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Standard commuting dilations and liftings

Santanu Dey (2012)

Colloquium Mathematicae

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.

Standard exact projective resolutions relative to a countable class of Fréchet spaces

P. Domański, J. Krone, D. Vogt (1997)

Studia Mathematica

We will show that for each sequence of quasinormable Fréchet spaces ( E n ) there is a Köthe space λ such that E x t 1 ( λ ( A ) , λ ( A ) = E x t 1 ( λ ( A ) , E n ) = 0 and there are exact sequences of the form . . . λ ( A ) λ ( A ) λ ( A ) λ ( A ) E n 0 . If, for a fixed ℕ, E n is nuclear or a Köthe sequence space, the resolution above may be reduced to a short exact sequence of the form 0 λ ( A ) λ ( A ) E n 0 . The result has some applications in the theory of the functor E x t 1 in various categories of Fréchet spaces by providing a substitute for non-existing projective resolutions.

Standard ideals in convolution Sobolev algebras on the half-line

José E. Galé, Antoni Wawrzyńczyk (2011)

Colloquium Mathematicae

We study the relation between standard ideals of the convolution Sobolev algebra ( n ) ( t ) and the convolution Beurling algebra L¹((1+t)ⁿ) on the half-line (0,∞). In particular it is proved that all closed ideals in ( n ) ( t ) with compact and countable hull are standard.

Stationary Quantum Markov processes as solutions of stochastic differential equations

Jürgen Hellmich, Claus Köstler, Burkhard Kümmerer (1998)

Banach Center Publications

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...

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