On some topological \(*\)-algebra of power series of \(N\) anti-commuting generators (Q2767628)

The \(*\)-algebra \(A\) is called topological \(*\)-algebra if \(A\) is a topological linear space over the field of real or complex numbers and the mapping \(A\ni f\to f^{*}\in A\) (involution) is continuous and the mapping \(A\times A\ni (f,g)\to f\cdot g\in A\) (product) is separately continuous. An element \(f\in A\) is called entire element if for any \(\lambda>0\) the set \(\{(\lambda f)^{n}/n!, n=1,2,\ldots\}\) is bounded in \(A\). The author construct a topological \(*\)-algebra of power series \(\sum_{(k_1,\ldots,k_{N})\in I}a_{(k_1,\ldots,k_{N})}u_1^{k_1}u_2^{k_2}\ldots u_{N}^{k_{N}}\), \(a_{(k_1,\ldots,k_{N})}\in C\) of \(N\) anti-commuting Hermite generators \(u_1,\ldots, u_{N}\) \((u_{i}u_{j}=-u_{j}u_{i}\), \(u_{i}^{*}=u_{i}\), \(i\neq j)\). It is proved that the constructed algebra consists of entire elements and the center of this algebra is described.