Theorem. The fixed-point algebra is the averaged image [fdip-008X]
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  • November 2, 2026

Let \(K\) be \(\mathbb R\) or \(\mathbb C\), and let \(A\) be a unital associative \(K\)-algebra. A symmetry of a linear functional \(\omega \) on \(A\) is a unital \(K\)-algebra automorphism \(\alpha \) with \(\omega (\alpha (a))=\omega (a)\) for every \(a\). Let \(G\) be a finite group, and let \(g\mapsto \alpha _g\) be a homomorphism from \(G\) into the unital \(K\)-algebra automorphisms of \(A\), so that \(\alpha _{hg}=\alpha _h\circ \alpha _g\). The order \(|G|\) is invertible in \(K\). Define \[E(a)=\frac 1{|G|}\sum _{g\in G}\alpha _g(a).\] Write \(A^G\) for the fixed-point algebra, the elements fixed by every \(\alpha _g\). Then \(E\) is linear, \(E(1)=1\), the image of \(E\) is \(A^G\), and \(E(b_1ab_2)=b_1E(a)b_2\) whenever \(b_1\) and \(b_2\) are fixed. If every \(\alpha _g\) preserves \(\omega \), then \(\omega (E(a))=\omega (a)\). These are the identities of a conditional expectation of \(\omega \) onto \(A^G\). If every \(\alpha _g\) preserves an involution of \(A\), then \(E(a^*)=E(a)^*\).

When the automorphisms are conjugations \(\alpha _g(R)=\rho (g)R\rho (g)^{-1}\) by a representation \(\rho \) on a finite-dimensional space, the fixed-point algebra is the commutant of \(\rho (G)\). On the half-turn of Proposition [fdip-003T], with \(P=\operatorname {diag}(I_2,-I_2)\) and \(R\) written in those blocks as \(\begin {pmatrix}A&B\\C&D\end {pmatrix}\), the average is the block-diagonal operator \[E(R)=\begin {pmatrix}A&0\\0&D\end {pmatrix}.\] That note identifies this commutant with \(M_2(\mathbb R)\times M_2(\mathbb R)\) and records that the image of the group algebra is the smaller algebra of scalar blocks.

The neutral Gauss constraint for a finite gauge group asks for the link functions fixed by every group element, and its projector is this same average. An invariant functional therefore satisfies \(\omega =\omega \circ E\): its value depends on an observable only through the averaged observable. The charged space remains the intertwiner space of Theorem [fdip-008B]. The multiplicity matrices of a real orthogonal action remain the commutant description of Theorem [fdip-004Z].

Proof. 

  • November 2, 2026

Each \(\alpha _g\) is \(K\)-linear, so \(E\) is linear. A unital automorphism fixes \(1\), and therefore \(E(1)=1\).

For \(h\in G\), \[\alpha _h(E(a)) =\frac 1{|G|}\sum _{g\in G}\alpha _h(\alpha _g(a)) =\frac 1{|G|}\sum _{g\in G}\alpha _{hg}(a).\] As \(g\) runs through \(G\), so does \(hg\). Hence \(\alpha _h(E(a))=E(a)\), and the image of \(E\) lies in \(A^G\).

If \(b\) is fixed by every \(\alpha _g\), every summand equals \(b\), so \(E(b)=b\). Thus \(A^G\) is contained in the image, and \(E\) is the identity on \(A^G\). The two inclusions give \[A^G=\operatorname {im}E.\]

Let \(b_1\) and \(b_2\) be fixed. Multiplicativity gives \(\alpha _g(b_1ab_2)=\alpha _g(b_1)\alpha _g(a)\alpha _g(b_2) =b_1\alpha _g(a)b_2\). Averaging both sides yields \(E(b_1ab_2)=b_1E(a)b_2\).

If every \(\alpha _g\) preserves \(\omega \), then \(\omega (\alpha _g(a))=\omega (a)\). Averaging the scalar values gives \(\omega (E(a))=\omega (a)\). If every automorphism preserves an involution, \(\alpha _g(a^*)=\alpha _g(a)^*\), and the same average gives \(E(a^*)=E(a)^*\).

For a conjugation action, \(\alpha _h(E(R))=E(R)\) is the statement that \(E(R)\) commutes with every \(\rho (h)\). If \(R\) already lies in the commutant, the preceding paragraph gives \(E(R)=R\). Thus the image is exactly the commutant.

In the half-turn coordinates the conjugation by \(P=\operatorname {diag}(I_2,-I_2)\) sends \(\begin {pmatrix}A&B\\C&D\end {pmatrix}\) to \(\begin {pmatrix}A&-B\\-C&D\end {pmatrix}\). The average of the two operators is the displayed block-diagonal matrix. The identification of that commutant, and the comparison with the group-algebra image, are the calculations of Proposition [fdip-003T].

Tomiyama's norm-one projection theorem has not been read, and no norm on \(A\) is used. The argument gives the finite average only. It does not identify the sectors with the centre of the commutant, and it does not replace the charged intertwiner description.