C*- Algebra representation and Matrices

Question

Let $A\subset M_7$ be a unital $C^*$-algebra, where $I_7 \in A$.

Suppose $dim(A)=5$.

1. What are the possible factor and multiplicity values?

2. What are the possible dimensions of $A'$? Here $A'$ is the commutant.

3. What are the values of $dim(A')$ that uniquely determine $A$ up to *-isomorphism?

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I want to somehow use the theorem:

Let $A \subset M_n$ be a $C^*$- algebra with $I_n \in A$. Then there are factor dimensions $n_1, \cdots, n_k$ and multiplicities $m_1, \cdots, m_k$ and a unitary $U \in M_n$ such that

• $A$ is *- isomorphic to $M_{n_1} \bigoplus \cdots \bigoplus M_{n_k}$
• $U^* AU$ is the set of block diagonal matrices, where $A_i \in M_{n_i}$ is repeated $m_i$ times
• $dim(A) = n_1^2 + \cdots n_k^2$
• $n= n_1m_1+ \cdots + n_km_k$
• $A' \simeq M_{m_1} \bigoplus \cdots \bigoplus M_{m_k}$ is the finite dimensional $C^*$ algebra with factor dimensions $m_1 , \cdots, m_k$ and multiplicities $n_1, \cdots, n_k$.
• $dim(A') = m_1^2 + \cdots m_k^2$

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By above, we have

• $5= n_1^2 + \cdots n_k^2$
• $7= n_1m_1+ \cdots + n_km_k$.

So the factor dimensions could be $\{1,1,1,1,1 \}$, $\{1,2,0,0,0 \}$.

And the multiplicities could be $\{7, 0, 0, 0, 0 \}, \{1,3,0,0,0 \}, \{3,2,0,0,0 \}$

I'm not sure if I'm on the right track, and also a bit lost on what to do.

Any help will be appreciated!

Answer 1

You are on track, although I don't really see how you got your multiplicities.

If all the blocks have size 1, you can make your $A$ as $\mathbb C^4\oplus \mathbb C I_3$. Or $\mathbb C^3\oplus \mathbb C I_2\oplus\mathbb C I_2$.

If you allow a block of size 2, that already gives you dimension 4, so there cannot be another one. That gives you $M_2(\mathbb C)\oplus \mathbb C I_5$. Other possibilities if you play with the multiplicities are $\big[M_2(\mathbb C)\otimes I_2\big]\oplus \mathbb C I_3$ and $\big[M_2(\mathbb C)\otimes I_3\big]\oplus \mathbb C I_1$. Here the tensor notation denotes the multiplity, i.e. $$ M_2(\mathbb C)\otimes I_2=\Big\{\begin{bmatrix} A&0\\0&A\end{bmatrix}:\ A\in M_2(\mathbb C)\Big\}. $$ The three $M_2(\mathbb C)\oplus\mathbb C$ cases have multiplicities $\{1,5\}$, $\{2,3\}$, and $\{3,1\}$ respectively.