Blends and Alloys

C. R. Math. Rep. Acad. Sci. Canada Vol. 35 (3) 2013, pp. 77–113
(Received: 2012-05-09 , Revised: 2013-07-24)

R. Exel, Departamento de Matem ́atica, Universidade Federal de Santa Catarina, 88040-970 Florian ́opolis SC, Brazil; e-mail:


Given two algebras $A$ and $B$, sometimes assumed to be C\*-algebras, we consider the question of putting algebra or C\*-algebra structures on the tensor product $A\otimes B$. In the C\*-case, assuming $B$ to be two-dimensonal, we characterize all possible such C\*-algebra structures in terms of an action of the cyclic group ${\mathbb Z}_2$. An example related to commuting squares is also discussed.

Si $A$ et $B$ sont deux algèbres (resp. deux C\*-algèbres), nous étudions dans cette note les structures possibles d’algèbre (resp. de C\*-algèbre) qui peuvent être définies sur le produit tensoriel $A\otimes B$. Si $A$ est une C\*-algèbre, nous caractérisons toutes les structures de C\*-algèbre sur le produit tensoriel $A\otimes \mathbb{C}^2$ par une action du groupe cyclique $\mathbb{Z}_2$. Nous présentons aussi un exemple associé aux carrés commutatifs.

Keywords: Algebra, C*-algebra, Jones’ basic construction, algebra structure, alloy, blend, commuting square, conditional expectation, crossed product, index finite type, tensor product

AMS Subject Classification: None of the above; but in this section, 16S99, 46L04

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