Finite-dimensional absolute-valued algebras
A real (not necessarily associative) algebra \(A\neq 0\) is said to be an \textit{absolute-valued algebra} if \(A\) is a normed vector space such that \(\| xy\| = \| x\| \, \| y\| \) for all \(x, \, y \in A.\) The case that \(A\) has finite dimension has been considered by A. A. Albert. First he proves that a finite-dimensional absolute-valued algebra having a unit element is necessarily isomorphic to \({\mathbb R}\), \({\mathbb C}\), \({\mathbb H}\) or \({\mathbb O}\). Furthermore, in the finite-dimensional (but not necessarily unital) case, he takes a norm-one element \(a\in A\) and he introduces a new product \(\bullet \) in \(A\) defined by \(x\bullet y = R_a^{-1}(x)L_a^{-1}(y)\), where \(R_a\) (resp. \(L_a\)) is the linear operator of right (resp. left) multiplication \(R_a : x\mapsto xa\) (resp. \(L_a : x\mapsto ax\)). Now he observes that \((A, \bullet )\) is an absolute valued algebra with a unit element (namely, \(a^2\)). In consequence, he obtains that if \(C\) is a finite-dimensional absolute-valued algebra, with product denoted by \(\diamond \), then there are two isometric linear maps \(\varphi \) and \(\psi\) and a new product in the normed vector space \(C\), denoted by juxtaposition, which makes of this vector space a unitary absolute-valued algebra \(B\) such that \(x\diamond y = \varphi (x)\psi (y)\) for all \(x, \, y\in C\). On the other hand, the existence of nonzero idempotents in every finite-dimensional absolute-valued algebra implies easily that \(\varphi \) and \(\psi\) can be chosen satisfying, in addition, \(\varphi (1) = 1=\psi (1)\). This characterization of \(C\) in terms of the unital absolute-valued algebra \(B\) has many interesting consequences on the structure of \(C.\) For example, it makes possible an easy determination of the center \(Z(C)\) of \(C,\) and it allows to establish a criterion for isomorphism of two of these absolute-valued algebras \(C_1\) and \(C_2 .\) Well-known examples of no-unitary absolute valued algebras are \(^\star {\mathbb C}\), \(^\star {\mathbb H}\), \(^\star {\mathbb O}\), \({\mathbb C}^\star \), \({\mathbb H}^\star \), \({\mathbb O}^\star \), \(\overset {*}{\mathbb C}\), \(\overset {*}{\mathbb H}\) and \(\overset {*}{\mathbb O}\). They are obtained by a certain modification of the product of the corresponding unitary absolute valued algebra \(B\) of the same dimension (nevertheless, relative to the new product the unit \(1\) of \(B\) does not remain a unit element in \(^\star {\mathbb C}\), \(^\star {\mathbb H}\), \(^\star {\mathbb O}\), \({\mathbb C}^\star \), \({\mathbb H}^\star \), \({\mathbb O}^\star \), \(\overset {*}{\mathbb C}\), \(\overset {*}{\mathbb H}\) or \(\overset {*}{\mathbb O}\), but only a nonzero idempotent). The starting point of the paper under review deals with the possibility of considering only positive linear isometries \(\varphi \) and \(\psi\) fixing \(1\) in the above description of \(C\), but \(B\) being either a unitary absolute-valued algebra or an absolute valued algebra isomorphic to \(^\star {\mathbb C}\), \(^\star {\mathbb H}\), \(^\star {\mathbb O}\), \({\mathbb C}^\star \), \({\mathbb H}^\star \), \({\mathbb O}^\star \), \(\overset {*}{\mathbb C}\), \(\overset {*}{\mathbb H}\) or \(\overset {*}{\mathbb O}\). A suitable reformulation of this possibility can be stated in the following way: the product \(\diamond \) of a finite-dimensional absolute valued algebra \(C\) can be given in terms of that of a unitary absolute valued algebra \(B\) and two \(1\)-preserving \textit{positive} isometric linear maps \(\varphi \) and \(\psi \) in some of the following ways: \[ (\text{I})\;x\diamond y = \varphi (x)\psi (y);\quad (\text{II})\;x\diamond y = \varphi (\bar{x})\psi (y);\quad (\text{III})\;x\diamond y = \varphi (x)\psi (\bar{y});\quad (\text{IV})\;x\star y = \varphi (\bar{x})\psi (\bar{y}). \] Furthermore, it is possible to show that if \(\dim C\geq 2\) then two finite-dimensional absolute-valued algebras of different types (I)--(IV) cannot be isomorphic. In order to treat the last assertion the authors first consider a \textit{relation of homotopy} between the different absolute-valued products that can be defined on a same vector space. Roughly speaking two of these products are \textit{homotopic} if one of them can be deformed to the other throughout a uniparametric family of absolute-valued products. Nevertheless, the same conclusion can be obtained considering the positive or negative character of the operators of left and right multiplication by the nonzero elements of \(C\) (see [\textit{A. Elduque} and \textit{J. M. Pérez}, J. Algebra 190, No. 2, 372--404 (1997; Zbl 0873.17005)]; also [\textit{E. Darpö} and \textit{E. Dieterich}, Math. Nachr. 285, No. 13, 1635--1642 (2012; Zbl 1326.17001)]). In section 6 the authors prove that the class of the \(8\)-dimensional absolute having a \(4\)-dimensional subalgebra agrees with the class of the absolute valued algebras that can be obtained from \(4\)-dimensional absolute valued algebras by a certain generalization of the Cayley-Dickson duplication process.
scientific article; zbMATH DE number 6108356
scientific article