Operator approach to quantization of semigroups.

*(English. Russian original)*Zbl 1319.46047
Sb. Math. 205, No. 3, 319-342 (2014); translation from Mat. Sb. 205, No. 3, 15-40 (2014).

For an abelian cancellative (discrete) semigroup with an identity \(S\), the authors construct and study a cocommutative compact quantum semigroup that is the analog of the quantum dual of a discrete group.

To elaborate, recall that every locally compact group \(\Lambda\) induces the Kac algebra \((C_{0}(\Lambda),\Delta)\), where \(\Delta:C_{0}(\Lambda)\to C_{0}(\Lambda)\otimes C_{0}(\Lambda)\) is the comultiplication defined by \((\Delta(f))(x,y)=f(xy)\). Its dual as a Kac algebra is \((C_{\mathrm{red}}^{*}(\Lambda),\Delta)\), where \(C_{\mathrm{red}}^{*}(\Lambda)\) is the reduced \(C^{*}\)-algebra of \(\Lambda\) and the comultiplication is now formally given by \(\Delta(T_{a})=T_{a}\otimes T_{a}\), where \(\left(T_{a}\right)_{a\in\Lambda}\) are the translation operators. When \(\Lambda\) is abelian, its dual is unitarily equivalent to the Kac algebra induced by the Pontryagin dual of \(\Lambda\); when \(\Lambda\) is discrete, its dual is a compact quantum group in the sense of Woronowicz.

In the paper under review, the authors consider the \(C^{*}\)-algebra \(C_{\mathrm{red}}^{*}(S)\) associated to \(S\), which is the one generated by the translation operators \(\left(T_{a}\right)_{a\in S}\) over \(\ell^{2}(S)\). They prove that the map \(T_{a}\mapsto T_{a}\otimes T_{a}\) extends to a comultiplication \(\Delta\) over \(C_{\mathrm{red}}^{*}(S)\), yielding a compact quantum semigroup \((C_{\mathrm{red}}^{*}(S),\Delta)\). Then they proceed to prove several results, including the following.

(1) Analogously to the theory of compact quantum groups, \((C_{\mathrm{red}}^{*}(S),\Delta)\) contains a canonical dense weak Hopf \(*\)-algebra. Weak Hopf algebras are a generalization of Hopf algebras introduced in [F. Li, J. Algebra 208, No. 1, 72–100 (1998; Zbl 0916.16020)] and studied by F. Li and S. Duplij [Commun. Math. Phys. 225, No.1, 191–217 (2002; Zbl 1032.17026)] and by others.

(2) Letting \(\Gamma\) be the Grothendieck group generated by \(S\) and \(G\) be the abelian compact Pontryagin dual of \(\Gamma\), \(C_{\mathrm{red}}^{*}(S)\) decomposes as the direct sum of a copy of \(C(G)\) and the commutator ideal \(K\) of \(C_{\mathrm{red}}^{*}(S)\). This leads to the identification of \(G\) as a compact quantum subgroup of \((C_{\mathrm{red}}^{*}(S),\Delta)\). The induced action and space of left cosets are described explicitly.

(3) \((C_{\mathrm{red}}^{*}(S),\Delta)\) carries a Haar state.

(4) The natural map from the category of the compact quantum semigroups constructed in the paper into the category of abelian cancellative semigroups is an injective functor.

As an application, a comultiplication on the Toeplitz algebra that is different from the one studied by the same authors [M. A. Aukhadiev et al., Lobachevskii J. Math. 32, No. 4, 304–316 (2011; Zbl 1271.46053)] is constructed.

Reviewer’s remarks. Two small translation/typing mistakes should be noted. The first sentence in Section 3 should read: “Let \(G\) be the compact abelian group …to the additive group \(\Gamma\)”, and the equation in Corollary 3 should be “\(\Delta(A)\circ\gamma=\gamma\circ A\)”.

To elaborate, recall that every locally compact group \(\Lambda\) induces the Kac algebra \((C_{0}(\Lambda),\Delta)\), where \(\Delta:C_{0}(\Lambda)\to C_{0}(\Lambda)\otimes C_{0}(\Lambda)\) is the comultiplication defined by \((\Delta(f))(x,y)=f(xy)\). Its dual as a Kac algebra is \((C_{\mathrm{red}}^{*}(\Lambda),\Delta)\), where \(C_{\mathrm{red}}^{*}(\Lambda)\) is the reduced \(C^{*}\)-algebra of \(\Lambda\) and the comultiplication is now formally given by \(\Delta(T_{a})=T_{a}\otimes T_{a}\), where \(\left(T_{a}\right)_{a\in\Lambda}\) are the translation operators. When \(\Lambda\) is abelian, its dual is unitarily equivalent to the Kac algebra induced by the Pontryagin dual of \(\Lambda\); when \(\Lambda\) is discrete, its dual is a compact quantum group in the sense of Woronowicz.

In the paper under review, the authors consider the \(C^{*}\)-algebra \(C_{\mathrm{red}}^{*}(S)\) associated to \(S\), which is the one generated by the translation operators \(\left(T_{a}\right)_{a\in S}\) over \(\ell^{2}(S)\). They prove that the map \(T_{a}\mapsto T_{a}\otimes T_{a}\) extends to a comultiplication \(\Delta\) over \(C_{\mathrm{red}}^{*}(S)\), yielding a compact quantum semigroup \((C_{\mathrm{red}}^{*}(S),\Delta)\). Then they proceed to prove several results, including the following.

(1) Analogously to the theory of compact quantum groups, \((C_{\mathrm{red}}^{*}(S),\Delta)\) contains a canonical dense weak Hopf \(*\)-algebra. Weak Hopf algebras are a generalization of Hopf algebras introduced in [F. Li, J. Algebra 208, No. 1, 72–100 (1998; Zbl 0916.16020)] and studied by F. Li and S. Duplij [Commun. Math. Phys. 225, No.1, 191–217 (2002; Zbl 1032.17026)] and by others.

(2) Letting \(\Gamma\) be the Grothendieck group generated by \(S\) and \(G\) be the abelian compact Pontryagin dual of \(\Gamma\), \(C_{\mathrm{red}}^{*}(S)\) decomposes as the direct sum of a copy of \(C(G)\) and the commutator ideal \(K\) of \(C_{\mathrm{red}}^{*}(S)\). This leads to the identification of \(G\) as a compact quantum subgroup of \((C_{\mathrm{red}}^{*}(S),\Delta)\). The induced action and space of left cosets are described explicitly.

(3) \((C_{\mathrm{red}}^{*}(S),\Delta)\) carries a Haar state.

(4) The natural map from the category of the compact quantum semigroups constructed in the paper into the category of abelian cancellative semigroups is an injective functor.

As an application, a comultiplication on the Toeplitz algebra that is different from the one studied by the same authors [M. A. Aukhadiev et al., Lobachevskii J. Math. 32, No. 4, 304–316 (2011; Zbl 1271.46053)] is constructed.

Reviewer’s remarks. Two small translation/typing mistakes should be noted. The first sentence in Section 3 should read: “Let \(G\) be the compact abelian group …to the additive group \(\Gamma\)”, and the equation in Corollary 3 should be “\(\Delta(A)\circ\gamma=\gamma\circ A\)”.

Reviewer: Ami Viselter (Haifa)

##### MSC:

46L65 | Quantizations, deformations for selfadjoint operator algebras |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

46L05 | General theory of \(C^*\)-algebras |

16T20 | Ring-theoretic aspects of quantum groups |