Description:
We introduce the notion of star-fundamental algebra over a field of characteristic zero. We prove that in the framework of the theory of polynomial identities, these algebras are the building blocks of a finite dimensional algebra with involution *. To any star-algebra A is attached a numerical sequence c_n^*(A), n\ge 1, called the sequence of *-codimensions of A. Its asymptotic is an invariant giving a measure of the *-polynomial identities satisfied by A. It is well known that for a PI-algebra such a sequence is exponentially bounded and \exp ^*(A)=\lim _{n\to \infty }\sqrt [n]{c_n^*(A)} can be explicitly computed. Here we prove that if A is a star-fundamental algebra, | \displaystyle C_1n^t\exp ^*(A)^n\le c_n^*(A)\le C_2n^t \exp ^*(A)^n, | where C_1>0,C_2, t are constants and t is explicitly computed as a linear function of the dimension of the skew semisimple part of A and the nilpotency index of the Jacobson radical of A. We also prove that any finite dimensional star-algebra has the same *-identities as a finite direct sum of star-fundamental algebras. As a consequence, by the main result in [J. Algebra 383 (2013), pp. 144-167] we get that if A is any finitely generated star-algebra satisfying a polynomial identity, then the above still holds and, so, \lim _{n\to \infty }\log _n \frac {c_n^*(A)}{\exp ^*(A)^n} exists and is an integer or half an integer.
Language:
English
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