Herz-Schur Multipliers of Fell Bundles and the Nuclearity of the Full C*-Algebras
Issue:
Volume 7, Issue 2, April 2021
Pages:
17-29
Received:
11 January 2021
Accepted:
18 March 2021
Published:
7 April 2021
Abstract: In this paper we develop the notion of Schur multipliers and Herz-Schur multipliers to the context of Fell bundle, as a generalization of the theory of multipliers of locally compact groups and crossed products. We prove a characterization theorem of this generalized Schur multiplier in terms of the representation of Fell bundles. In order to prove this characterization theorem we define a new class of completely bounded maps; and discuss in detail of its properties. In this process, by the way, we give a new proof of Stinpring’s Theorem of non-unital version. Then we investigate the transference theorem of Schur multipliers and Herz-Schur multipliers, which is a generalization of the transference theorem well-known either in the group case or crossed products. We use the notion of multipliers to define an approximation property of Fell bundles. Then we give a necessary and sufficient condition if the reduced cross-sectional algebra of a Fell bundle over a discrete groups is nuclear in terms of this generalized notion. This is a generalization of the classical theorem concerning the amenability of locally compact groups. As an application, we prove that for a Fell bundle, if its cross-sectional algebra is nuclear, then for any subgroup of the group on which the Fell bundle is defined, the cross-sectional algebra of the restricted Fell bundle on this subgroup is nuclear.
Abstract: In this paper we develop the notion of Schur multipliers and Herz-Schur multipliers to the context of Fell bundle, as a generalization of the theory of multipliers of locally compact groups and crossed products. We prove a characterization theorem of this generalized Schur multiplier in terms of the representation of Fell bundles. In order to prove...
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Subsets of Scales in Compositions Constructed by Similarity
Issue:
Volume 7, Issue 2, April 2021
Pages:
30-39
Received:
14 April 2021
Accepted:
5 May 2021
Published:
27 May 2021
Abstract: We present techniques in musical composition based on subsets of scales, built on theoretical notions, together with a number of examples. The techniques we describe are for constructing compositions with reference to memory, via similarity. We begin with some technical elements: after introducing the technique of intersecting accompaniments, we describe similarity concatenation compositions, which are special compositions constructed via similarity. We then outline a method to solve the problem of approximating scales with frequency ratios generated by rational numbers with small numerators and denominators, via equal temperament. As well as the standard solution via 12 tone equal temperament, we present a solution via 31 tone equal temperament. We then introduce the notion of a connected triheptad, generalising the tonic, subdominant and dominant of the major scale. We next present some examples of the notions previously introduced. Example 1 features a connected triheptad, and Example 2 features a connected triheptad, a similarity concatenation composition, and an intersecting accompaniment. There follows a section on cubist sets, featuring a returning similarity concatenation composition. We then move a conceptual level higher: we consider the concatenation of similarity concatenation compositions via similarity. This is reminiscent of higher dimensional algebra, and there follows a formal approach to higher dimensional relations, together with an example in 31 tone equal temperament using the formalism described earlier. We use the formalism of braids for our higher dimensional relations. We end with a section on musical applications of paths in graphs, generalising the chromatic scale.
Abstract: We present techniques in musical composition based on subsets of scales, built on theoretical notions, together with a number of examples. The techniques we describe are for constructing compositions with reference to memory, via similarity. We begin with some technical elements: after introducing the technique of intersecting accompaniments, we de...
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