The Index Theory Associated to a Non-Finite Trace on a C*-Algebra

2005 ◽  
Vol 48 (2) ◽  
pp. 251-259 ◽  
Author(s):  
G. J. Murphy
Keyword(s):  
Compact Group ◽  
Representation Theorem ◽  
Toeplitz Operators ◽  
Von Neumann Algebra ◽  
Index Theory ◽  
Index Theorem ◽  
Fredholm Index ◽  
Von Neumann ◽  
C Algebra ◽  
Finite Trace

AbstractThe index theory considered in this paper, a generalisation of the classical Fredholm index theory, is obtained in terms of a non-finite trace on a unitalC*-algebra. We relate it to the index theory of M. Breuer, which is developed in a von Neumann algebra setting, by means of a representation theorem. We show how our new index theory can be used to obtain an index theorem for Toeplitz operators on the compact group U(2), where the classical index theory does not give any interesting result.

scholarly journals Centre-valued Index for Toeplitz Operators with Noncommuting Symbols

2016 ◽  
Vol 68 (5) ◽  
pp. 1023-1066
Author(s):  
John Phillips ◽  
Iain Raeburn
Keyword(s):  
Toeplitz Operators ◽  
Adjoint Operator ◽  
Index Theorem ◽  
Scalar Case ◽  
Fredholm Operators ◽  
Negative Spectrum ◽  
Von Neumann ◽  
C Algebra ◽  
Making Sense ◽  
Algebra Theory

AbstractWe formulate and prove a “winding number” index theorem for certain “Toeplitz” operators in the same spirit as Gohberg–Krein, Lesch and others. The “number” is replaced by a self-adjoint operator in a subalgebra Z ⊆ Z(A) of a unital C*-algebra, A. We assume a faithful Z-valued trace τ on A left invariant under an action α:R → Aut(A) leaving Z pointwise fixed. If δ is the infinitesimal generator of α and u is invertible in dom(δ), then the “winding operator” of u is . By a careful choice of representations we extend (A, Z, τ, α) to a von Neumann setting where and . Then , the von Neumann crossed product, and there is a faithful, dual -trace on . If P is the projection in corresponding to the non-negative spectrum of the generator of R inside and is the embedding, then we define and show it is Fredholm in an appropriate sense and the -valued index of Tu is the negative of the winding operator. In outline the proof follows that of the scalar case done previously by the authors. The main difficulty is making sense of the constructions with the scalars replaced by in the von Neumann setting. The construction of the dual -trace on requires the nontrivial development of a -Hilbert algebra theory. We show that certain of these Fredholm operators fiber as a “section” of Fredholm operators with scalar-valued index and the centre-valued index fibers as a section of the scalar-valued indices.

scholarly journals Local and 2-local derivations on noncommutative Arens algebras

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
Shavkat Ayupov ◽  
Karimbergen Kudaybergenov ◽  
Berdakh Nurjanov ◽  
Amir Alauadinov
Keyword(s):  
Von Neumann Algebra ◽  
Inner Derivation ◽  
Von Neumann ◽  
Local Derivation ◽  
Finite Von Neumann Algebra ◽  
Finite Trace

AbstractThe paper is devoted to so-called local and 2-local derivations on the noncommutative Arens algebra L ω(M,τ) associated with a von Neumann algebra M and a faithful normal semi-finite trace τ. We prove that every 2-local derivation on L ω(M,τ) is a spatial derivation, and if M is a finite von Neumann algebra, then each local derivation on L ω(M,τ) is also a spatial derivation and every 2-local derivation on M is in fact an inner derivation.

scholarly journals LOCAL DERIVATIONS ON ALGEBRAS OF MEASURABLE OPERATORS

2011 ◽  
Vol 13 (04) ◽  
pp. 643-657 ◽  
Author(s):  
S. ALBEVERIO ◽  
SH. A. AYUPOV ◽  
K. K. KUDAYBERGENOV ◽  
B. O. NURJANOV
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Continuity Condition ◽  
Type I ◽  
Atomic Lattice ◽  
Von Neumann ◽  
Measurable Operators ◽  
Finite Trace ◽  
Necessary And Sufficient

The paper is devoted to local derivations on the algebra [Formula: see text] of τ-measurable operators affiliated with a von Neumann algebra [Formula: see text] and a faithful normal semi-finite trace τ. We prove that every local derivation on [Formula: see text] which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras, they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the continuity condition on local derivations in the above results is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra [Formula: see text] for the algebra [Formula: see text] to admit local derivations which are not derivations.

