Frobenius–Rieffel norms on finite-dimensional C*-algebras

A main goal of noncommutative metric geometry is to establish the convergence of spaces arising in the physics or operator-algebra literature [24, 25, 14, 11, 12, 17]. To accomplish this, one must equip operator algebras with compact quantum metrics, which were introduced by Rieffel [22, 23] and motivated by work of Connes [4, 5]. Then, convergence of compact quantum metric spaces is proven with quantum analogues of the Gromov–Hausdorff distance [24, 16, 15, 13, 18, 28].

In [2], the first author and Latrémolière recently exhibited the convergence of quantum metric spaces built from approximately finite-dimensional C*-algebras (AF algebras) and, in particular, convergence of the Effros–Shen algebras [8] with respect to their irrational parameters. Quantum metric spaces are obtained by endowing unital C*-algebras with a type of seminorm whose properties are inspired by the Lipschitz seminorm. A property, which is not needed in [2], but appears desirable in other context [26], is called the strongly Leibniz property. A seminorm $s$ satisfies the strongly Leibniz property on an operator algebra ${\mathcal{A}}$ if

for all invertible $A\in{\mathcal{A}}$, in which $\|\cdot\|_{\mathrm{op}}$ is the operator norm. This can be seen as a noncommutative analogue of the quotient rule for derivatives. Although the authors of [2] were able to prove their results without this property, Rieffel’s work on module convergence over the sphere [26] uses the strong Leibniz property, and it can be expected to play a role in the study of module convergence in general.

Let $T$ be a topological space and let $t\in T$. The reason that the seminorms in [2] do not likely satisfy the strongly Leibniz rule is because the seminorms are of the form

where $A$ is an element of the C*-algebra ${\mathcal{A}}$, ${\mathcal{B}}$ is a C*-subalgebra of ${\mathcal{A}}$, and $P_{t}:{\mathcal{A}}\rightarrow{\mathcal{B}}$ is a certain surjective linear map called a faithful conditional expectation. But conditional expectations are rarely multiplicative (otherwise, the strongly Leibniz property of this seminorm would come for free). Rather than replace $P_{t}$, which provides crucial estimates, Rieffel provided another option in [27, Section 5] following his previous work in [21]: replace the operator norm with one induced by $P_{t}$ and the subalgebra ${\mathcal{B}}$. For $A\in{\mathcal{A}}$, the Frobenius–Rieffel norm is

where $A^{*}$ is the adjoint of $A$. If we define

then $s_{P_{t}}$ is a seminorm that is strongly Leibniz [27, Theorem 5.5].

However, this replacement comes at a cost. Following [2], we want the family of maps $(s_{P_{t}})_{t\in T}$ to vary continuously (pointwise) on a particular subset of ${\mathcal{A}}$ with respect to $\|\cdot\|_{\mathrm{op}}$. Thankfully, in the setting of [2], one need only verify this continuity when ${\mathcal{A}}$ is finite dimensional. In this case, $\|\cdot\|_{P_{t}}$ and $\|\cdot\|_{\mathrm{op}}$ are equivalent on ${\mathcal{A}}$, meaning there exist constants $\kappa_{t}^{+},\kappa_{t}^{-}>0$ such that

for each $t\in T$. Therefore, we can replace $s_{P_{t}}$ with

which is strongly Leibniz. However, the constants $\kappa_{t}^{+},\kappa_{t}^{-}$ need not change continuously with respect to $t$. Therefore, our aim in this paper is to find explicit equivalence constants for the operator norm and Frobenius–Rieffel norms on finite-dimensional C*-algebras, so that we may prove the continuity of the constants $\kappa^{\pm}_{t}$ with respect to $t\in T$. In fact, one of our main results (Theorem 5.2) shows that there exist explicit equivalence constants for the finite-dimensional C*-algebras that form the Effros–Shen algebras which vary continuously with respect to the irrational parameters that determine these algebras.

