Product isometries and automorphisms of the car algebra

1978 ◽  
pp. 173-185
Author(s):  
Richard V Kadison
Keyword(s):  
Car Algebra

Fourier transformation and Weyl relations in the CAR Algebra

1973 ◽  
Vol 6 (8) ◽  
pp. 302-304 ◽  
Author(s):  
R. M. Lovely ◽  
F. J. Bloore
Keyword(s):  
Fourier Transformation ◽  
Car Algebra

scholarly journals MARKOV STATES AND CHAINS ON THE CAR ALGEBRA

2007 ◽  
Vol 10 (02) ◽  
pp. 165-183 ◽  
Author(s):  
LUIGI ACCARDI ◽  
FRANCESCO FIDALEO ◽  
FARRUH MUKHAMEDOV
Keyword(s):  
Markov Chains ◽  
Tensor Product ◽  
Markov Processes ◽  
Car Algebra ◽  
Quantum Markov Processes ◽  
Canonical Anticommutation Relations ◽  
Satisfactory Theory ◽  
Quantum Markov Chains ◽  
General Method

We introduce the notion of Markov states and chains on the Canonical Anticommutation Relations algebra over ℤ, emphasizing some remarkable differences with the infinite tensor product case. We describe the structure of the Markov states on this algebra and show that, contrarily to the infinite tensor product case, not all these states are diagonalizable. A general method to construct nontrivial quantum Markov chains on the CAR algebra is also proposed and illustrated by some pivotal examples. This analysis provides a further step for a satisfactory theory of quantum Markov processes.

scholarly journals IMPLEMENTATION OF ENDOMORPHISMS OF THE CAR ALGEBRA

1995 ◽  
Vol 07 (06) ◽  
pp. 833-869 ◽  
Author(s):  
CARSTEN BINNENHEI
Keyword(s):  
Hilbert Space ◽  
Connected Components ◽  
Square Root ◽  
Fredholm Index ◽  
Fock States ◽  
Car Algebra ◽  
Bogoliubov Transformations ◽  
Particle Operator ◽  
Explicit Expressions

The implementation of non-surjective Bogoliubov transformations in Fock states over CAR algebras is investigated. Such a transformation is implementable by a Hilbert space of isometries if and only if the well-known Shale-Stinespring condition is met. In this case, the dimension of the implementing Hilbert space equals the square root of the Watatani index of the associated inclusion of CAR algebras, and both are determined by the Fredholm index of the corresponding one-particle operator. Explicit expressions for the implementing operators are obtained, and the connected components of the semigroup of implementable transformations are described.

Entropy of $C^\ast$-dynamical systems defined by bitstreams

1998 ◽  
Vol 18 (4) ◽  
pp. 859-874 ◽  
Author(s):  
V. YA. GOLODETS ◽  
ERLING ST&\Oslash;RMER
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Positive Entropy ◽  
Completely Positive ◽  
Car Algebra

We study automorphisms of the CAR-algebra obtained from binary shifts. We consider cases when the $C^\ast$-dynamical system is asymptotically abelian, is proximally asymptotically abelian, is an entropic $K$-system or has completely positive entropy. The entropy is computed in several cases.

scholarly journals UHF FLOWS AND THE FLIP AUTOMORPHISM

2001 ◽  
Vol 13 (09) ◽  
pp. 1163-1181 ◽  
Author(s):  
A. KISHIMOTO
Keyword(s):  
Automorphism Group ◽  
Conjugacy Class ◽  
Adjoint Action ◽  
Restricted Class ◽  
C Algebra ◽  
Type Formula ◽  
Car Algebra ◽  
Suitable Sense ◽  
Cocycle Conjugacy

A UHF algebra is a C*-algebra A of the type [Formula: see text] for some sequence (ni) with ni≥2, where Mn is the algebra of n×n matrices, while a UHF flow α is a flow (or a one-parameter automorphism group) on the UHF algebra A obtained as [Formula: see text], where [Formula: see text] for some [Formula: see text]. This is the simplest kind of flows on the UHF algebra we could think of; yet there seem to have been no attempts to characterize the cocycle conjugacy class of UHF flows so that we might conclude, e.g., that the non-trivial quasi-free flows on the CAR algebra are beyond that class. We give here one attempt, which is still short of what we have desired, using the flip automorphism of A⊗A. Our characterization for a somewhat restricted class of flows (approximately inner and absorbing a universal UHF flow) says that the flow α is cocycle conjugate to a UHF flow if and only if the flip is approximated by the adjoint action of unitaries which are almost invariant under α⊗α. Another tantalizing problem is whether we can conclude that a flow is cocycle conjugate to a UHF flow if it is close to a UHF flow in a suitable sense. We give a solution to this, as a corollary, for the above-mentioned restricted class of flows. We will also discuss several kinds of flows to clarify the situation.

scholarly journals On Subalgebras of the Car-Algebra

1995 ◽  
Vol 129 (1) ◽  
pp. 35-63 ◽  
Author(s):  
E. Kirchberg
Keyword(s):  
Car Algebra
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