scholarly journals Matrix geometries emergent from a point

2014 ◽  
Vol 26 (09) ◽  
pp. 1450017
Author(s):  
Francesco D'Andrea ◽  
Fedele Lizzi ◽  
Pierre Martinetti
Keyword(s):  
Equivalence Classes ◽  
Unitary Equivalence ◽  
Categorical Approach ◽  
Spectral Triples ◽  
Moyal Plane

We describe a categorical approach to finite noncommutative geometries. Objects in the category are spectral triples, rather than unitary equivalence classes as in other approaches. This enables us to treat fluctuations of the metric and unitary equivalences on the same footing, as representatives of particular morphisms in this category. We then show how a matrix geometry (Moyal plane) emerges as a fluctuation from one point, and discuss some geometric aspects of this space.

scholarly journals Unitary equivalence classes of split-step quantum walks

2021 ◽  
Vol 20 (11) ◽  
Author(s):  
Akihiro Narimatsu ◽  
Hiromichi Ohno ◽  
Kazuyuki Wada
Keyword(s):  
Quantum Walks ◽  
Equivalence Classes ◽  
Unitary Equivalence

scholarly journals On Ultraproduct Embeddings and Amenability for Tracial von Neumann Algebras

2020 ◽  
Author(s):  
Scott Atkinson ◽  
Srivatsav Kunnawalkam Elayavalli
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Analogous Result ◽  
Equivalence Classes ◽  
Embedding Problem ◽  
Unitary Equivalence ◽  
Von Neumann ◽  
Sofic Groups

Abstract We define the notion of self-tracial stability for tracial von Neumann algebras and show that a tracial von Neumann algebra satisfying the Connes embedding problem (CEP) is self-tracially stable if and only if it is amenable. We then generalize a result of Jung by showing that a separable tracial von Neumann algebra that satisfies the CEP is amenable if and only if any two embeddings into $R^{\mathcal{U}}$ are ucp-conjugate. Moreover, we show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$om$(N, \prod _{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of embeddings is separable if and only $N$ is hyperfinite. This resolves a question of Popa for Connes embeddable factors. These results hold when we further ask that the pairs of embeddings commute, admitting a nontrivial action of $\textrm{Out}(N\otimes N)$ on ${\mathbb{H}}\textrm{om}(N\otimes N, \prod _{k\to \mathcal{U}}M_k)$ whenever $N$ is non-amenable. We also obtain an analogous result for commuting sofic representations of countable sofic groups.

scholarly journals Spectral distance on Lorentzian Moyal plane

2020 ◽  
Vol 17 (06) ◽  
pp. 2050089
Author(s):  
Anwesha Chakraborty ◽  
Biswajit Chakraborty
Keyword(s):  
Noncommutative Geometry ◽  
Coherent States ◽  
Euclidean Case ◽  
Spectral Triples ◽  
Spectral Distance ◽  
Moyal Plane ◽  
Pure States ◽  
Math Phys ◽  
Distance Formula

We present here a completely operatorial approach, using Hilbert–Schmidt operators, to compute spectral distances between time-like separated “events”, associated with the pure states of the algebra describing the Lorentzian Moyal plane, using the axiomatic framework given by [N. Franco, The Lorentzian distance formula in noncommutative geometry, J. Phys. Conf. Ser. 968(1) (2018) 012005; N. Franco, Temporal Lorentzian spectral triples, Rev. Math. Phys. 26(8) (2014) 1430007]. The result shows no deformations of non-commutative origin, as in the Euclidean case, if the pure states are constructed out of Glauber–Sudarshan coherent states.

scholarly journals Symmetric states: local unitary equivalence via stabilizers

2010 ◽  
Vol 10 (11&12) ◽  
pp. 1029-1041
Author(s):  
Curt D. Cenci ◽  
David W. Lyons ◽  
Laura M. Snyder ◽  
Scott N. Walck
Keyword(s):  
Degrees Of Freedom ◽  
Rotation Group ◽  
Equivalence Classes ◽  
Unitary Equivalence ◽  
Finite Subgroups ◽  
Local Unitary Equivalence ◽  
Symmetric States

We classify local unitary equivalence classes of symmetric states via a classification of their local unitary stabilizer subgroups. For states whose local unitary stabilizer groups have a positive number of continuous degrees of freedom, the classification is exhaustive. We show that local unitary stabilizer groups with no continuous degrees of freedom are isomorphic to finite subgroups of the rotation group $SO(3)$, and give examples of states with discrete stabilizers.

UNIVERSAL ALGEBRA OF SECTORS

2009 ◽  
Vol 19 (03) ◽  
pp. 347-371 ◽  
Author(s):  
KATSUNORI KAWAMURA
Keyword(s):  
Field Theory ◽  
Universal Algebra ◽  
Equivalence Classes ◽  
Cuntz Algebra ◽  
Quantum Field ◽  
Unitary Equivalence ◽  
C Algebra ◽  
Branching Laws ◽  
Common Unit ◽  
Algebraic Formulation

We show that a nontrivial example of universal algebra appears in quantum field theory. For a unital C *-algebra [Formula: see text], a sector is a unitary equivalence class of unital *-endomorphisms of [Formula: see text]. We show that the set [Formula: see text] of all sectors of [Formula: see text] is a universal algebra with an N-ary sum which is not reduced to any binary sum when [Formula: see text] includes the Cuntz algebra [Formula: see text] as a C *-subalgebra with common unit for N ≥ 3. Next we explain that the set [Formula: see text] of all unitary equivalence classes of unital *-representations of [Formula: see text] is a right module of [Formula: see text]. An essential algebraic formulation of branching laws of representations is given by using submodules of [Formula: see text]. As an application, we show that the action of [Formula: see text] on [Formula: see text] distinguishes elements of [Formula: see text].

Weak Containment and Induced Representations of Groups

1962 ◽  
Vol 14 ◽  
pp. 237-268 ◽  
Author(s):  
J. M. G. Fell
Keyword(s):  
Dual Space ◽  
Locally Compact Group ◽  
Equivalence Classes ◽  
Unitary Representations ◽  
Unitary Equivalence ◽  
Locally Compact ◽  
Induced Representations ◽  
Weak Containment ◽  
Irreducible Unitary Representations ◽  
Special Case

Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G. An important tool for investigating the group algebra of G is the so-called hull-kernel topology of G†, which is discussed in (3) as a special case of the relation of weak containment. The question arises: Given a group G, how do we determine G† and its topology? For many groups G, Mackey's theory of induced representations permits us to catalogue all the elements of G†. One suspects that by suitably supplementing this theory it should be possible to obtain the topology of G† at the same time. It is the purpose of this paper to explore this possibility. Unfortunately, we are not able to complete the programme at present.

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