Let $A$ be a unital algebra over $\mathbb{C}$, $M$ be an $A$ bimodule, $C^n(A, M)$ be a space of all $n$-linear maps $f \colon A^n \to M$ (to be called $n$-cochains) and define $b \colon C^n(A, M) \to C^{n + 1}(A, M)$ by the formula \begin{align*} (bf)(a_1, \dotsc, a_n, a_{n+1}) :={}& a_1 f(a_2, \dotsc, a_{n+1}) \\ {}& + \sum_{j=1}^n (-1)^j f(a_1, \dotsc,a_{j-1}, a_j a_{j+1}, a_{j+2}, \dotsc, a_{n+1}) \\ {}&+(-1)^{n+1}f(a_1, \dotsc, a_n) a_{n+1} \,. \end{align*} One checks that $b^2 = 0$ so we get a cochain complex – the cohomology of this complex is called Hochschild cohomology. Hochschild homology is defined in the following way: $C_0(A, M) = M$ and $C_n(A, M) = M \otimes A^{\otimes n}$ and \begin{align*} b(m \otimes a_1 \otimes \dotsb \otimes a_n) :={}& m a_1 \otimes a_2 \otimes \dotsb \otimes a_n \\ {}& + \sum_{j=1}^{n-1} (-1)^j m \otimes a_1 \otimes \dotsb \otimes a_{j-1} \otimes a_j a_{j+1} \otimes a_{j+2} \otimes \dotsb \otimes a_n \\ {}& + (-1)^n a_n m \otimes a_1 \otimes \dotsb \otimes a_{n-1} \,. \end{align*} Again $b^2 = 0$ and therefore we get chain complex: the homology of this complex is Hochschild homology. For an algebra map $f \colon A \to B$ (or more general, for a chain map) we get natural induced maps in both homology and cohomology.
Suppose that I know that some map $f \colon A \to A$ induces identity in Hochschild homology: is it also true that then it induces identity in Hochschild cohomology?
If it would happen that it is not true for general bimodule $M$, I would like to know whether is it true in the simplest case $M = A$ and $M = A^*$ (the dual module) for homology and cohomology respectively.
The second part is quite similar, but instead of Hochschild (co)homology I would like to know the same for the so called cyclic (co)homology. For the cyclic theory one assumes $M = A$ for homology and $M = A^*$ for cohomology: using some natural identifications for $\mathrm{Hom}(A^{\otimes n}, A^*) \cong \mathrm{Hom}(A^{\otimes (n + 1)}, \mathbb{C})$ the formula for the differential (for example for cohomology) reads: \begin{align*} (bf)(a_0, a_1, \dotsc, a_{n+1}) ={}& \sum_{j=0}^n f(a_0, \dotsc, a_{j-1}, a_j a_{j+1}, a_{j+2}, \dotsc, a_{n+1}) \\ {}& + (-1)^{n+1} f(a_{n+1}a_0, a_1, \dotsc, a_n) \,. \end{align*} Now we introduce the (zero degree) cyclic operator $\lambda$ where $$ (\lambda f)(a_0, \dotsc, a_n) := (-1)^n f(a_n, a_0, \dotsc, a_{n-1}) \,. $$ We consider cochains $f$ such that $\lambda f = f$ – these are called cyclic cochains and the space of cyclic cochains forms a subcomplex. Therefore we can consider its cohomology which by definition is the cyclic cohomology. The cyclic homology is defined in the same manner. One can again show that any algebra map induces maps on the level of homology and cohomology. And my question is:
Suppose that we know that some map $f \colon A \to A$ induces identity in cyclic homology. Is it true that it also induces identity in cyclic cohomology?
My question is inspired by the following discussion: https://mathoverflow.net/questions/243593/inner-automorphisms-acts-as-identity-on-hochschild-homology