Theorem. Finite *-algebras, states and GNS [fdip-008T]
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  • November 2, 2026

A state in the sense of the real and complex algebraic states is a normalized positive functional. That note assumes no norm and no operator representation. On a finite-dimensional *-algebra the functional itself determines both. The squared length of a class is the value of the state on \(\,a^*a\), and the dimension of the resulting space is the rank of one Gram matrix.

Let \(K=\mathbb R\) or \(K=\mathbb C\), and let \(A\) be a finite-dimensional unital associative \(K\)-algebra with an involution of the kind fixed for a real algebraic state or a complex algebraic state. Let \(\omega \) be a real or complex algebraic state on \(A\), according to the scalar field: \(K\)-linear, normalized in the sense of a normalized linear functional, Hermitian positive, and satisfying \(\omega (a^*a)\ge 0\). Define \(\langle x,y\rangle =\omega (x^*y)\). Let \(N\) be the set of \(x\) with \(\langle x,x\rangle =0\). The quotient \(H_\omega =A/N\) carries the descended inner product, the operator \(\pi _\omega (a)\) sends the class of \(b\) to the class of \(ab\), and \(\Omega \) is the class of \(1\).

The form is positive semidefinite and Hermitian, and \(N\) is a left ideal. For the cyclic vector, \[\langle \pi _\omega (a)\Omega ,\pi _\omega (b)\Omega \rangle =\omega (a^*b).\] For any basis the Gram matrix \(G_{ij}=\omega (e_i^*e_j)\) has \(\dim H_\omega =\operatorname {rank} G\). The squared length of the class of \(a\) is \(\omega (a^*a)\).

Let \(A=\bigoplus _i M_{m_i}(K)\), with componentwise product and with conjugate transpose acting on each block. Suppose each \(\rho _i\) has the form \(\rho _i=c_i^*c_i\), and \(\sum _i\mathrm {Tr}(\rho _i)=1\). The functional \(\omega (a)=\sum _i\mathrm {Tr}(\rho _i a_i)\) is a state, and \[\dim H_\omega =\sum _i m_i\operatorname {rank}\rho _i.\] On \(M_2\), the density \(\mathrm {diag}(2/3,1/3)\) has rank \(2\) and gives dimension \(4\). The density \(\mathrm {diag}(1,0)\) has rank \(1\) and gives dimension \(2\).

Let \(X\) be a finite set and let \(A\) be the algebra of functions \(X\to K\), with pointwise product and pointwise conjugation. Let \(w:X\to \mathbb R_{\geq 0}\) satisfy \(\sum _x w(x)=1\), and set \(\omega (f)=\sum _x w(x)f(x)\). Write \(\operatorname {supp} w\) for the labels with \(w(x)>0\). Then \(H_\omega \) is the space of all functions on that support, with the inner product weighted by \(w\). On a finite set that space is \(\ell ^2(\operatorname {supp} w)\).

Equivariant precision and real types records, inside a symmetry sector, a matrix over the reals, the complexes or the quaternions. This section does not prove that those matrices are the blocks of a decomposition of the observable algebra.

Proof. 

  • November 2, 2026

The form is linear in the second argument because \(\omega \) is.

It is conjugate-linear in the first because \((\lambda x)^*=\overline \lambda \,x^*\).

It is Hermitian because \(\langle y,x\rangle \) equals \(\omega ((x^*y)^*)\), and Hermitian positivity makes that the conjugate of \(\omega (x^*y)\). The square \(\langle x,x\rangle =\omega (x^*x)\) is nonnegative by the state.

The inequality of Cauchy and Schwarz is \(\lvert \langle x,y\rangle \rvert ^2\le \langle x,x\rangle \langle y,y\rangle \). Over \(\mathbb C\), if \(\langle x,y\rangle \) is not zero, choose a scalar \(u\) of modulus \(1\) with \(u\langle x,y\rangle =\lvert \langle x,y\rangle \rvert \), and replace \(y\) by \(uy\). The new square \(\langle y,y\rangle \) is unchanged, and the new inner product is real and nonnegative. Over \(\mathbb R\) no phase is required. For every real \(t\), \[\langle x+ty,x+ty\rangle =\langle x,x\rangle +2t\langle x,y\rangle +t^2\langle y,y\rangle \ge 0.\] If \(\langle y,y\rangle =0\), the values \(2t\langle x,y\rangle \) cannot stay above \(- \langle x,x\rangle \) for every real \(t\) unless \(\langle x,y\rangle =0\). If \(\langle y,y\rangle >0\), complete the square. The same quadratic is \[\langle y,y\rangle \Bigl (t+\frac {\langle x,y\rangle }{\langle y,y\rangle }\Bigr )^2 +\langle x,x\rangle -\frac {\langle x,y\rangle ^2}{\langle y,y\rangle }.\] The squared term can be made zero, so the remaining term is nonnegative. Thus the inequality holds. Over \(\mathbb C\), putting the unimodular factor back into \(y\) multiplies \(\langle x,y\rangle \) by a scalar of modulus \(1\) and does not change \(\langle y,y\rangle \).

If \(\langle x,x\rangle =0\), the inequality forces \(\langle x,y\rangle =0\) for every \(y\). Conversely the case \(y=x\) shows that the radical is exactly \(N\). It is a subspace. If \(x,y\in N\) and \(\lambda \in K\), then \[\begin {aligned} \langle x+\lambda y,x+\lambda y\rangle ={}&\langle x,x\rangle +\lambda \langle x,y\rangle \\ &+\overline {\lambda }\langle y,x\rangle +\lvert \lambda \rvert ^2\langle y,y\rangle . \end {aligned}\] The two squares vanish because \(x,y\in N\). The two cross terms vanish because an element of \(N\) is orthogonal to every vector. It is a left ideal. For \(b\in A\) and \(x\in N\), \(\langle bx,bx\rangle =\omega (x^*b^*bx)=\langle x,(b^*b)x\rangle =0\).

