Irrational rotational algebra

The irrational rotational algebras are considered in the mathematical sub-area of functional analysis. It is a class of C * -algebras that result from the C * -algebra of continuous , complex-valued functions on the unit circle together with a rotation of this unit circle by an irrational angle.

construction

V rotates the domain of functions ${\displaystyle \mathbb {T} \to \mathbb {C} }$

The following is a firmly chosen irrational number. Consider the - Hilbert space of square integrable functions , where as usual the circle group is identified by means of the unit circle , and then the two unitary operators and defined as follows : ${\displaystyle \theta }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle L^{2}(\mathbb {R} /\mathbb {Z} )}$ ${\displaystyle \mathbb {R} /\mathbb {Z} }$ ${\displaystyle t\mapsto e^{2\pi it}}$ ${\displaystyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}}$ ${\displaystyle U}$ ${\displaystyle V}$

${\displaystyle Uf\,:=\,zf}$, in which ${\displaystyle z(t)=e^{2\pi it}}$

and

${\displaystyle Vf(t)\,:=\,f(t-\theta )\,.}$

${\displaystyle U}$is a multiplication operator and rotates a function by the angle . ${\displaystyle V}$ ${\displaystyle \theta }$

The C * -algebra generated by and is therefore called the irrational rotation algebra to the angle and is denoted by. ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle C^{*}(U,V)\subset B(L^{2}(\mathbb {T} ))}$ ${\displaystyle \theta }$ ${\displaystyle A_{\theta }}$

properties

• It is easy to confirm that it is indeed ${\displaystyle UV=e^{2\pi i\theta }VU}$

${\displaystyle UVf(t)=U(Vf)(t)=z(t)Vf(t)}$

${\displaystyle =z(t)f(t-\theta )=e^{2\pi i\theta }z(t-\theta )f(t-\theta )=e^{2\pi i\theta }(zf)(t-\theta )}$

${\displaystyle =e^{2\pi i\theta }(Uf)(t-\theta )=e^{2\pi i\theta }VUf(t)}$.

• The irrational rotation algebra has the following universal property that characterizes it up to isomorphism: If a C * -algebra is generated by two unitary operators and that satisfy the relation , then there is exactly one * -isomorphism with and . ${\displaystyle A}$ ${\displaystyle {\tilde {U}}}$ ${\displaystyle {\tilde {V}}}$ ${\displaystyle {\tilde {U}}{\tilde {V}}=e^{2\pi i\theta }{\tilde {V}}{\tilde {U}}}$ ${\displaystyle A_{\theta }\rightarrow A}$ ${\displaystyle U\mapsto {\tilde {U}}}$ ${\displaystyle V\mapsto {\tilde {V}}}$

• ${\displaystyle A_{\theta }}$is simple, that is, the algebra does not contain any two-sided * ideals other than and itself. ${\displaystyle \{0\}}$

• There is a clear trace , that is, there is exactly one linear functional with for all , for all and , where the one element is in . ${\displaystyle \tau :A_{\theta }\rightarrow \mathbb {C} }$ ${\displaystyle \tau :A_{\theta }\rightarrow \mathbb {C} }$ ${\displaystyle \tau (a^{*}a)\geq 0}$ ${\displaystyle a\in A_{\theta }}$ ${\displaystyle \tau (ab)=\tau (ba)}$ ${\displaystyle a,b\in A_{\theta }}$ ${\displaystyle \tau (I)=1}$ ${\displaystyle I}$ ${\displaystyle A_{\theta }}$

• The group of invertible elements is close to . ${\displaystyle A_{\theta }}$

• The irrational rotational algebras are nuclear .

Alternative construction

Here an alternative construction of the irrational rotational algebra on the sequence space with the orthonormal basis is presented. Define the unitary operators by: ${\displaystyle \ell ^{2}=\ell ^{2}(\mathbb {Z} )}$ ${\displaystyle (e_{n})_{n_{\in }\mathbb {Z} }}$ ${\displaystyle U,V\in B(\ell ^{2})}$

${\displaystyle Ue_{n}\,=\,e_{n+1}}$( two-sided shift ),

${\displaystyle Ve_{n}\,=\,e^{-2\pi in\theta }e_{n}}$(infinite diagonal matrix ).