APPROXIMATELY INNER FLOWS ON SEPARABLE C*-ALGEBRAS

2002 ◽  
Vol 14 (07n08) ◽  
pp. 649-673 ◽  
Author(s):  
AKITAKA KISHIMOTO
Keyword(s):  
Irreducible Representation ◽  
Von Neumann Algebra ◽  
Automorphism Groups ◽  
Unitary Groups ◽  
Type I ◽  
Von Neumann ◽  
C Algebra ◽  
C Algebras ◽  
Inner Automorphism ◽  
Uniformly Continuous

We present two types of result for approximately inner one-parameter automorphism groups (referred to as AI flows hereafter) of separable C*-algebras. First, if there is an irreducible representation π of a separable C*-algebra A such that π(A) does not contain non-zero compact operators, then there is an AI flow α such that π is α-covariant and α is far from uniformly continuous in the sense that α induces a flow on π(A) which has full Connes spectrum. Second, if α is an AI flow on a separable C*-algebra A and π is an α-covariant irreducible representation, then we can choose a sequence (hn) of self-adjoint elements in A such that αt is the limit of inner flows Ad eithn and the sequence π(eithn) of one-parameter unitary groups (referred to as unitary flows hereafter) converges to a unitary flow which implements α in π. This latter result will be extended to cover the case of weakly inner type I representations. In passing we shall also show that if two representations of a separable simple C*-algebra on a separable Hilbert space generate the same von Neumann algebra of type I, then there is an approximately inner automorphism which sends one into the other up to equivalence.

The connection between the universal enveloping C*-algebra and the universal enveloping von Neumann algebra of a JW-algebra

1992 ◽  
Vol 112 (3) ◽  
pp. 575-579 ◽  
Author(s):  
Fatmah B. Jamjoom
Keyword(s):  
Von Neumann Algebra ◽  
Main Result Theorem ◽  
Von Neumann ◽  
C Algebra ◽  
Result Theorem ◽  
The Relationship

AbstractThis article aims to study the relationship between the universal enveloping C*-algebra C*(M) and the universal enveloping von Neumann algebra W*(M), when M is a JW-algebra. In our main result (Theorem 2·7) we show that C*(M) can be realized as the C*-subalgebra of W*(M) generated by M.

scholarly journals Remarks on complemented subspaces of von Neumann algebras

1992 ◽  
Vol 121 (1-2) ◽  
pp. 1-4 ◽  
Author(s):  
G. Pisier
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
C Algebra ◽  
Know How ◽  
Complemented Subspaces ◽  
Reflexive Subspace

SynopsisIn this note we include two remarks about bounded (not necessarily contractive) linear projections on a von Neumann algebra. We show that if M is a von Neumann subalgebra of B(H) which is complemented in B(H) and isomorphic to M⊗M, then M is injective (or equivalently M is contractively complemented). We do not know how to get rid of the second assumption on M. In the second part, we show that any complemented reflexive subspace of a C*-algebra is necessarily linearly isomorphic to a Hilbert space.

scholarly journals Averaging operators in non commutative Lp spaces I

1983 ◽  
Vol 24 (1) ◽  
pp. 71-74 ◽  
Author(s):  
Christopher Barnett
Keyword(s):  
Von Neumann Algebra ◽  
Measure Theory ◽  
Continuous Functions ◽  
Unbounded Operators ◽  
Averaging Operators ◽  
Von Neumann ◽  
Lp Spaces ◽  
Finite Von Neumann Algebra ◽  
Spaces Of Continuous Functions ◽  
Finite Trace

The origin of the theory of averaging operators is explained in [1]. The theory has been developed on spaces of continuous functions that vanish at infinity by Kelley in [3] and on the Lp spaces of measure theory by Rota [5]. The motivation for this paper arose out of the latter paper. The aim of this paper is to prove a generalisation of Rota's main representation theorem (every average is a conditional expectation) in the context of a ‘non commutative integration’. This context is as follows. Let be a finite von Neumann algebra and ϕ a faithful normal finite trace on such that ϕ(I) = 1, where I is the identity of . We can construct the Banach spaces Lp (, ϕ), where 1 ≤ p < °, with norm ∥x∥p = ϕ(÷x÷p)1/p, of possibly unbounded operators affiliated with , as in [9]. We note that is dense in Lp(, ϕ). These spaces share many of the features of the Lp spaces of measure theory; indeed if is abelian then Lp(,ϕ) is isometrically isomorphic to Lp of some measure space.

scholarly journals SMOOTH PATHS OF CONDITIONAL EXPECTATIONS

2011 ◽  
Vol 22 (07) ◽  
pp. 1031-1050
Author(s):  
ESTEBAN ANDRUCHOW ◽  
GABRIEL LAROTONDA
Keyword(s):  
Conditional Expectation ◽  
Von Neumann Algebra ◽  
Boundedness Condition ◽  
Derivative Operator ◽  
Von Neumann ◽  
Conditional Expectations ◽  
Finite Trace

Let [Formula: see text] be a von Neumann algebra with a finite trace τ, represented in [Formula: see text], and let [Formula: see text] be sub-algebras, for t in an interval I (0 ∈ I). Let [Formula: see text] be the unique τ-preserving conditional expectation. We say that the path t ↦ Et is smooth if for every [Formula: see text] and [Formula: see text], the map [Formula: see text] is continuously differentiable. This condition implies the existence of the derivative operator [Formula: see text] If this operator satisfies the additional boundedness condition, [Formula: see text] for any closed bounded subinterval J ⊂ I, and CJ > 0 a constant depending only on J, then the algebras [Formula: see text] are *-isomorphic. More precisely, there exists a curve [Formula: see text], t ∈ I of unital, *-preserving linear isomorphisms which intertwine the expectations, [Formula: see text] The curve Gt is weakly continuously differentiable. Moreover, the intertwining property in particular implies that Gt maps [Formula: see text] onto [Formula: see text]. We show that this restriction is a multiplicative isomorphism.

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