I am trying to calculate the pull-back of a cohomology class on $\Omega K(\mathbb{C})$ along the H-space map of $K(\mathbb{C}).$

Let $b_k\in \tilde{H}^{2k}(\Omega K(\mathbb{C});\mathbb{R})$ be a reduced cohomology class on the loopspace of the complex K theory space (it is in fact derived from the Borel regulator classes, but this should not matter). The H space structure on $K(\mathbb{C})$ gives a map $$ \Omega K(\mathbb{C})\wedge \Omega K(\mathbb{C})\to \Omega K(\mathbb{C})$$ (By forming sums of chain complexes.) The Kunneth formula tells us that $$\tilde{H}^{2k}(\Omega K(\mathbb{C})\wedge \Omega K(\mathbb{C}); \mathbb{R})\cong \bigoplus_{i+j=2k}\tilde{H}^{i}(\Omega K(\mathbb{C});\mathbb{R})\otimes \tilde{H}^{j}(\Omega K(\mathbb{C});\mathbb{R}).$$

In this scenario I want to calculate $f^*b_k\in\tilde{H}^{2k}(\Omega K(\mathbb{C})\wedge \Omega K(\mathbb{C});\mathbb{R}).$ I am hoping that $f^*b_k=b_k\otimes 1+1\otimes b_k.$ Has someone done this or a similar calculation before?