## Metrics on states from actions of compact groups (1998)

Venue: | Doc. Math |

Citations: | 44 - 5 self |

### BibTeX

@ARTICLE{Rieffel98metricson,

author = {Marc A. Rieffel},

title = {Metrics on states from actions of compact groups},

journal = {Doc. Math},

year = {1998},

pages = {215--229}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Let a compact Lie group act ergodically on a unital C ∗-algebra A. We consider several ways of using this structure to define metrics on the state space of A. These ways involve length functions, norms on the Lie algebra, and Dirac operators. The main thrust is to verify that the corresponding metric topologies on the state space agree with the weak-∗ topology. Connes [Co1, Co2, Co3] has shown us that Riemannian metrics on non-commutative spaces (C ∗-algebras) can be specified by generalized Dirac operators. Although in this setting there is no underlying manifold on which one then obtains an ordinary metric, Connes has shown that one does obtain in a simple way an ordinary metric on the state space of the C ∗-algebra, generalizing the Monge-Kantorovich metric on probability measures [Ra] (called the “Hutchinson metric ” in the theory of fractals [Ba]). But an aspect of this matter which has not received much attention so far [P] is the question of when the metric topology (that is, the topology from the metric coming from a Dirac operator) agrees with the underlying weak- ∗ topology on the state space. Note that for locally compact spaces their topology agrees with the weak- ∗ topology coming from viewing points as linear functionals (by evaluation) on the algebra of continuous functions vanishing at infinity.

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Citation Context ... simple way an ordinary metric on the state space of the C ∗ -algebra, generalizing the Monge-Kantorovich metric on probability measures [Ra] (called the “Hutchinson metric” in the theory of fractals =-=[Ba]-=-). But an aspect of this matter which has not received much attention so far [P] is the question of when the metric topology (that is, the topology from the metric coming from a Dirac operator) agrees... |

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Citation Context ...ith ‖a‖ ≤ k‖â‖∞ for a ∈ L. This condition clearly holds when A is a C∗-algebra, L is dense in A, and S is the state space of A, so that we are dealing with the usual Kadison functional representation =-=[KR]-=-. But we remark that even in this case the constant k above cannot always be taken to be 1 (bottom of page 263 of [KR]). This suggests that in using formula (1.1) one might want to restrict to using j... |

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Citation Context ...ence of further hypotheses it can easily happen that ρ(µ, ν) = +∞. For one interesting situation where this sometimes happens see the end of the discussion of the second example following axiom 4’ of =-=[C3]-=-.) The semi-norm L is an example of a general Lipschitz semi-norm, that is [BC, Cu, P, Wv1, Wv2], a semi-norm L on a dense subalgebra L of A satisfying the Leibniz property: (1.2) L(ab) ≤ L(a)‖b‖ + ‖a... |

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Citation Context ... an ordinary metric, Connes has shown that one does obtain in a simple way an ordinary metric on the state space of the C ∗ -algebra, generalizing the Monge-Kantorovich metric on probability measures =-=[Ra]-=- (called the “Hutchinson metric” in the theory of fractals [Ba]). But an aspect of this matter which has not received much attention so far [P] is the question of when the metric topology (that is, th... |

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Citation Context ...or. Then the metric topology on the state space of A defined by the metric from D agrees with the weak-∗ topology. An important case to which this theorem applies consists of the non-commutative tori =-=[Rf]-=-, since they carry ergodic actions of ordinary tori [OPT]. The metric geometry of noncommutative tori has recently become of interest in connection with string theory [CDS, RS, S]. We begin by showing... |

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Citation Context ...n which a quantum group acts ergodically [Bo, Wn]. This would be especially interesting since for non-commutative compact groups there is only a sparse collection of known examples of ergodic actions =-=[Ws]-=-, whereas in [Wn] a rich collection of ergodic actions of compact quantum groups is constructed. Closely related is the setting of ergodic coactions of discrete groups [N, Q]. But I have not explored ... |

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Citation Context ... (iii) of Theorem 1.9. For this we need the unobvious fact [HLS, Bo] that because G is compact and α is ergodic, each irreducible representation of G occurs with at most finite multiplicity in A. (In =-=[HLS]-=- it is also shown that η is a trace, but we do not need this fact here.) The following lemma is undoubtedly well-known, but I do not know a reference for it. 132.5 Lemma. Let α be a (strongly continu... |

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Citation Context ...ined by the metric from D agrees with the weak-∗ topology. An important case to which this theorem applies consists of the non-commutative tori [Rf], since they carry ergodic actions of ordinary tori =-=[OPT]-=-. The metric geometry of noncommutative tori has recently become of interest in connection with string theory [CDS, RS, S]. We begin by showing in the first section of this paper that the mechanism fo... |

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Citation Context ... group acts ergodically [Bo, Wn]. This would be especially interesting since for non-commutative compact groups there is only a sparse collection of known examples of ergodic actions [Ws], whereas in =-=[Wn]-=- a rich collection of ergodic actions of compact quantum groups is constructed. Closely related is the setting of ergodic coactions of discrete groups [N, Q]. But I have not explored any of these poss... |

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Citation Context ...ng the Monge-Kantorovich metric on probability measures [Ra] (called the “Hutchinson metric” in the theory of fractals [Ba]). But an aspect of this matter which has not received much attention so far =-=[P]-=- is the question of when the metric topology (that is, the topology from the metric coming from a Dirac operator) agrees with the underlying weak-∗ topology on the state space. Note that for locally c... |

7 |
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Citation Context ...the results of the previous sections to prove our main theorem, stated above, for the metrics which come from Dirac operators. It is natural to ask about actions of non-compact groups. Examination of =-=[Wv4]-=- suggests that there may be very interesting phenomena there. The considerations of the present paper also make one wonder whether there is an appropriate analogue of length functions for compact quan... |

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Citation Context ... as in the previous section, we see that L ⊆ L0 and L0 ≤ L. Thus we are exactly in position to apply the Comparison Lemma 1.10 to obtain the desired conclusion. □ We remark that Weaver (theorem 24 of =-=[Wv1]-=-) in effect proved for this setting the total boundedness of B1 for the particular case of non-commutative 2-tori, by different methods. 4. Metrics from Dirac operators Suppose again that G is a compa... |

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Citation Context ...generally, for 0 < r < 1 we could define Lr by L r (a) = sup{‖αx(a) − a‖/(ℓ(x)) r : x ̸= e} along the lines considered in [Ro1, Ro2]. For actions on the non-commutative torus this has been studied in =-=[Wv2]-=-, but we will not pursue this here.) It is not so clear whether L is carried into itself by α, but we do not need this fact here. (For Lie groups see theorem 4.1 of [Ro1] or the comments after theorem... |

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1 |
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(Show Context)
Citation Context ...orus this has been studied in [Wv2], but we will not pursue this here.) It is not so clear whether L is carried into itself by α, but we do not need this fact here. (For Lie groups see theorem 4.1 of =-=[Ro1]-=- or the comments after theorem 6.1 of [Ro2].) Let us consider, however, the α-invariance of L. We find that L(αz(a)) = sup{‖αz(αz −1xz(a) − a)‖/ℓ(x) : x ̸= e} = sup{‖αx(a) − a‖/ℓ(zxz −1 ) : x ̸= e}. T... |