Stable rank of Fourier algebras and an application to Korovkin theory.
We identify how the standard commuting dilation of the maximal commuting piece of any row contraction, especially on a finite-dimensional Hilbert space, is associated to the minimal isometric dilation of the row contraction. Using the concept of standard commuting dilation it is also shown that if liftings of row contractions are on finite-dimensional Hilbert spaces, then there are strong restrictions on properties of the liftings.
We will show that for each sequence of quasinormable Fréchet spaces there is a Köthe space λ such that and there are exact sequences of the form . If, for a fixed ℕ, is nuclear or a Köthe sequence space, the resolution above may be reduced to a short exact sequence of the form . The result has some applications in the theory of the functor in various categories of Fréchet spaces by providing a substitute for non-existing projective resolutions.
We study the relation between standard ideals of the convolution Sobolev algebra and the convolution Beurling algebra L¹((1+t)ⁿ) on the half-line (0,∞). In particular it is proved that all closed ideals in with compact and countable hull are standard.
From the operator algebraic approach to stationary (quantum) Markov processes there has emerged an axiomatic definition of quantum white noise. The role of Brownian motion is played by an additive cocycle with respect to its time evolution. In this report we describe some recent work, showing that this general structure already allows a rich theory of stochastic integration and stochastic differential equations. In particular, if a quantum Markov process is represented by a unitary cocycle, we can...