Spectral Truncations in Noncommutative Geometry and Operator Systems

Abstract

In this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in momentum space) and with tolerance relations which provide a coarse grain approximation of geometric spaces at a finite resolution. In our new approach the traditional role played by $$C^*$$ C ∗ -algebras is taken over by operator systems. As part of the techniques we treat $$C^*$$ C ∗ -envelopes, dual operator systems and stable equivalence. We define a propagation number for operator systems, which we show to be an invariant under stable equivalence and use to compare approximations of the same space. We illustrate our methods for concrete examples obtained by spectral truncations of the circle. These are operator systems of finite-dimensional Toeplitz matrices and their dual operator systems which are given by functions in the group algebra on the integers with support in a fixed interval. It turns out that the cones of positive elements and the pure state spaces for these operator systems possess a very rich structure which we analyze including for the algebraic geometry of the boundary of the positive cone and the metric aspect i.e. the distance on the state space associated to the Dirac operator. The main property of the spectral truncation is that it keeps the isometry group intact. In contrast, if one considers the other finite approximation provided by circulant matrices the isometry group becomes discrete, even though in this case the operator system is a $$C^*$$ C ∗ -algebra. We analyze this in the context of the finite Fourier transform on the cyclic group. The extension of noncommutative geometry to operator systems allows one to deal with metric spaces up to finite resolution by considering the relation $$d(x,y)< \varepsilon $$ d ( x , y ) < ε between two points, or more generally a tolerance relation which naturally gives rise to an operator system.

On Multimatrix Models Motivated by Random Noncommutative Geometry I: The Functional Renormalization Group as a Flow in the Free Algebra

“…• The exact RG-flow should consider operators that are not pure traces of elements in the free algebra, but that are smeared with functions resulting from the IR-regulator. 22 Mind the flipped sign convention. Also that the couplings of the operators A 4 and B 4 have to coincide.…”

“…In this paper, the model of space(time) we focus on is an abstraction of fuzzy spaces [27,30,71], whose elements were later assembled into a spectral triple (the spin geometry object in NCG) called fuzzy geometry [8,12]. For the future, in a broader NCG context, it would be desirable to relate the FRGE to the newly investigated truncations in the spectral NCG formalism [22,42,43] (see [29] for a preceding related idea), but for initial investigations fuzzy geometries are interesting enough and also in line with them, e.g., for the case of the sphere [76,Sect. 3.3].…”

The operator system of Toeplitz matrices

“…Proof. The first two assertions are given by Propositions 4.2 and 4.3 of [9], while the third assertion is derived from Proposition 6.3 of [20]. Definition 2.2.…”

“…, s n−1 in an operator system S. The present paper considers such matrices x when S is an operator system, and addresses the issue of positivity for these Toeplitz matrices, particularly in the cases S = M m (C), the C * -algebra of m × m complex matrices, and S = C(S 1 ) (m) , the operator system of m × m complex Toeplitz matrices. The work in this paper is strongly motivated by recent results of Connes and van Suijlekom [9] which, among other things, identify the operator system of Toeplitz matrices with the dual space of a function system of trigonometric polynomials. To explain the contributions of the present paper and set the notation, let C(S 1 ) denote the unital abelian C * -algebra of all continuous functions f : S 1 → C, where S 1 ⊂ C is the unit circle.…”

Asymptotic commutativity of quantized spaces: The case of CPp,q

“…In agreement with results in [12,13], these corrections are seen to vanish in the asymptotic limit. The Wick-Voros product lends itself naturally to a matrix approximation, and considering finite matrices is tantamount to the imposition of a cutoff geometry [20,21], which provides both an ultraviolet and an infrared cutoff. We do not do the finite matrix approximation here.…”