3,002 research outputs found
Higher Weak Derivatives and Reflexive Algebras of Operators
Let D be a self-adjoint operator on a Hilbert space H and x a bounded
operator on H. We say that x is n-times weakly D-differentiable, if for any
pair of vectors a, b from H the function is n-times
differentiable. We give several characterizations of this property, among which
one is original. The results are used to show, that for a von Neumann algebra M
on H, the sub-algebra of n-times weakly D-differentiable operators has a
representation as a reflexive algebra of operators on a bigger Hilbert space.Comment: This version acknowledges results from the litterature, which the
first edition was unaware of. The result on the existence of a representation
with a reflexive image is ne
Finite von Neumann algebra factors with property Gamma
Techniques introduced by G. Pisier in his proof that finite von Neumann
factors with property gamma have length at most 5 are modified to prove that
the length is 3. It is proved that if such a factor is a complemented subspace
of some larger C*-algebra then there exists a projection of norm one from the
larger onto the smaller algebra. A new proof of the fact that the second
continuous Hochschild cohomology group of such an algebra with coefficients in
the algebra vanishes, is also included.Comment: 12 page
On the complete boundedness of the Schur block product
We give a Stinespring representation of the Schur block product, say (*), on
pairs of square matrices with entries in a C*-algebra as a completely bounded
bilinear operator of the form: A:=(a_{ij}), B:= (b_{ij}): A (*) B :=
(a_{ij}b_{ij}) = V* pi(A) F pi(B) V, such that V is an isometry, pi is a
*-representation and F is a self-adjoint unitary. This implies an inequality
due to Livshits and two apparently new ones on diagonals of matrices. ||A (*)
B|| \leq ||A||_r ||B||_c operator, row and column norm; - diag(A*A) \leq A* (*)
A \leq diag(A*A), and for all vectors f, g: | |^2 \leq < diag(AA*)
g, g> .Comment: 10 p, revised, expanded and to appear in Proc. AM
Commutator inequalities via Schur products
For a self-adjoint unbounded operator D on a Hilbert space H, a bounded
operator y on H and some complex Borel functions g(t) we establish inequalities
of the type
||[g(D),y]|| \leq A|||y|| + B||[D,y]|| + ...+ X|[D, [D,...[D, y]...]]||.
The proofs take place in a space of infinite matrices with operator entries,
and in this setting it is possible to approximate the matrix associated to
[g(D), y] by the Schur product of a matrix approximating [D,y] and a scalar
matrix. A classical inequality of Bennett on the norm of Schur products may
then be applied to obtain the results.Comment: 16 page
Sums of two dimensional spectral triples
We study countable sums of two dimensional modules for the continuous complex
functions on a compact metric space and show that it is possible to construct a
spectral triple which gives the original metric back. This spectral triple will
be finitely summable for any positive parameter. We also construct a sum of two
dimensional modules which reflects some aspects of the topological dimensions
of the compact metric space, but this will only give the metric back
approximately. We make an explicit computation of the last module for the unit
interval. The metric is recovered exactly, the Dixmier trace induces a multiple
of the Lebesgue integral and the number N(K) of eigenvalues bounded by K
behaves, such that N(K)/K is bounded, but without limit for K growing.Comment: 27 page
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