Opposite algebra

A right action of $A$ is equivalently a left action of $A^{\mathrm{op}}$: writing $\xi a$ as $\rho(a^{\mathrm{op}})\xi$ converts $(\xi a)b=\xi(ab)$ into the homomorphism law for $\rho$. This is why bimodules are represented by commuting left actions of $A$ and $A^{\mathrm{op}}$. If $A$ is commutative, the canonical product-reversing map is an isomorphism because reversing multiplication changes nothing.

For a real spectral triple with representation $\pi$ and antiunitary real structure $J$, the standard right representation is

The involution inside this formula makes $\pi^{\mathrm{op}}$ complex-linear under the usual antiunitary convention. The order-zero condition requires $[\pi(a),\pi^{\mathrm{op}}(b^{\mathrm{op}})]=0$. The first-order condition additionally requires $\bigl[\, [D,\pi(a)],\pi^{\mathrm{op}}(b^{\mathrm{op}})\,\bigr]=0$, where the inner commutator uses its bounded extension. These conventions follow Connes, §2.

For matrix algebras, transposition gives a complex-linear $*$-isomorphism $M_n(\mathbb C)^{\mathrm{op}}\cong M_n(\mathbb C)$. More generally, a $C^*$-algebra may be isomorphic to its opposite without a preferred isomorphism, and such an isomorphism is extra structure. The notations $a^{\mathrm{op}}$, $a^\circ$, and $a^0$ vary across the literature; the last two often denote the represented right action rather than the abstract element itself.

1. Alain Connes, “Noncommutative Geometry and Reality,” Journal of Mathematical Physics 36 (1995), 6194–6231. DOI record. Relevant: §2 on the opposite-algebra representation and order conditions.
2. Alain Connes and Matilde Marcolli, Noncommutative Geometry, Quantum Fields and Motives, American Mathematical Society, 2008. DOI record. Relevant: Chapter 1 on real spectral triples and bimodule conventions.