hat {\alpha }}{\hat {G}}\cong M_{n}(A)}$ ${\displaystyle A\ltimes _{\alpha }\!G\subset M_{n}(A)}$

If the two-element group is a concrete example , then and is an automorphism with . One obtains with the above isomorphism ${\displaystyle G=\mathbb {Z} _{2}=\{0,1\}}$ ${\displaystyle \alpha _{0}=\mathrm {id} _{A}}$ ${\displaystyle \alpha _{1}\in \mathrm {Aut} (A)}$ ${\displaystyle \alpha _{1}\circ \alpha _{1}=\mathrm {id} _{A}}$

${\displaystyle A\ltimes _{\alpha }\!\mathbb {Z} _{2}\cong \left\{{\begin{pmatrix}a&b\\\alpha _{1}(b)&\alpha _{1}(a)\end{pmatrix}}|a,b\in A\right\}\subset M_{2}(A)}$.

In order to get from this, one has to consider the dual operation from on according to the above sentence . is of course the identity on the cross product and ${\displaystyle M_{2}(A)}$ ${\displaystyle {\hat {\alpha }}}$ ${\displaystyle {\hat {\mathbb {Z} _{2}}}=\mathbb {Z} _{2}=\{0,1\}}$ ${\displaystyle A\ltimes _{\alpha }\!\mathbb {Z} _{2}}$ ${\displaystyle {\hat {\alpha }}_{0}}$

${\displaystyle {\hat {\alpha }}_{1}({\begin{pmatrix}a&b\\\alpha _{1}(b)&\alpha _{1}(a)\end{pmatrix}})={\begin{pmatrix}a&-b\\-\alpha _{1}(b)&\alpha _{1}(a)\end{pmatrix}}}$.

If one applies the same embedding in the matrix algebra , one obtains a total of a subalgebra of which one can show that it is too isomorphic. ${\displaystyle M_{2}(A\ltimes _{\alpha }\!\mathbb {Z} _{2})}$ ${\displaystyle M_{4}(A)}$ ${\displaystyle M_{2}(A)}$

Individual evidence

1. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , theorem 10.1.2
2. ^ H. Takai: On a duality for crossed products of C * -algebras , Journal of Functional Analysis, Volume 19 (1975), pages 25-39
3. ↑ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0125494505 , sentence 7.9.3