# A common lifting of a positive element?

Suppose $A$ is a C$^{*}$-algebra and that we have finite-dimensional irreducible representations $\varphi_{1},\ldots,\varphi_{n}$ of $A$. It follows from the question C$^{*}$-algebra acting irreducibly on the finite-dimensional space $\mathbb{C}^{n}$ must be $M_{n}(\mathbb{C})$ that each $\varphi_{i}$ is surjective. Suppose we are given positive elements $b_{i}\in\varphi_{i}(A)$ for $1\leq i\leq n$. Due to surjectivity, we can find postive elements $a_{1},\ldots,a_{n}\in A$ such that $\varphi_{i}(a_{i})=b_{i}$. My question is:

Is it possible to find a common positive lifting of the $b_{i}'s$? I.e., does there exist a positive $z\in A$ such that $\varphi_{i}(z)=b_{i}$ for all $1\leq i\leq n$?

I am reading an old paper of Fell's and, if this is not possible in general, I was wondering how he accomplishes what I've underlined:

As Martin points out, the question obviously has a negative answer if two of the representations are equivalent.

However, Fell considers non-equivalent irreducible representations, and the question has a positive answer if all representations are pairwise non-equivalent.

Note that by Corollary 4.1.10 in Dixmier's book, two finite dimensional irreducible representations are equivalent if and only if they have the same kernel. Hence the positive answer follows from the following more general result. The proof uses basic theory of primitive ideal spaces, see for instance Sections 2.9, 2.10 and 2.11 in Dixmier's book.

Lemma. Let $A, B_1,\dots, B_n$ be $C^\ast$-algebras with $B_i$ simple for $i=1,\dots, n$, and suppose that $\pi_i \colon A \to B_i$ are surjective $\ast$-homomorphisms for $i=1,\dots, n$ such that $\mathrm{ker}\, \pi_i \neq \mathrm{ker} \, \pi_j$ for $i\neq j$. Then the $\ast$-homomorphism $\pi := \bigoplus_{i=1}^n \pi_i \colon A \to \bigoplus_{i=1}^n B_i$ is surjective.

Proof. Let $I_i = \mathrm{ker}\, \pi_i$ for $i=1,\dots, n$ which is a maximal two-sided, closed ideal in $A$. Hence $\{ I_i\}$ are distinct, closed (one-point) subset of the primitive ideal space $\mathrm{Prim}\, A$ for $i=1,\dots, n$, and the two-sided, closed ideal $I_i$ corresponds to the open subset $U_i := \mathrm{Prim}\, A \setminus \{I_i\}$ in $\mathrm{Prim}\, A$.

Let $I : = \bigcap_{i=1}^n I_i$. The open subset in $\mathrm{Prim}\, A$ corresponding to $I$ is $\bigcap_{i=1} U_i = \mathrm{Prim}\, A \setminus\{ I_1,\dots, I_n\}$. Hence $A/I$ has a primitive ideal space (canonically) homeomorphic to $\{ I_1,\dots, I_n\} \subseteq \mathrm{Prim}\, A$ in the subspace topology. As each $\{I_i\}$ is closed in $\mathrm{Prim}\, A$, it follows that $\mathrm{Prim}\, A/I$ is a discrete $n$-point space. Hence we obtain canonical isomorphisms $$A/I \cong \bigoplus_{i=1}^n (A/I)/(I_i/I) \cong \bigoplus_{i=1}^n A/I_i \cong \bigoplus_{i=1}^n B_i.$$ That $\pi$ is surjective is an easy consequence. QED

• Thank you very much. Would you be able to explain how you use the fact that $\textrm{Prim }A/I$ is a discrete $n$-point space to obtain the canonical isomorphism?
– ervx
Commented Aug 31, 2018 at 13:48
• If the primitive ideal space of a C*-algebra is discrete, then the C*-algebra is canonically isomorphic to the direct sum of all its simple quotients. Commented Aug 31, 2018 at 22:08
• For anyone interested: Theorem 8.1 in Kaplansky's "Normed Algebras" contains a proof of this fact.
– ervx
Commented Sep 19, 2018 at 12:58

No, it's not possible. Take $A=M_2(\mathbb C)$, $\varphi_1(a)=a$, $\varphi_2(a)=uau^*$, where $$U=\begin{bmatrix} 0&1\\1&0\end{bmatrix}.$$ Both $\varphi_1$ and $\varphi_2$ are bijective.

Take $$b_1=I,\ \ b_2=\begin{bmatrix} 1&0\\0&2\end{bmatrix}.$$ Thus if $\varphi_1(z)=I$, then $z=I$, while $\varphi_2(z)=b_2$ implies $$z=\begin{bmatrix} 2&0\\0&1\end{bmatrix}.$$

As for what is Fell doing, I cannot tell without knowing at least what $A$ is, which your text does not say.

• Thank you for the counterexample!
– ervx
Commented Aug 31, 2018 at 13:48