This thesis has two parts. The first part deals with some questions in amenability. We show that for a Banach algebra A with a bounded approximate identity, the amenability of the projective tensor product of A with itself, the amenability of the projective tensor product of A with A^op and the amenability of A are equivalent. Also if A is a closed ideal in a commutative Banach algebra B, then the (weak) amenability of the projective tensor product of A and B implies the (weak) amenability of A. Finally, we show that if the Banach algebra A is amenable through multiplication π then is also amenable through any multiplication ρ such that the norm of π-ρ is less than 1/( 11). The second part deals with questions in generalized notions of amenability such as approximate amenability and bounded approximate amenability. First we prove some new results about approximately amenable Banach algebras. Then we state a characterization of approximately amenable Banach algebras and a characterization of boundedly approximately amenable Banach algebras. Finally, we prove that B(l^p (E)) is not approximately amenable for Banach spaces E with certain properties. As a corollary of this part, we give a new proof that B(l^2) is not approximately amenable.

This thesis has two parts. The first part deals with some questions in amenability. We show that for a Banach algebra A with a bounded approximate identity, the amenability of the projective tensor product of A with itself, the amenability of the projective tensor product of A with A^op and the amenability of A are equivalent. Also if A is a closed ideal in a commutative Banach algebra B, then the (weak) amenability of the projective tensor product of A and B implies the (weak) amenability of A. Finally, we show that if the Banach algebra A is amenable through multiplication π then is also amenable through any multiplication ρ such that the norm of π-ρ is less than 1/( 11). The second part deals with questions in generalized notions of amenability such as approximate amenability and bounded approximate amenability. First we prove some new results about approximately amenable Banach algebras. Then we state a characterization of approximately amenable Banach algebras and a characterization of boundedly approximately amenable Banach algebras. Finally, we prove that B(l^p (E)) is not approximately amenable for Banach spaces E with certain properties. As a corollary of this part, we give a new proof that B(l^2) is not approximately amenable.

Abstract harmonic analysis is mainly concerned with the study of locally compact groups, their unitary representations, and the function spaces associated with them. The Fourier and Fourier-Stieltjes algebras are two of the most important function spaces associated with a locally compact group. The Rajchman algebra associated with a locally compact group is defined to be the set of all elements of the Fourier-Stieltjes algebra which vanish at infinity. This is a closed, complemented ideal in the Fourier-Stieltjes algebra that contains the Fourier algebra. In the Abelian case, the Rajchman algebras can be identified with the algebra of Rajchman measures on the dual group. Such measures have been widely studied in the classical harmonic analysis. In contrast, for non-commutative locally compact groups little is known about these interesting algebras. In this thesis, we investigate certain Banach algebra properties of Rajchman algebras associated with locally compact groups. In particular, we study various amenability properties of Rajchman algebras, and observe their diverse characteristics for different classes of locally compact groups. We prove that amenability of the Rajchman algebra of a group is equivalent to the group being compact and almost Abelian, a property that is shared by the Fourier-Stieltjes algebra. In contrast, we also present examples of large classes of locally compact groups, such as non-compact Abelian groups and infinite solvable groups, for which Rajchman algebras are not even operator weakly amenable. Moreover, we establish various extension theorems that allow us to generalize the previous result to all non-compact connected SIN-groups. Finally, we investigate the spectral behavior of Rajchman algebras associated with Abelian locally compact groups, and construct point derivations at certain elements of their spectrum using Varopoulos’ decompositions for Rajchman algebras. Having constructed similar decompositions, we obtain analytic discs around certain idempotent characters of Rajchman algebras. These results, and others that we obtain, illustrate the inherent distinction between the Rajchman algebra and the Fourier algebra of many locally compact groups.

