Thesis (M. Sc.)--School of Mathematics and Statistics, Faculty of Science, University of Sydney, 2006. / Bibliography: leaves 53-54.

Thesis (M. Sc.)--University of Sydney, 2006. / Title from title screen (viewed 30 May 2008). Submitted in fulfilment of the requirements for the degree of Master of Science to the School of Mathematics and Statistics, Faculty of Science. Degree awarded 2006; thesis submitted 2005. Includes bibliographical references. Also available in print form.

Much work has been done on the ℓ¹-algebras of groups, but much less on ℓ¹-algebras of semigroups. This thesis studies those of inverse semigroups, also known as generalised groups, with emphasis on the involutive structure. Where results extend to the semigroup ring, I extend them. I determine the characters of a semilattice in terms of its order structure. The simplest suffice to separate its ℓ¹-algebra. I also determine the algebra's minimal idempotents. I introduce a generalisation of Banach *-algebras which has good hereditary properties and includes the inverse semi groups rings. These latter have an ultimate identity which can be used to test for representability. Involutive semigroups with s*s an idempotent yield inverse semi groups when quotiented by the congruence induced by their algebras' *-radical. The left regular *-representation of inverse seroigroups is faithful and acts like that of groups. The corresponding idea of amenability coincides with the traditional one. Brandt semi groups have the weak containment property iff the associated group does. The relationship of ideals to weak containment is studied, and inverse semigroups with well ordered semilattices are shown to have the property if all their subgroups do. The converse is extended for Clifford semigroups. Symmetry and related ideas are considered, and basic results proved for the above mentioned generalisation, and a better version for a possibly more restricted generalisation. The symmetry of an ℓ¹-algebra of an E-unitary inverse semi group is shown to depend on the symmetry of the ℓ¹-algebra of its maximal group homomorphic image if the semilattice has a certain structure or the semigroup is a Clifford semigroup. Inverse semi groups with well ordered semilattices are shown to have symmetric ℓ¹-algebra if all the subgroups do. Finally, some topologically simple ℓ¹-algebras and simple semigroup rings are constructed, extending results on simple inverse semigroup rings.

No description available.

Thesis (M.A.)--Kutztown University of Pennsylvania, 1994. / Source: Masters Abstracts International, Volume: 45-06, page: 3173. Abstract precedes thesis as [2] preliminary leaves. Typescript. Includes bibliographical references (leaves 52-53).

An integral domain D is weakly integrally closed if whenever there is an element x in the quotient field of D and a nonzero finitely generated ideal J of D such that xJ J2, then x is in D. We define weakly integrally closed numerical monoids similarly. If a monoid algebra is weakly integrally closed, then so is the monoid. A pattern F of finitely many 0's and 1's is forbidden if whenever the characteristic binary string of a numerical monoid M contains F, then M is not weakly integrally closed. Any stretch of the pattern 11011 is forbidden. A numerical monoid M is weakly integrally closed if and only if it has a forbidden pattern. For every finite set S of forbidden patterns, there exists a monoid that is not weakly integrally closed and that contains no stretch of a pattern in S. It is shown that particular monoid algebras are weakly integrally closed. / by Mary E. Hopkins. / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.

by Chen Yuqun. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references and index. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

