Find an explicit orthonormal basis of the separable Hilbert space

by applying the Gram-Schmidt procedure to the polynomials [math]f_n=x^n[/math], with [math]n\in\mathbb N[/math].

Given a Hilbert space [math]H[/math], prove that we have embeddings of [math]*[/math]-algebras as follows, which are both proper, unless [math]H[/math] is finite dimensional:

Also, prove that in this picture the adjoint operation [math]T\to T^*[/math] takes a very simple form, namely [math](M^*)_{ij}=\overline{M}_{ji}[/math] at the level of the associated matrices.

Prove that for the usual matrices [math]A,B\in M_N(\mathbb C)[/math] we have

where [math]\sigma^+[/math] denotes the set of eigenvalues, taken with multiplicities.

Prove that an operator [math]T\in B(H)[/math] satisfies the condition

for any [math]x\in H[/math] precisely when it is positive in our sense, [math]\sigma(T)\in[0,\infty)[/math].

Prove that the Pontrjagin dual of [math]\mathbb Z_N[/math] is this group itself

and work out the details of the subsequent isomorphism [math]C^*(\mathbb Z_N)\simeq C(\mathbb Z_N)[/math].

Find a discrete group [math]\Gamma[/math] such that the quotient map

is not an isomorphism.