After some background on C*-algebras and the construction of the Frobenius–Rieffel norms, we provide some basic facts in the next section. Then, we find equivalence constants when ${\mathcal{A}}$ is the space of complex $n\times n$-matrices. This provides a framework for the general case of finite-dimensional C*-algebras, which we tackle next. Our main method is to represent the conditional expectations as means of unitary conjugates for some standard subalgebras, and then extend these results to all unital C*-subalgebras by showing that although the Frobenius–Rieffel norms are not unitarily invariant, their equivalence constants are.

The facts we state about C*-algebras in this section can be found in standard texts such as [7, 19, 20]. A C*-algebra $({\mathcal{A}},\|\cdot\|)$ is a Banach algebra over ${\mathds{C}}$ equipped with a conjugate-linear anti-multiplicative involution $*:{\mathcal{A}}\rightarrow{\mathcal{A}}$ called the adjoint satisfying the C*-identity (i.e., $\|A^{*}A\|=\|A\|^{2}$ for all $A\in{\mathcal{A}}$). We say that ${\mathcal{A}}$ is unital if it has a multiplicative identity. If two C*-algebras ${\mathcal{A}},{\mathcal{B}}$ are *-isomorphic, then we denote this by ${\mathcal{A}}\cong{\mathcal{B}}$. Let $n\in{\mathds{N}}=\{1,2,3,\ldots\}$. We denote the space of complex $n\times n$ matrices by ${\mathcal{M}}_{n}$ and its C*-norm by $\|\cdot\|_{\mathrm{op}}$, the operator norm induced by the 2-norm on ${\mathds{C}}^{n}$. We denote the $n\times n$ identity matrix by $I_{n}$. For $A\in{\mathcal{M}}_{n}$, we let $A_{i,j}\in{\mathds{C}}$ denote the $(i,j)$-entry of $A$ for all $i,j\in[N]$, where $[N]=\{1,2,\ldots,N\}$.

The following maps are needed for the construction of Frobenius–Rieffel norms.

We can now define norms induced by faithful conditional expectations.

The terminology for these norms is due to two facts: Rieffel introduced these norms [27, Section 5] using his work that introduced spaces called Hilbert C*-modules [21] and we show in Theorem 3.16 that one can recover the Frobenius norm using a particular C*-subalgebra.

One of the main results that makes our work in this paper possible is the fact that we can express our conditional expectations as orthogonal projections. The key property that allows this is the preservation of faithful tracial states. A state on a C*-algebra ${\mathcal{A}}$ is a positive linear functional $\varphi:{\mathcal{A}}\rightarrow{\mathds{C}}$ of norm $1$. We say that $\varphi$ is faithful if $\varphi(A^{*}A)=0\iff A=0$ and tracial if $\varphi(AB)=\varphi(BA)$ for all $A,B\in{\mathcal{A}}$. If ${\mathcal{B}}$ is a unital C*-subalgebra and $P:{\mathcal{A}}\rightarrow{\mathcal{B}}$ is a conditional expectation onto ${\mathcal{B}}$, then we say that $P$ is $\varphi$-preserving if $\varphi\circ P=\varphi$.

A faithful tracial state allows us to define an inner product on ${\mathcal{A}}$.

The following fact is well known.

In this case, we denote the associated Frobenius–Rieffel norm on ${\mathcal{A}}$ by $\|\cdot\|_{\tau,{\mathcal{B}}}$. Now, let ${\mathcal{A}}=\bigoplus_{k=1}^{N}{\mathcal{M}}_{d_{k}}$, let ${\mathcal{B}}\subseteq{\mathcal{A}}$ be a unital C*-subalgebra, and let $\mathbf{v}=(v_{1},v_{2},\ldots,v_{N})\in(0,1)^{N}$ such that $\sum_{k=1}^{N}v_{k}=1$.

1. (1)

   We denote the conditional expectation of Theorem 2.6 induced by the faithful tracial state $\tau_{\mathbf{v}}$ of Example 2.4 by $P^{\mathbf{v}}_{\mathcal{B}}$. We denote the associated Frobenius–Rieffel norm by $\|\cdot\|_{\mathbf{v},{\mathcal{B}}}$.

