Polynomially Dependent Homomorphisms and Frobenius $n$-Homomorphisms

We introduce and study polynomially dependent homomorphisms, which are special linear maps between associative algebras with identity. The multiplicative structure is much involved in the definition of such homomorphisms (we consider only the case of maps $f\colon A\to B$ with commutative $B$). The most important particular case of these maps are the Frobenius $n$-homomorphisms, which were introduced by V. M. Buchstaber and E. G. Rees in 1996–1997. A 1-homomorphism $f\colon A\to B$ is just an algebra homomorphism (the algebra $B$ is commutative). A typical example of an $n$-homomorphism is given by the sum of $n$ algebra homomorphisms, $f=f_1+\dots+f_n$, $f_i\colon A\to B$, $1\leq i\leq n$. Another example is the trace of $n\times n$ matrices over a field $R$ of characteristic zero, $\mathrm{tr}\colon M_n(R)\to R$, and, more generally, the character of any $n$-dimensional representation, $\mathrm{tr}\rho\colon A\to R$, $\rho\colon A\to M_n(R)$. The properties of $n$-homomorphisms (some of which were proved by Buchstaber and Rees under additional conditions) are derived, and a general theory of polynomially dependent homomorphisms is developed. One of the main results of the paper is a uniqueness theorem, which distinguishes the classes of $n$-homomorphisms among all polynomially dependent homomorphisms by a single natural completeness condition. As a topological application of $n$-homomorphisms, we consider the theory of $n$-homomorphisms between commutative $C^*$-algebras with identity. We prove that the norm of any such $n$-homomorphism is equal to $n$ and describe the structure of all such $n$-homomorphisms, which generalizes the classical Gelfand transform (the case of $n=1$). An interesting fact discovered is that the Gelfand transform, which is a functorial bijection between appropriate spaces of maps, becomes a homeomorphism after considering natural topologies on these spaces.