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P. M. Sołtan

Publications and source records attributed to P. M. Sołtan.

10 recordsLinked to original sources
arXiv2017

Quantum groups with projection and extensions of locally compact quantum groups

The main result of the paper is the characterization of those locally compact quantum groups with projection, i.e. non-commutative analogs of semidirect products, which are extensions as defined by L. Vainerman and S. Vaes. It turns out that quantum groups with projection are usually not extensions. We discuss several examples including the quantum $\mathrm{U}_q(2)$. The major tool used to obtain these results is the co-duality for coideals in algebras of functions on locally compact quantum groups and the concept of a normal coideal.

math.OA↗
arXiv2007

Quantum bohr compactification

We introduce a non commutative analog of the Bohr compactification. Starting from a general quantum group G we define a compact quantum group bG which has a universal property such as the universal property of the classical Bohr compactification for topological groups. We study the object bG in special cases when G is a classical locally compact group, dual of a classical group, discrete or compact quantum group as well a quantum group arising from a manageable multiplicative unitary. We also use our construction to give new examples of compact quantum groups.

math.OA↗
arXiv2006

Analysis on a Homogeneous Space of a Quantum Group

A detailed account of the construction of a homogeneous space for the quantum "az+b" group is presented. The homogeneous space is described by a commutative C*-algebra which means that it is a classical space. Then a covariant differential calculus on the homogeneous space is constructed and studied. A covariant measure and an analogue of the exponential function are used to introduce elements of Fourier analysis.

math.OA↗
arXiv2006

A remark on manageable multiplicative unitaries

We propose a weaker condition for multiplicative unitary operators related to quantum groups, than the condition of manageability introduced by S.L. Woronowicz. We prove that all the main results of the theory of manageable multiplicative unitaries remain true under this weaker condition. We also show that multiplicative unitaries arising naturally in the construction of some recent examples of non-compact quantum groups satisfy our condition, but fail to be manageable.

math.OA↗
arXiv2006

A note on multipliers of discrete quantum groups

We investigate the problem whether a given multiplier of a tensor product of two algebras belongs to the tensor product of multiplier algebras. We give a characterization of such multipliers in the case when one of the algebras is the algebra of functions on a discrete quantum group.

math.QA↗
arXiv2006

New quantum "az+b" groups

We construct quantum "az+b" groups for new values of the deformation parameter. Along the way we introduce new special functions and study their analytic properties as well as analyze the commutation relations determined by the choice of parameter.

math.OA↗
arXiv2006

Quantum bohr compactification of discrete quantum groups

We introduce the analog of Bohr compactification for discrete quantum groups on C*-algebra level. The cases of unimodular and general C*-algebraic discrete quantum groups are treated separately. The passage from the former case to the latter is done with help of the concept of canonical Kac quotient for compact quantum groups\footnotemark. Several examples are presented and passage to the Bohr compactification is investigated.

math.OA↗
arXiv2006

Compactifications of discrete quantum groups

Given a discrete quantum group A we construct a certain Hopf *-algebra AP which is a unital *-subalgebra of the multiplier algebra of A. The structure maps for AP are inherited from M(A) and thus the construction yields a compactification of A which is analogous to the Bohr compactification of a locally compact group. This algebra has the expected universal property with respect to homomorphisms from multiplier Hopf algebras of compact type (and is therefore unique). This provides an easy proof of the fact that for a discrete quantum group with an infinite dimensional algebra the multiplier algebra is never a Hopf algebra.

math.QA↗
arXiv2006

A remark on the spectrum of the analytic generator

It is shown that although the spectrum of the analytic generator of a one--parameter group of isometries of a Banach space may be equal to the whole complex plane, a simple operation of ampliating the analytic generator onto its graph locates its spectrum in the positive halfline.

math.FA↗