*-orderable semigroups

Fix a *-orderable field k. We introduce the class of *-orderable semigroups as those semigroups with involution S for which the semigroup algebra kS endowed with the canonical involution admits a *-ordering. It is shown that this class is a quasivariety that is locally and residually closed. A cancellative nilpotent semigroup with involution is proved to be *-orderable if and only if it has unique extraction of roots. In general this equivalence fails, although every *-orderable semigroup has unique extraction of roots.     

On C*-quasiregular semigroups *

In this paper, we introduce an important subclass of quasiregular semigroups, namely the class of C^*-quasiregular semigroups. This class of semigroups contains the classes of Clifford semigroups, quasi Clifford semigroups, C-quasiregular semigroups and their generalizations as its subclasses. Some characterization theorems for such semigroups are obtained. The structure of this kind of quasiregular semigroups is investigated by using the generalized ?-product of some semigroups on a semilattice Y. Construction techniques of such classes of semigroups are particularly demonstrated.     

Inverse semigroups and combinatorial C*-algebras

We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case. *Partially supported by CNPq.     

Onσ-pure submodules of QTAG-modules

Archiv der Mathematik -     

On q *-bisimple IC semigroups of type-E

A semigroup is called type-E if the band of its idempotents can be expressed as a direct product of a rectangular band and an ω-chain. For brevity, we call an IC ^*-bisimple quasi-adequate semigroup of type-E a q ^*-bisimple IC semigroup of type-E. In this paper, we characterize q ^*-bisimple semigroups by using some kind of generalized Bruck-Reilly extensions. As a consequence, some results concerning *-bisimple type-A ω-semigroups given by Asibong-Ibe (Semigroup Forum 31:99–117, 1985) are generalized.     

Decomposition of operator semigroups on W*-algebras

We consider semigroups of operators on a W^∗-algebra and prove, under appropriate assumptions, the existence of a Jacobs-DeLeeuw-Glicksberg type decomposition. This decomposition splits the algebra into a “stable” and “reversible” part with respect to the semigroup and yields, among others, a structural approach to the Perron-Frobenius spectral theory for completely positive operators on W^∗-algebras.     

On ℘-regular semigroups having regular *-transversals

Our aim is to investigate the properties and structure of the ℘-regular semigroups having regular *-transversals. We first give two kinds of definitions for regular *-transversals and prove that they are equivalent. A condition for two regular *-transversals of a ℘-regular semigroup to be isomorphic is given. The *-congruences on the ℘-regular semigroups having regular *-transversals are characterized by using the *-congruence triples. In particular, we describe the idempotent separating congruences and the group congruences on such ℘-regular semigroups.     

Long-time asymptotic properties of dynamical semigroups onW *-algebras

Mathematische Zeitschrift -     

Some notes on strongly E*-unitary inverse semigroups

In this paper, we give a direct proof that every strongly inverse semigroup can be embedded into a 0-semidirect product of a semilattice with zero by a group. As a corollary, we obtain a new proof of the structure theory of strongly inverse semigroups described in [1]. We also prove that the strongly inverse semigroups are precisely inverse semigroups equipped with a , idempotent pure prehomomorphism to a primitive inverse semigroup.     

A class of regular semigroups with regular *- transversals

Let S be a regular semigroup. If there is a subsemigroup S ^* of S and a unary operation * in S satisfying: (1) x ^* ∈ S ^* \cap V_ S ^* (x) for all x∈ S; (2) (x ^* ) ^* =x for all x∈ S ^* ; (3) (x ^* y) ^* =y ^* x ^** and (xy ^* ) ^* =y ^** x ^* for all x,y∈ S, then S ^* is called a regular *- transversal of S ; if (3) is replaced with (xy) ^* =y ^* x ^* for all x,y∈ S, then S ^* is called a strongly regular *- transversal of S. In this paper we consider the class of regular semigroups with a strongly regular *- transversal. It is proved that these semigroups are P - regular semigroups. We characterize the structure of the regular semigroups with a strongly regular *- transversal.     

Reduced C*-algebras of Fell bundles over inverse semigroups

We construct a weak conditional expectation from the section C*-algebra of a Fell bundle over a unital inverse semigroup to its unit fibre. We use this to define the reduced C*-algebra of the Fell bundle. We study when the reduced C*-algebra for an inverse semigroup action on a groupoid by partial equivalences coincides with the reduced groupoid C*-algebra of the transformation groupoid, giving both positive results and counterexamples.     

0-范畴E-酉逆半群的结构

本证明了每一个0-范畴E-酉逆半群能被嵌入到半格与本原逆半群的λ一半直积中。     

一类变种半群上的格林*关系

研究了一类变换半群POPE(X;θ)上的格林*关系,利用格林*关系的定义,得到了半群POPE(X;θ)上元素之间存在格林*关系的条件,这些结果推广了这类变换半群上的格林关系.     

Quotient semigroups and extension semigroups

We discuss properties of quotient semigroup of abelian semigroup from the viewpoint of C ^*-algebra and apply them to a survey of extension semigroups. Certain interrelations among some equivalence relations of extensions are also considered.     

Invariant semigroups of orthodox semigroups

We consider, in a right inverse semigroupS with a multiplicative inverse transversalS ^o, the notion of anS ^o-invariant subsemigroup and use this to describe all the left amenable orders definable onS. The results obtained, together with their duals, are used to prove that ifS is an orthodox semigroup with a multiplicative inverse transversalS ^o, then every amenable order onS ^o can be extended to a unique amenable order onS. NATO Collaborative Research Grant 910765 is gratefully acknowledged. The second-named author also gratefully acknowledges support from the Calouste Gulbenkian Foundation, Lisbon.