Fuglede–Kadison determinant: theme and variations

Harpe, Pierre de la

doi:10.1073/pnas.1202059110

Cited by 17 publications

(15 citation statements)

References 60 publications

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“…Note that det tr can take on complex values, not just positive numbers. And it is shown in [26] that…”

Section: Fuglede-kadison Determinantmentioning

confidence: 99%

“…In particular, the notion was extended to C * -algebras in [29]. We refer the readers to [26] for a recent survey, and make the following definition to proceed.…”

Section: Trace Of Maurer-cartan Formmentioning

confidence: 99%

“…For convenience, we shall call these points tr-singular is well defined for C * -algebras with trace (cf. [26,29]). Note that det tr can take on complex values, not just positive numbers.…”

Section: Fuglede-kadison Determinantmentioning

confidence: 99%

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Joint spectrum and the infinite dihedral group

Grigorchuk

Yang

2017

Proc. Steklov Inst. Math.

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For a tuple A = (A 1 , A 2 , ..., A n ) of elements in a unital Banach algebra B, its projective joint spectrum P (A) is the collection of z ∈ C n such that the multiparameter pencil A(z) = z 1 A 1 + z 2 A 2 + · · · + z n A n is not invertible.If B is the group C * -algebra for a discrete group G generated by A 1 , A 2 , ..., A n with respect to a representation ρ, then P (A) is an invariant of (weak) equivalence for ρ. This paper computes the joint spectrum of R = (1, a, t) for the infinite dihedral group D ∞ =< a, t | a 2 = t 2 = 1 > with respect to the left regular representation λ D∞ , and gives an in-depth analysis on its properties. A formula for the Fuglede-Kadison determinant of the pencil R(z) = z 0 + z 1 a + z 2 t is obtained, and it is used to compute the first singular homology group of the joint resolvent set P c (R). The joint spectrum gives new insight into some earlier studies on groups of intermediate growth, through which the corresponding joint spectrum of (1, a, t) with respect to the Koopman representation ρ (constructed through a self-similar action of D ∞ on a binary tree) can be computed. It turns out that the joint spectra with respect to the two representations coincide. Interestingly, this fact leads to a self-similar realization of the group C * -algebra C * (D ∞ ).This self-similarity of C * (D ∞ ) is manifested by some dynamical properties of the joint spectrum.

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“…Note that det tr can take on complex values, not just positive numbers. And it is shown in [26] that…”

Section: Fuglede-kadison Determinantmentioning

confidence: 99%

“…In particular, the notion was extended to C * -algebras in [29]. We refer the readers to [26] for a recent survey, and make the following definition to proceed.…”

Section: Trace Of Maurer-cartan Formmentioning

confidence: 99%

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Joint spectrum and the infinite dihedral group

Grigorchuk

Yang

2017

Proc. Steklov Inst. Math.

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“…FK-determinant has been well-studied in many papers, and we refer readers to [23] for a survey. In particular, it was generalized to C * -algebras in [27] as follows.…”

Section: Completenessmentioning

confidence: 99%

“…defines a notion of determinant for invertible elements x. One sees that det * may take on complex values, and it is shown in [23] that…”

Section: Completenessmentioning

confidence: 99%

Hermitian Geometry on Resolvent Set

Douglas

Yang

2018

Operator Theory: Advances and Applications

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For a tuple A = (A 1 , A 2 , ..., A n ) of elements in a unital Banach algebra B, its projective joint spectrum P (A) is the collection of z ∈ C n such thatcontains much topological information about the joint resolvent set P c (A). This paper defines Hermitian metric on P c (A) through the B-valued fundamental form Ω A = −ω * A ∧ ω A and its coupling with faithful states φ on B. The connection between the tuple A and the metrics is the main subject of this paper. A notable feature of this metric is that it has singularities at the joint spectrum P (A). So completeness of the metric is an important issue.When this construction is applied to a single operator V , the metric on the resolvent set ρ(V ) adds new ingredients to functional calculus. An interesting example is when V is quasi-nilpotent, in which case the metric live on the punctured complex plane. It turns out that the blow up rate of the metric at the origin 0 is directly linked with V 's lattice of hyper-invariant subspaces.2010 Mathematics Subject Classification: 47A13(primary), 53A35(secondary) Key words and phrases: Maurer-Cartan form, projective joint spectrum, Hermitian metric, Ricci tensor, quasi-nilpotent operator.

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The $$C^*$$-Algebra of the Infinite Dihedral Group $$D_{\infty }$$

Yang

2023

A Spectral Theory of Noncommuting Operators

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Fuglede–Kadison determinant: theme and variations

Cited by 17 publications

References 60 publications

Joint spectrum and the infinite dihedral group

Joint spectrum and the infinite dihedral group

Hermitian Geometry on Resolvent Set

The $$C^*$$-Algebra of the Infinite Dihedral Group $$D_{\infty }$$

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