Subsemivarieties of $Q$-algebras

A variety is a class of Banach algebras $V$, for which there exists a family of laws $\{\Vert P\Vert\le K_p\}_P$ such that $V$ is precisely the class of all Banach algebras $A$ which satisfies all of the laws (i.e. for all $P$, $\Vert P\Vert _A\le K_p)$. We say that $V$ is an $H$-variety if all of the laws are homogeneous. A semivariety is a class of Banach algebras $W$, for which there exists a family of homogeneous laws $\{\Vert P\Vert\le K_P\}_P$ such that $W$ is precisely the class of all Banach algebras $A$, for which there exists $K>0$ such that for all homogeneous polynomials $P$, $\Vert P\Vert _A\le K^i\cdot K_P$, where $i=\deg(P)$. However, there is no variety between the variety of all $IQ$-algebras and the variety of all $IR$-algebras, which can be defined by homogeneous laws alone. So the theory of semivarieties and the theory of varieties differ significantly. In this paper we shall construct uncountable chains and antichains of semivarieties which are not varieties.