by Au Yun-Nam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 133-137). / Introduction --- p.iv / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- Some Semigroup Properties --- p.1 / Chapter 1.2 --- General Properties of Semigroup Algebras --- p.5 / Chapter 1.3 --- Group Algebras --- p.7 / Chapter 1.3.1 --- Some Basic Properties of Groups --- p.7 / Chapter 1.3.2 --- General Properties of Group Algebras --- p.8 / Chapter 1.3.3 --- Δ-Method for Group Algebras --- p.10 / Chapter 1.4 --- Graded Algebras --- p.12 / Chapter 1.5 --- Crossed Products and Smash Products --- p.14 / Chapter 2 --- Radicals of Graded Rings --- p.17 / Chapter 2.1 --- Jacobson Radical of Crossed Products --- p.17 / Chapter 2.2 --- Graded Radicals and Reflected Radicals --- p.18 / Chapter 2.3 --- Radicals of Group-graded Rings --- p.24 / Chapter 2.4 --- Algebras Graded by Semilattices --- p.26 / Chapter 2.5 --- Algebras Graded by Bands --- p.27 / Chapter 2.5.1 --- Hereditary Radicals of Band-graded Rings --- p.27 / Chapter 2.5.2 --- Special Band-graded Rings --- p.30 / Chapter 3 --- Radicals of Semigroup Algebras --- p.34 / Chapter 3.1 --- Radicals of Polynomial Rings --- p.34 / Chapter 3.2 --- Radicals of Commutative Semigroup Algebras --- p.36 / Chapter 3.2.1 --- Commutative Cancellative Semigroups --- p.37 / Chapter 3.2.2 --- General Commutative Semigroups --- p.39 / Chapter 3.2.3 --- The Nilness and Semiprimitivity of Commutative Semigroup Algebras --- p.45 / Chapter 3.3 --- Radicals of Cancellative Semigroup Algebras --- p.48 / Chapter 3.3.1 --- Group of Fractions of Cancellative Semigroups --- p.48 / Chapter 3.3.2 --- Jacobson Radical of Cancellative Semigroup Algebras --- p.54 / Chapter 3.3.3 --- Subsemigroups of Polycyclic-by-Finite Groups --- p.57 / Chapter 3.3.4 --- Nilpotent Semigroups --- p.59 / Chapter 3.4 --- Radicals of Algebras of Matrix type --- p.62 / Chapter 3.4.1 --- Properties of Rees Algebras --- p.62 / Chapter 3.4.2 --- Algebras Graded by Elementary Rees Matrix Semigroups --- p.65 / Chapter 3.5 --- Radicals of Inverse Semigroup Algebras --- p.68 / Chapter 3.5.1 --- Properties of Inverse Semigroup Algebras --- p.69 / Chapter 3.5.2 --- Radical of Algebras of Clifford Semigroups --- p.72 / Chapter 3.5.3 --- Semiprimitivity Problems of Inverse Semigroup Algebras --- p.73 / Chapter 3.6 --- Other Semigroup Algebras --- p.76 / Chapter 3.6.1 --- Completely Regular Semigroup Algebras --- p.76 / Chapter 3.6.2 --- Separative Semigroup Algebras --- p.77 / Chapter 3.7 --- Radicals of Pi-semigroup Algebras --- p.80 / Chapter 3.7.1 --- PI-Algebras --- p.80 / Chapter 3.7.2 --- Permutational Property and Algebras of Permutative Semigroups --- p.80 / Chapter 3.7.3 --- Radicals of PI-algebras --- p.82 / Chapter 4 --- Finiteness Conditions on Semigroup Algebras --- p.85 / Chapter 4.1 --- Introduction --- p.85 / Chapter 4.1.1 --- Preliminaries --- p.85 / Chapter 4.1.2 --- Semilattice Graded Rings --- p.86 / Chapter 4.1.3 --- Group Graded Rings --- p.88 / Chapter 4.1.4 --- Groupoid Graded Rings --- p.89 / Chapter 4.1.5 --- Semigroup Graded PI-Algebras --- p.91 / Chapter 4.1.6 --- Application to Semigroup Algebras --- p.92 / Chapter 4.2 --- Semiprime and Goldie Rings --- p.92 / Chapter 4.3 --- Noetherian Semigroup Algebras --- p.99 / Chapter 4.4 --- Descending Chain Conditions --- p.107 / Chapter 4.4.1 --- Artinian Semigroup Graded Rings --- p.107 / Chapter 4.4.2 --- Semilocal Semigroup Algebras --- p.109 / Chapter 5 --- Dimensions and Second Layer Condition on Semigroup Algebras --- p.119 / Chapter 5.1 --- Dimensions --- p.119 / Chapter 5.1.1 --- Gelfand-Kirillov Dimension --- p.119 / Chapter 5.1.2 --- Classical Krull and Krull Dimensions --- p.121 / Chapter 5.2 --- The Growth and the Rank of Semigroups --- p.123 / Chapter 5.3 --- Dimensions on Semigroup Algebras --- p.124 / Chapter 5.4 --- Second Layer Condition --- p.128 / Notations and Abbreviations --- p.132 / Bibliography --- p.133

An n-tuple of operators V=(V_1,…,V_n) acting on a Hilbert space H is said to be isometric if the corresponding row operator is an isometry. A free semigroup algebra is the weakly closed algebra generated by an isometric n-tuple V. The structure of a free semigroup algebra contains a great deal of information about V. Thus it is natural to study this algebra in order to study V. A free semigroup algebra is said to be analytic if it is isomorphic to the noncommutative analytic Toeplitz algebra, which is a higher-dimensional generalization of the classical algebra of bounded analytic functions on the complex unit disk. This notion of analyticity is of central importance in the general theory of free semigroup algebras. A vector x in H is said to be wandering for an isometric n-tuple V if the set of words in the entries of V map x to an orthonormal set. As in the classical case, the analytic structure of the noncommutative analytic Toeplitz algebra is determined by the existence of wandering vectors for the generators of the algebra. In the first part of this thesis, we prove the following dichotomy: either an isometric n-tuple V has a wandering vector, or the free semigroup algebra it generates is a von Neumann algebra. This implies the existence of wandering vectors for every analytic free semigroup algebra. As a consequence, it follows that every free semigroup algebra is reflexive, in the sense that it is completely determined by its invariant subspace lattice. In the second part of this thesis we prove a decomposition for an isometric tuple of operators which generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. The key result is an operator-algebraic characterization of an absolutely continuous isometric tuple in terms of analyticity. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.

An n-tuple of operators V=(V_1,…,V_n) acting on a Hilbert space H is said to be isometric if the corresponding row operator is an isometry. A free semigroup algebra is the weakly closed algebra generated by an isometric n-tuple V. The structure of a free semigroup algebra contains a great deal of information about V. Thus it is natural to study this algebra in order to study V. A free semigroup algebra is said to be analytic if it is isomorphic to the noncommutative analytic Toeplitz algebra, which is a higher-dimensional generalization of the classical algebra of bounded analytic functions on the complex unit disk. This notion of analyticity is of central importance in the general theory of free semigroup algebras. A vector x in H is said to be wandering for an isometric n-tuple V if the set of words in the entries of V map x to an orthonormal set. As in the classical case, the analytic structure of the noncommutative analytic Toeplitz algebra is determined by the existence of wandering vectors for the generators of the algebra. In the first part of this thesis, we prove the following dichotomy: either an isometric n-tuple V has a wandering vector, or the free semigroup algebra it generates is a von Neumann algebra. This implies the existence of wandering vectors for every analytic free semigroup algebra. As a consequence, it follows that every free semigroup algebra is reflexive, in the sense that it is completely determined by its invariant subspace lattice. In the second part of this thesis we prove a decomposition for an isometric tuple of operators which generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. The key result is an operator-algebraic characterization of an absolutely continuous isometric tuple in terms of analyticity. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.