The form therefore descends to the quotient. If \(x-x'\) and \(y-y'\) lie in \(N\), then \(\langle x,y\rangle -\langle x',y'\rangle =\langle x-x',y\rangle +\langle x',y-y'\rangle \). The first summand vanishes because \(x-x'\) is in the radical. The second is the conjugate of \(\langle y-y',x'\rangle \), which vanishes for the same reason. The descended form is positive definite, and finite dimensionality makes the quotient a Hilbert space with no further completion. Its dimension is \(\dim A-\dim N\).

On a basis \(e_1,\ldots ,e_n\), the matrix \(G_{ij}=\langle e_i,e_j\rangle =\omega (e_i^*e_j)\) is Hermitian. A vector \(v\) represents \(x=\sum _i v_i e_i\). Hermitian symmetry gives \(G_{ji}=\overline {G_{ij}}\). The number \(\langle x,e_j\rangle =\sum _i\overline {v_i}G_{ij}\) vanishes for every \(j\) if and only if \(x\in N\): the forward direction was proved for every vector, and the converse is linearity in the second argument. Conjugating those sums and replacing \(\overline {G_{ij}}\) by \(G_{ji}\) produces \((Gv)_j=0\). Rank-nullity gives \(\operatorname {rank} G=n-\dim N=\dim H_\omega \). The class of \(a\) has squared length \(\omega (a^*a)\). The operator \(\pi _\omega (a)\) is well defined because \(N\) is a left ideal, and \(\langle \pi _\omega (a)\Omega ,\pi _\omega (b)\Omega \rangle =\langle a,b\rangle =\omega (a^*b)\).

On the direct sum, the product and the involution act blockwise, so \(\omega (a^*a)=\sum _i\mathrm {Tr}(\rho _i a_i^*a_i)\). The trace is invariant under a cyclic permutation, and \(\mathrm {Tr}(b^*b)\) is the sum of the squared moduli of the entries of \(b\). Hence \[\mathrm {Tr}(\rho _i a_i^*a_i) =\mathrm {Tr}(c_i a_i^*a_i c_i^*) =\mathrm {Tr}\bigl ((a_i c_i^*)^*(a_i c_i^*)\bigr ),\] which is nonnegative. The functional is therefore positive. It is normalized because \(\omega (1)=\sum _i\mathrm {Tr}(\rho _i)=1\). Each \(\rho _i=c_i^*c_i\) is self-adjoint. The trace of a conjugate transpose is the conjugate of the trace, and \((a_i\rho _i)^*=\rho _i a_i^*\), so \(\mathrm {Tr}(\rho _i a_i^*)=\overline {\mathrm {Tr}(\rho _i a_i)}\). Thus \(\omega \) is Hermitian.

The same identity shows that \(\omega (a^*a)=0\) if and only if \(a_i c_i^*=0\) for every \(i\). Different blocks occupy complementary summands, so \(\dim N=\sum _i\dim \{a_i:a_i c_i^*=0\}\). A matrix vanishes on a subspace precisely when it factors through the quotient by that subspace. The image of \(c_i^*\) has dimension \(\operatorname {rank} c_i^*\). The matrices vanishing on it are determined by their values on a complement, so their space has dimension \(m_i\bigl (m_i-\operatorname {rank} c_i^*\bigr )\). The quotient of that block therefore has dimension \(m_i\operatorname {rank} c_i^*\). Summing over the blocks gives \(\dim H_\omega =\sum _i m_i\operatorname {rank} c_i^*\).

It remains to match that rank with \(\operatorname {rank}\rho _i\). For the standard inner product, \(\|c_i v\|^2=\langle c_i^*c_i v,v\rangle \). If \(c_i^*c_i v=0\), the right-hand side vanishes, so \(c_i v=0\). The converse is immediate, and rank-nullity gives \(\operatorname {rank}\rho _i=\operatorname {rank} c_i\). A minor of a transpose is the transpose of a minor, and the determinant is unchanged by transposition. A minor of a conjugate transpose is the conjugate of a minor of the transpose. The largest size of a nonzero minor is therefore the same for \(c_i\), \(c_i^T\) and \(c_i^*\). Hence \(\operatorname {rank} c_i^*=\operatorname {rank}\rho _i\), and \(\dim H_\omega =\sum _i m_i\operatorname {rank}\rho _i\). On \(M_2\), \(\mathrm {diag}(2/3,1/3)\) is the square of \(\mathrm {diag}(\sqrt {2/3},\sqrt {1/3})\), so its rank is \(2\). \(\mathrm {diag}(1,0)\) is the square of \(\mathrm {diag}(1,0)\), and its only nonzero minor has size \(1\). The formula gives dimensions \(4\) and \(2\).

On functions, the indicator basis is orthogonal for the Gram form. Indeed \(\omega (1_x^*1_y)=\delta _{xy}w(x)\), so the Gram matrix is diagonal with entries \(w(x)\). Its rank is the number of strictly positive weights. A function lies in \(N\) exactly when it vanishes on \(\operatorname {supp} w\), because \(\omega (f^*f)=\sum _x w(x)\lvert f(x)\rvert ^2\). Two functions determine the same class exactly when they agree on that support, and every function on the support arises in this way. The descended inner product is \[\langle f,g\rangle =\sum _{w(x)>0}w(x)\overline {f(x)}\,g(x).\] Every function on a finite set is square-summable, so the space is \(\ell ^2(\operatorname {supp} w)\).