The structure of some induced representations

Abstract. The curvature of the noncommutative torus T 2 θ (θ ∈ R \ Q) endowed with a noncommutative conformal metric has been the focus of attention of several recent works. Continuing the approach taken in the paper [A. Connes and H. Moscovici, Modular curvature for noncommutative two-tori, J. Amer. Math. Soc. 27 (2014), 639-684] we extend the study of the curvature to twisted Dirac spectral triples constructed out of Heisenberg bimodules that implement the Morita equivalence of the C * -algebraIn the enlarged context the conformal metric on T 2 θ is exchanged with an arbitrary Hermitian metric on the HeisenbergWe prove that the Ray-Singer log-determinant of the corresponding Laplacian, viewed as a functional on the space of all Hermitian metrics on E ′ , attains its extremum at the unique Hermitian metric whose corresponding connection has constant curvature. The gradient of the log-determinant functional gives rise to a noncommutative analogue of the Gaussian curvature. The genuinely new outcome of this paper is that the latter is shown to be independent of any Heisenberg bimodule, and in this sense it is Morita invariant. To prove the above results we extend Connes' pseudodifferential calculus to Heisenberg modules. The twisted version, which offers more flexibility even in the case of trivial coefficients, could potentially be applied to other problems in the elliptic theory on noncommutative tori. A noteworthy technical feature is that we systematize the computation of the resolvent expansion for elliptic differential operators on noncommutative tori to an extent which makes the (previously employed) computer assistance unnecessary.

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“…the action α. Let ω : G × G → T be a smooth cocycle (aka multiplier [Kle62]). That is, ω is a smooth map satisfying…”

Section: Effective Pseudodifferential Operators and Resolventmentioning

confidence: 99%

Modular Curvature and Morita Equivalence

Lesch

Moscovici

2016

Geom. Funct. Anal.

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“…Let a be a normalized multiplier on the discrete group G (cf. [1,7,9]). An element x E Gis a-regular if o(x, a) = a(a, x) for all a which commute with x.…”

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confidence: 99%

“…An element x E Gis a-regular if o(x, a) = a(a, x) for all a which commute with x. If x is a-regular so is every conjugate of x [7,Lemma 3], and we may speak of the a-regular conjugacy classes in G. Let A0 = A0(a) be the set of elements lying in finite a-regular conjugacy classes, let A = A(a) be the subgroup generated by A0, and let A' be its commutator subgroup. The left regular a representation Aa = AG acts on l2(G) by K(x)f(y) = o(x-x, y)fix-xy), fEl2(G), x,yEG.…”

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confidence: 99%

“…It is a finite representation [7,Theorem 9]. We say that the multiplier a is type I if all the primary a representations are type I.…”

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confidence: 99%

“…3. To prove that (a), (b), (c) in Theorem 1 are necessary, we begin by observing that if the set A0 is a subgroup of infinite index, then by [7,Theorem 4] Xa is type II. However, the proof of that theorem is valid also in the case that the subgroup A generated by A0 is of infinite index.…”

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confidence: 99%

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