scholarly journals The Mackey bijection for complex reductive groups and continuous fields of reduced group C*-algebras

2020 ◽  
Vol 24 (20) ◽  
pp. 580-602
Author(s):  
Nigel Higson ◽  
Angel Román
Keyword(s):  
Reductive Groups ◽  
C Algebras ◽  
Reduced Group ◽  
Continuous Fields

scholarly journals MACROSCOPIC OBSERVABLES AND THE BORN RULE, I: LONG RUN FREQUENCIES

2008 ◽  
Vol 20 (10) ◽  
pp. 1173-1190 ◽  
Author(s):  
N. P. LANDSMAN
Keyword(s):  
Quantum Mechanics ◽  
Quantum System ◽  
Single Case ◽  
Born Rule ◽  
C Algebra ◽  
C Algebras ◽  
Long Run ◽  
Frequency Interpretation ◽  
Continuous Fields ◽  
The One

We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr's doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutative C*-algebra of observables is empirically accessible only through associated commutative C*-algebras. The Born probabilities emerge as the relative frequencies of outcomes in long runs of measurements on a quantum system; it is not necessary to adopt the frequency interpretation of single-case probabilities (which will be the subject of a sequel paper). Our derivation of the Born rule uses ideas from a program begun by Finkelstein [17] and Hartle [21], intending to remove the Born rule as a separate postulate of quantum mechanics. Mathematically speaking, our approach refines previous elaborations of this program — notably the one due to Farhi, Goldstone, and Gutmann [15] as completed by Van Wesep [50] — in replacing infinite tensor products of Hilbert spaces by continuous fields of C*-algebras. Furthermore, instead of relying on the controversial eigenstate-eigenvalue link in quantum theory, our derivation just assumes that pure states in classical physics have the usual interpretation as truthmakers that assign sharp values to observables.

scholarly journals Parabolic induction and restriction via -algebras and Hilbert -modules

2016 ◽  
Vol 152 (6) ◽  
pp. 1286-1318 ◽  
Author(s):  
Pierre Clare ◽  
Tyrone Crisp ◽  
Nigel Higson
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Inner Product ◽  
Reductive Groups ◽  
Hilbert Modules ◽  
Parabolic Induction ◽  
Full Treatment ◽  
Reduced Group ◽  
The Right ◽  
Tempered Representation

This paper is about the reduced group $C^{\ast }$-algebras of real reductive groups, and about Hilbert $C^{\ast }$-modules over these $C^{\ast }$-algebras. We shall do three things. First, we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced $C^{\ast }$-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced $C^{\ast }$-algebra to determine the structure of the Hilbert $C^{\ast }$-bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in a sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction in the context of tempered unitary Hilbert space representations.

scholarly journals Fiberwise KK-equivalence of continuous fields of C*-algebras

2008 ◽  
Vol 3 (2) ◽  
pp. 205-219 ◽  
Author(s):  
Marius Dadarlat
Keyword(s):  
Compact Space ◽  
Metrizable Space ◽  
Cuntz Algebra ◽  
Hilbert Cube ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Finite Dimensional ◽  
Compact Metrizable Space ◽  
Continuous Fields

AbstractLet A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element σ of the parametrized Kasparov group KKX(A,B) is invertible if and only all its fiberwise components σx ∈ KK(A(x),B(x)) are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra . Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of a separable nuclear continuous C(X)-algebra A over a finite dimensional locally compact space X satisfies the UCT, then A satisfies the UCT.

scholarly journals Discrete groups and simple C*-algebras

1991 ◽  
Vol 109 (3) ◽  
pp. 521-537 ◽  
Author(s):  
Erik Bédos
Keyword(s):  
Discrete Group ◽  
Free Groups ◽  
Free Subgroup ◽  
Discrete Groups ◽  
Full Group ◽  
C Algebras ◽  
Reduced Group

Let G denote a discrete group and let us say that G is C*-simple if the reduced group C*-algebra associated with G is simple. We notice immediately that there is no interest in considering here the full group C*-algebra associated with G, because it is simple if and only if C is trivial. Since Powers in 1975 [26] proved that all nonabelian free groups are C*-simple, the class of C*-simp1e groups has been considerably enlarged (see [1, 2, 6, 7, 12, 13, 14, 16, 24] as a sample!), and two important subclasses are so-called weak Powers groups ([6, 13]; see Section 4 for definition and examples) and the groups of Akemann-Lee type [1, 2], which are groups possessing a normal non-abelian free subgroup with trivial centralizer.

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