The modular automorphism group and the Kubo-Martin-Schwinger boundary condition

1970 ◽  
pp. 63-72
Author(s):  
M. Takesaki
Keyword(s):  
Boundary Condition ◽  
Automorphism Group ◽  
Modular Automorphism

Twisted cyclic theory and an index theory for the gauge invariant KMS state on the Cuntz algebra On

2009 ◽  
Vol 6 (2) ◽  
pp. 339-380 ◽  
Author(s):  
A.L. Carey ◽  
J. Phillips ◽  
A. Rennie
Keyword(s):  
Automorphism Group ◽  
General Theory ◽  
Index Theory ◽  
Index Theorem ◽  
Cuntz Algebra ◽  
Spectral Flow ◽  
Gauge Invariant ◽  
Gauge Action ◽  
Cuntz Algebras ◽  
Modular Automorphism

AbstractThis paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on each Cuntz algebra. We introduce a modified K1-group for each Cuntz algebra which has an index pairing with this twisted cocycle. This index pairing for Cuntz algebras has an interpretation in terms of Araki's notion of relative entropy.

One-Dimensional Fin-Tip Boundary Condition Correction II

2001 ◽  
Vol 22 (5) ◽  
pp. 35-40 ◽  
Author(s):  
D. C. Look Jr ◽  
Arvind Krishnan
Keyword(s):  
Boundary Condition ◽  
One Dimensional

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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