Then one easily confirms what follows. Because of the universal property of the irrational rotational algebra mentioned above, one obtains from it . ${\displaystyle UVe_{n}=e^{-2\pi in\theta }e_{n+1}=e^{2\pi i\theta }VUe_{n}}$ ${\displaystyle UV=e^{2\pi i\theta }VU}$ ${\displaystyle A_{\theta }\cong C^{*}(U,V)\subset B(\ell ^{2})}$

K theory

After a set of Marc Rieffel , for every projection with , with the unique track on was. ${\displaystyle \alpha \in (\mathbb {Z} +\theta \mathbb {Z} )\cap [0,1]}$ ${\displaystyle P\in A_{\theta }}$ ${\displaystyle \tau (P)=\alpha }$ ${\displaystyle \tau }$ ${\displaystyle A_{\theta }}$

Since there is an imperforate, scaled, commutative group with Riesz's decomposition property (for these terms see Ordered Abelian Group ), there is exactly one AF-C * algebra according to the Effros-Handelman-Shen theorem, apart from isomorphism , which this group as K ₀ group , and it stands to reason that the C * -algebra , which is not itself an AF-C * -algebra, is associated with. Indeed, M. Pimsner and D. Voiculescu were able to construct an embedding . From this it follows first and then: ${\displaystyle (\mathbb {Z} +\theta \mathbb {Z} ,(\mathbb {Z} +\theta \mathbb {Z} )\cap \mathbb {R} ^{+},(\mathbb {Z} +\theta \mathbb {Z} )\cap [0,1])}$ ${\displaystyle {\mathcal {A}}_{\theta }}$ ${\displaystyle A_{\theta }}$ ${\displaystyle {\mathcal {A}}_{\theta }}$ ${\displaystyle A_{\theta }\rightarrow {\mathcal {A}}_{\theta }}$ ${\displaystyle K_{0}(A_{\theta })\cong \mathbb {Z} +\theta \mathbb {Z} }$

• Two irrational rotation algebras and are isomorphic if and only if is. ${\displaystyle A_{\theta }}$ ${\displaystyle A_{\eta }}$ ${\displaystyle \eta =\pm \theta \,\mathrm {mod} \,\mathbb {Z} }$

Cross product

The irrational rotational algebra is the prototype of the cross product of a C * -dynamic system . Is by defined, and is , it is a C * -Dynamic system and it is . ${\displaystyle \alpha \in \mathrm {Aut} (C(\mathbb {T} ))}$ ${\displaystyle (\alpha (f))(t)\,:=\,f(t-\theta )}$ ${\displaystyle \sigma :\mathbb {Z} \rightarrow \mathrm {Aut} (C(\mathbb {T} )),\,n\mapsto \alpha ^{n}}$ ${\displaystyle (C(\mathbb {T} ),\mathbb {Z} ,\sigma )}$ ${\displaystyle A_{\theta }\cong C(\mathbb {T} )\ltimes _{\sigma }\mathbb {Z} }$

Individual evidence

1. ↑ I. Putnam: The invertibles are dense in the irrational rotation C * -algebras , J. Pure Applied Mathematics, Volume 140 (1990), pages 160-166
2. ^ MA Rieffel: C * -algebras associated with irrational rotations , Pacific J. Math., Volume 93 (1981), pages 415-429
3. ↑ M. Pimsner, D. Voiculescu: imbedding the irrational rotation algebra into at AF algebra , Journal of Operator Theory, Volume 4 (1980), pages 93-118

KR Davidson: C * -Algebras by Example , American Mathematical Society (1996), ISBN 0-821-80599-1 :

1. ↑ Chapter VI: Irrational Rotation Algebra
2. ↑ Theorem VI.1.4
3. ↑ Sentence VI.1.3
4. ↑ Corollary VI.5.3
5. ↑ Example VIII.1.1