LEARNING DESIGN OPERATION MULTIPLICATION ALGEBRA USING WATER CONTEXT, OIL AND STONE STUDENTS CLASS V

The purpose of this study resulted in the learning path of students of algebraic multiplication materials by using water, oil and stone that evolved from informal to formal forms. The method used is a research design consisting of three stages, namely preliminary, design experiment and retrospective analysis. A series of lessons are designed and developed based on conjectures from the learning process and PMRI approach. This research was conducted in SMP N 5 Palembang involving 30 students of class VII. Learning Trajectory (LT) obtained includes three activities that are activities 1. Introduce the objects around that can be used in learning algebraic multiplication such as oil, stone and water, 2 Finding the Concept of Multiplication Algebra and 3. Solving problems related to daily life. The results of the experimental learning show that a series of activities that have been done helps to improve students' understanding of algebraic multiplication learning

Completely bounded bimodule maps and spectral synthesis

We initiate the study of the completely bounded multipliers of the Haagerup tensor product [Formula: see text] of two copies of the Fourier algebra [Formula: see text] of a locally compact group [Formula: see text]. If [Formula: see text] is a closed subset of [Formula: see text] we let [Formula: see text] and show that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] then [Formula: see text] is a set of local spectral synthesis for [Formula: see text]. Conversely, we prove that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] and [Formula: see text] is a Moore group then [Formula: see text] is a set of spectral synthesis for [Formula: see text]. Using the natural identification of the space of all completely bounded weak* continuous [Formula: see text]-bimodule maps with the dual of [Formula: see text], we show that, in the case [Formula: see text] is weakly amenable, such a map leaves the multiplication algebra of [Formula: see text] invariant if and only if its support is contained in the antidiagonal of [Formula: see text].

COMPLEMENTEDLY DENSE IDEALS: DECOMPOSABLE ALGEBRAS

An ideal I of a (non-associative) algebra A is dense if the multiplication algebra of A acts faithfully on I, and is complementedly dense if it is a direct summand of a dense ideal. We prove that every complementedly dense ideal of a semiprime algebra is a semiprime algebra, and determine its central closure and its extended centroid. We also prove that a semiprime algebra is an essential subdirect product of prime algebras if and only if, its extended centroid is a direct product of fields. This result is applied to discuss decomposable algebras with respect to some familiar closures for ideals.

REVISITING HOCHSCHILD COHOMOLOGY FOR ALGEBRA BUNDLES

Hochschild cohomology of an associative algebra bundle with coefficients in a bimodule bundle has been defined and studied in earlier paper. Here, by using cohomological methods, we establish that an algebra bundle is a semidirect product of its radical bundle and a semisimple subalgebra bundle. Further we define multiplication algebra bundle of an algebra bundle and representation of an algebra bundle. We study special representations of an algebra bundle using Hochschild cohomology of an associative algebra bundle with coefficients in a bimodule bundle. We observe that if a representation of an algebra bundle is special then its obstruction is zero. Further we show that a subgroup H of H2(ξ, N) is faithfully represented as a transitive group of translations operating on the set of those equivalence classes of algebra bundle extensions of ξ which determine a given representation [φ, K].

Totally multiplicatively prime algebras

We introduce the totally multiplicatively prime algebras as those normed algebras for which there exists a positive number K such that K‖F‖‖a‖ ≤ ‖WF,a‖ for all F in M(A) (the multiplication algebra of A) and a in A, where WF,a denotes the operator from M(A) into A defined by WF,a(T) = FT(a) for all T in M(A). These algebras are totally prime and their multiplication algebra is ultraprime. We get the stability of the class of totally multiplicatively prime algebras by taking central closure. We prove that prime H*-algebras are totally multiplicatively prime and that the ℓ1-norm is the only classical norm on the free non-associative algebras for which these are totally multiplicatively prime.