Fredholm theories in von Neumann algebras. II

Breuer, M.

doi:10.1007/bf01351884

Cited by 89 publications

(92 citation statements)

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“…If £ is the spectral projection for T corresponding to the interval 10, k], then £ e sé [4], and \\AE\\ = \\UTE\\| | U\\ || TE\\ _rc. Since the subspace £ contains NT [2], £ is not equivalent to JtifOrlAU-£)]. We claim that £ is a finite projection.…”

Section: Proofmentioning

confidence: 93%

Extensions of the index in factors of type ${\rm II}\sb{\infty }$

Gartenberg¹

1974

Proc. Amer. Math. Soc.

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Abstract.In this paper we show that the analytic index has no continuous extension to those operators in a factor of type II,, on a separable Hubert space which are not semi-Fredholm in the Breuer sense. A similar result has already been proved by Coburn and Lebow [3] for factors of type Ix. Here we use Breuer's generalized Fredholm theory to extend their result to the more general setting.1. Definitions and preliminaries. As usual, ^(Jf ) denotes the algebra of all bounded operators on the separable Hubert space Jf. A *-subalgebra of á?pf) that is closed in the weak operator topology is called a von Neumann algebra. If the center of sé consists precisely of scalar multiples of the identity, then sé is called a factor. For E, Fin 0*ise), the set of all projection operations in sé, we write E^FoEF=E.The equivalence relation ~ on 0>ise) is defined by £~F if and only if there is a partially isometric operator U in sé such that E=U*U and F=UU*. Finally, an order relation ^ on SPisé) is given by £;< F if and only if there is an F' in g?isé) such that E~F'^F.A projection operator E is said to be finite if it is not equivalent to any F e0>ise) where F^E and £#£. Otherwise, £ is said to be infinite. If the identity of a von Neumann algebra sé is a finite (infinite) projection, then sé is called finite (infinite).We follow Breuer's generalization of the concepts of compact and Fredholm operators to a von Neumann algebra sé. For B e sé NB = sup{£ e 0>isé) :BE = 0} and RB = inf{£ e 0>ise) :EB = S) are called the null projection and range projection of B, respectively. We call B finite if RB is a finite projection. If Jf is the norm closure of the

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Section: Proofmentioning

confidence: 93%

Extensions of the index in factors of type ${\rm II}\sb{\infty }$

Gartenberg¹

1974

Proc. Amer. Math. Soc.

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“…Using the index theory of M. Breuer [1], [2] and the fact that K(SX) = inj lim[A', GLn], it can be seen that this construction depends only on the homotopy class of / and the equivalence class of r, it respects the obvious inclusion of GLn into GLn +,, and it is a homomorphism.…”

Section: ^%(%)^And -^C(*)^0mentioning

confidence: 99%

Extensions relative to a 𝐼𝐼_{∞}-factor

Cho¹

1979

Proc. Amer. Math. Soc.

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Abstract.It will be shown that the equivalence classes of C*-algebra extensions of C(X) relative to a II^,-factor and Hom(A'(Jf), R) are isomorphic. This provides a proof for the result of Brown, Douglas _and Fillmore [5] on the isomorphism between the former group and Hom(A/^-Y), R).Let % be a separable infinite dimensional Hubert space, £(%) the algebra of all bounded linear operators on %, %(%) the ideal of compact operators, and &(%) the quotient algebra £(%)/%(%).In [3], [4], [5], ExtX was defined as the set of equivalence classes of C*-algebra extensions,

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“…Let F be, as above, a fundamental domain for the action of Γ in D. Then for every Γ− equivariant, bounded function g on D, having compact support in the interior of F , the Toeplitz operator T g ∈ A t is in L 1 (A t ) and has trace equal to a universal constant times the integral F g(z)(Im ) −2 dzdz Moreover we will show that the commutator of two Toeplitz opertors, having symbols that are smooth and continuous on the closure of F , belongs to trace ideal of the II ∞ factor. In particular if the symbol is invertible in the neighbourhood of the intersection of the boundary of D with the closure of F , the operator is Fredholm in A t in Breuer's sense ( [6]). …”

Section: Introductionmentioning

confidence: 99%

Index of Γ-Equivariant Toeplitz Operators

Nest

Rădulescu

2000

C*-Algebras

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Abstract. Let Γ be a discrete subgroup of P SL(2, R) of infinite covolume with infinite conjugacy classes. Let H t be the Hilbert space consisting of analytic functions in L 2 (D, (Im z) t−2 dzdz) and let, for t > 1, π t denote the corresponding projective unitary representation of P SL(2, R) on this Hilbert space. We denote by A t the II ∞ factor given by the commutant of π t (Γ) in B(H t ). Let F denote a fundamental domain for Γ in D and assume that t > 5. ∂M = ∂D ∩ F is given the topology of disjoint union of its connected components.Suppose that f is a continuous Γ-invariant function on D whose restriction to F extends to a continuous function on F and such that f | ∂M is an invertible element of C 0 (∂M )˜. Let T t f = P t M f P t denote the Toeplitz operator with symbol f . Then T t f is Fredholm, in the Breuer sense, with respect to the II ∞ factor A t and, moreover, its Breuer index is equal to the total winding number of f on ∂M .

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Fredholm theories in von Neumann algebras. II

Cited by 89 publications

References 3 publications

Extensions of the index in factors of type ${\rm II}\sb{\infty }$

Extensions of the index in factors of type ${\rm II}\sb{\infty }$

Extensions relative to a 𝐼𝐼_{∞}-factor

Index of Γ-Equivariant Toeplitz Operators

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