Abstract harmonic analysis is mainly concerned with the study of locally compact groups, their unitary representations, and the function spaces associated with them. The Fourier and Fourier-Stieltjes algebras are two of the most important function spaces associated with a locally compact group. The Rajchman algebra associated with a locally compact group is defined to be the set of all elements of the Fourier-Stieltjes algebra which vanish at infinity. This is a closed, complemented ideal in the Fourier-Stieltjes algebra that contains the Fourier algebra. In the Abelian case, the Rajchman algebras can be identified with the algebra of Rajchman measures on the dual group. Such measures have been widely studied in the classical harmonic analysis. In contrast, for non-commutative locally compact groups little is known about these interesting algebras. In this thesis, we investigate certain Banach algebra properties of Rajchman algebras associated with locally compact groups. In particular, we study various amenability properties of Rajchman algebras, and observe their diverse characteristics for different classes of locally compact groups. We prove that amenability of the Rajchman algebra of a group is equivalent to the group being compact and almost Abelian, a property that is shared by the Fourier-Stieltjes algebra. In contrast, we also present examples of large classes of locally compact groups, such as non-compact Abelian groups and infinite solvable groups, for which Rajchman algebras are not even operator weakly amenable. Moreover, we establish various extension theorems that allow us to generalize the previous result to all non-compact connected SIN-groups. Finally, we investigate the spectral behavior of Rajchman algebras associated with Abelian locally compact groups, and construct point derivations at certain elements of their spectrum using Varopoulos’ decompositions for Rajchman algebras. Having constructed similar decompositions, we obtain analytic discs around certain idempotent characters of Rajchman algebras. These results, and others that we obtain, illustrate the inherent distinction between the Rajchman algebra and the Fourier algebra of many locally compact groups.

No description available.

In this thesis we discuss amenability properties of the Banach algebra-valued continuous functions on a compact Hausdorff space X. Let A be a Banach algebra. The space of A-valued continuous functions on X, denoted by C(X,A), form a new Banach algebra. We show that C(X,A) has a bounded approximate diagonal (i.e. it is amenable) if and only if A has a bounded approximate diagonal. We also show that if A has a compactly central approximate diagonal then C(X,A) has a compact approximate diagonal. We note that, unlike C(X), in general C(X,A) is not a C*-algebra, and is no longer commutative if A is not so. Our method is inspired by a work of M. Abtahi and Y. Zhang. In addition to the above investigation, we directly construct a bounded approximate diagonal for C0(X), the Banach algebra of the closure of compactly supported continuous functions on a locally compact Hausdorff space X.

The first part of our work is a statistical and geometric study of properties of Thompson's Group F. We enumerate the number of elements of F which are represented by a reduced pair of n-caret trees, and give asymptotic estimates. We also discuss the effects on word length and number of carets of right multiplication by a standard generator x0 or x1. We enumerate the average number of carets along the left edge of an n-caret tree, and use an Euler transformation to make some conjectures relating to right multiplication by a generator. We describe a computer algorithm which produces Fordham's Table, and discuss using the computer algorithm to find a corresponding Fordham's Table for different generating sets for F. We expound upon the work of Cleary and Taback by completely classifying dead end elements of Thompson's group, and use the classification to discuss the spread of dead end elements and describe interesting elements we call deep roots. We discuss how deep roots may aid in answering the amenability problem for Thompson's group. The second part of our work deals with random facet pairings of simplexes. We show that a random endpoint pairings of segments most often results in a disconnected one-manifold, and relate this to a game called "The Human Knot." When the dimension of the simplexes is greater than 1, however, a random facet pairing most often results in a connected pseudo manifold. This result can be stated in terms of graph theory as follows. Most regular multi graphs are connected, as long as the common valence is at least three.

Let X be a space and let S ⊂ X with a measure of set size |S| and boundary size |∂S|. Fix a set C ⊂ X called the constraining set. The constrained isoperimetric problem asks when we can find a subset S of C that maximizes the Følner ratio FR(S) = |S|/|∂S|. We consider different measures for subsets of R^2,R^3,Z^2,Z^3 and describe the properties that must be satisfied for sets S that maximize the Folner ratio. We give explicit examples.

No description available.

Clinical research suggests that adolescent offenders often do not view their criminal behaviors as problematic and, therefore, are not motivated for treatment. Although customarily defined as a static characteristic, the transtheoretical model (TTM) proposes treatment amenability is dynamic and can be achieved through tailored interventions that motivate individuals for treatment. The current study examines the predictive validity of TTM measures for adolescent offenders at a maximum security correctional facility. In particular, the Stages of Change Scale (SOCS) and Decisional Balance for Adolescent Offenders (DBS-AO) were compared with a more traditional assessment tool utilized in evaluating treatment amenability of juvenile offenders (i.e., Risk-Sophistication-Treatment Inventory; RSTI). One hundred adolescent offenders from the Gainesville State School completed two waves of data collection with a 3-month time interval. Information was collected on offenders' treatment progress between waves. Consistent with TTM research, predictors of treatment progress included low scores on the Cons scale on the DBS-AO and on the Precontemplation scale on the SOCS. Participants in the most advanced levels of treatment also scored high on the Sophistication-Maturity scale on the RSTI and the Impression Management scale on the Paulhus Deception Scale.