This chapter uses categorical techniques to describe relations between various sets of operators on a Hilbert space, such as self-adjoint, positive, density, effect and projection operators. These relations, including various Hilbert-Schmidt isomorphisms of the form tr(A−), are expressed in terms of dual adjunctions, and maps between them. Of particular interest is the connection with quantum s...

We re-examine the appearance of semiheaps and (para-associative) ternary algebras in quantum mechanics. In particular, we review construction a semiheap on Hilbert space set bounded operators space. The new aspect this work is discussion how symmetries system induce homomorphisms relevant algebras.

We give geometric constructions of families of graded Gorenstein Artin algebras, some of which span a component of the space Gor(T ) parametrizing Gorenstein Artin algebras with a given Hilbert function T . This gives a lot of examples where Gor(T ) is reducible. We also show that the Hilbert function of a codimension four Gorenstein Artin algebra can have an arbitrarily long constant part with...

A new approach to the generalised Brownian motion introduced by M. Bożejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal’s notions of tensorial and combinatorial species. Any such species V gives rise to an endofunctor FV of the category of Hilbert spaces with contractions mapping a Hilbert space H to a symmetric Hilbe...

A new approach to the generalised Brownian motion introduced by M. Bo_ zejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal's notions of tensorial and combinatorial species. Any such species V gives rise to an endofunctor FV of the category of Hilbert spaces with contractions mapping a Hilbert space H to a symmetric Hil...

The Kadison-Kastler problem asks whether close C*-algebras on a Hilbert space must be spatially isomorphic. We establish this when one of the algebras is separable and nuclear. We also apply our methods to the study of near inclusions of C*-algebras.

We define and systematically study nonassociative C *-algebras as C *-algebras internal to a topological tensor category. We also offer a concrete approach to these C *-algebras, as G-invariant, norm closed *-subalgebras of bounded operators on a G-Hilbert space, with deformed composition product. Our central results are those of stabilization and Takai duality for (twisted) crossed products in...

We prove that a surjective isometry between the unit spheres of two uniform algebras is extended to real-linear algebras. It provides first positive solution Tingley's problem on Banach space, without being Hilbert consisting analytic functions.

Noncommutative multivariable versions of weighted shift operators arise naturally as ‘weighted’ left creation operators acting on the Fock space Hilbert space. We identify a natural notion of periodicity for these N tuples, and then find a family of inductive limit algebras determined by the periodic weighted shifts which can be regarded as noncommutative multivariable generalizations of the Bu...