scholarly journals Jones index theory for Hilbert C∗-bimodules and its equivalence with conjugation theory

2004 ◽  
Vol 215 (1) ◽  
pp. 1-49 ◽  
Author(s):  
Tsuyoshi Kajiwara ◽  
Claudia Pinzari ◽  
Yasuo Watatani
Keyword(s):  
Index Theory ◽  
Jones Index

A NOTE ON OCNEANU'S APPROACH TO JONES' INDEX THEORY

1993 ◽  
Vol 04 (05) ◽  
pp. 859-871 ◽  
Author(s):  
SHIGERU YAMAGAMI
Keyword(s):  
Index Theory ◽  
Frobenius Reciprocity ◽  
Jones Index ◽  
Basic Facts

Some of the basic facts in the Ocneanu's approach to Jones' index theory are proved. Among other things the biunitarity of connections in paragroups is proved generally by making use of Frobenius reciprocity.

scholarly journals Jones index theory by Hilbert C${}^{*}$-bimodules and K-theory

2000 ◽  
Vol 352 (8) ◽  
pp. 3429-3472 ◽  
Author(s):  
Tsuyoshi Kajiwara ◽  
Yasuo Watatani
Keyword(s):  
Index Theory ◽  
Jones Index ◽  
K Theory

An Index Theory for Semigroups of *-Endomorphisms of and Type II1 Factors.

1988 ◽  
Vol 40 (1) ◽  
pp. 86-114 ◽  
Author(s):  
Robert T. Powers
Keyword(s):  
Index Theory ◽  
Type Ii ◽  
Jones Index ◽  
Countable Infinity ◽  
Index 2 ◽  
Study Unit

In this paper we study unit preserving *-endomorphisms of and type II1 factors. A *-endomorphism α which has the property that the intersection of the ranges of αn for n = 1 , 2 , … consists solely of multiples of the unit are called shifts. In Section 2 it is shown that shifts of can be characterized up to outer conjugacy by an index n = ∞ 1, 2 , …. In Section 3 shifts of R the hyperfinite II1 factor are studied. An outer conjugacy invariant of a shift of R is the Jones index [R: α(R)]. In Section 3 a class of shifts of index 2 are studied. These are called binary shifts. It is shown that there are uncountably many binary shifts which are pairwise non conjugate and among the binary shifts there are at least a countable infinity of shifts which are pairwise not outer conjugate.

EXTENSION ALGEBRAS OF II1 FACTORS VIA *-ENDOMORPHISMS

1994 ◽  
Vol 05 (05) ◽  
pp. 635-655 ◽  
Author(s):  
MARIE CHODA
Keyword(s):  
Index Theory ◽  
Crossed Product ◽  
Jones Index

From the view of Jones index theory, we construct a III factor <A, γ>, for a pair {A, γ} of a II 1 factor A and a trace preserving *-endomorphism γ, through the method of the crossed product by a *-endomorphism. The AFD III λ factor M is isomorphic to <A, σ> for a basic *-endomorphism σ of the hyperfinite II 1 factor A if and only if λ−1 ∈ {4 cos 2(π/n : n ≥ 3} ⋃ [4, ∞). Ocneanu's canonical shift Γ for an inclusion N ⊂ M of II 1-factors with [M: N] < ∞ is extended to a *-endomorphism [Formula: see text] of <A, Γ> for A = M′ ∩ M∞. The extended *-endomorphism [Formula: see text] is Longo's canonical *-endomorphism.

Higher Index Theory

2020 ◽  
Author(s):  
Rufus Willett ◽  
Guoliang Yu
Keyword(s):  
Index Theory

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

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