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# $*$-Subvarieties of the Variety Generated by $\bigl( M_2(\mathbb{K}),t \bigr)$

Published:2003-02-01
Printed: Feb 2003
• Francesca Benanti
• Onofrio M. Di Vincenzo
• Vincenzo Nardozza
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## Abstract

Let $\mathbb{K}$ be a field of characteristic zero, and $*=t$ the transpose involution for the matrix algebra $M_2 (\mathbb{K})$. Let $\mathfrak{U}$ be a proper subvariety of the variety of algebras with involution generated by $\bigl( M_2 (\mathbb{K}),* \bigr)$. We define two sequences of algebras with involution $\mathcal{R}_p$, $\mathcal{S}_q$, where $p,q \in \mathbb{N}$. Then we show that $T_* (\mathfrak{U})$ and $T_* (\mathcal{R}_p \oplus \mathcal{S}_q)$ are $*$-asymptotically equivalent for suitable $p,q$.
 Keywords: algebras with involution, asymptotic equivalence
 MSC Classifications: 16R10 - $T$-ideals, identities, varieties of rings and algebras 16W10 - Rings with involution; Lie, Jordan and other nonassociative structures [See also 17B60, 17C50, 46Kxx] 16R50 - Other kinds of identities (generalized polynomial, rational, involution)