2. (2)

   If $N=1$, then $\tau=\frac{1}{d_{1}}\mathrm{Tr}$ is the unique faithful tracial state on ${\mathcal{M}}_{d_{1}}$, and we denote the conditional expectation of Theorem 2.6 induced by $\tau$ by $P_{\mathcal{B}}$. We denote the associated Frobenius–Rieffel norm by $\|\cdot\|_{{\mathcal{B}}}$.

In this section, we detail the subalgebras of $\bigoplus_{k=1}^{N}{\mathcal{M}}_{d_{k}}$ that we will be working with and the conditional expectations given by Theorem 2.6. We also explain why we use “Frobenius” in the name of the norms of Theorem 2.3.

A partition ${\lambda}$ of $n\in{\mathds{N}}$, denoted ${\lambda}\vdash n,$ is a tuple ${\lambda}=(n_{1},n_{2},\ldots,n_{L})\in{\mathds{N}}^{L}$, where $L\in{\mathds{N}}$ depends on $n$ and $n=\sum_{i=1}^{L}n_{i}$. We need the following refinement to describe certain subalgebras of ${\mathcal{M}}_{n}$.

For example, $(2^{2},2),(2^{3}),(2,2,1^{2}),(3,3),(6)$ are refined partitions of $6$ with, respectively, lengths $2,1,3,2,1$; multiplicity vectors $(2,1),(3),(1,1,2),(1,1),(1)$; and dimension vectors $(2,2),(2),(2,2,1),(3,3),(6)$.

In what follows, we use Kronecker products and direct sums. For example, by $(I_{2}\otimes{\mathcal{M}}_{2})\oplus(I_{1}\otimes{\mathcal{M}}_{1})\subset{\mathcal{M}}_{5}$, we mean the subalgebra

of ${\mathcal{M}}_{5}$, where $\mathrm{diag}(A,A,\mu)$ is the block-diagonal matrix

with $0$s in the entries not occupied by the $A$s and $\mu$.

The algebra of circulant matrices provides another example of a unital C*-subalgebra of ${\mathcal{M}}_{n}$ that is not of the form (3.2).

The next definition serves as a vital intermediate step in finding equivalence constants associated to all unital C*-subalgebras and faithful tracial states of $\bigoplus_{k=1}^{N}{\mathcal{M}}_{d_{k}}$.

Up to unitary equivalence, standard unital C*-subalgebras comprise all unital C*-subalgebras of $\bigoplus_{k=1}^{N}{\mathcal{M}}_{d_{k}}$. To be clear, let ${\mathcal{B}},{\mathcal{C}}\subseteq{\mathcal{A}}=\bigoplus_{k=1}^{N}{\mathcal{M}}_{d_{k}}$ be two unital C*-subalgebras. We say that ${\mathcal{B}}$ and ${\mathcal{C}}$ are unitarily equivalent (with respect to ${\mathcal{A}}$) if there exists a unitary $U\in{\mathcal{A}}$ such that $B\mapsto UBU^{*}$ is a bijection from ${\mathcal{B}}$ onto ${\mathcal{C}}$, in which case we write ${\mathcal{C}}=U{\mathcal{B}}U^{*}$. Sometimes the term spatially isomorphic is used for unitary equivalence, but spatially isomorphic is also sometimes used in a more general sense.

Unitary equivalence is stronger than *-isomorphism. For example, the unital C*-subalgebras

of ${\mathcal{M}}_{2}\oplus{\mathcal{M}}_{2}$ are *-isomorphic but not unitarily equivalent in ${\mathcal{M}}_{2}\oplus{\mathcal{M}}_{2}$ (they are unitarily equivalent in ${\mathcal{M}}_{4}$, but we are viewing them as subalgebras of ${\mathcal{M}}_{2}\oplus{\mathcal{M}}_{2}$). We now state the following well-known result.

For example, the *-algebra of circulant matrices of Example 3.5 is unitarily equivalent with respect to ${\mathcal{M}}_{n}$ to ${\mathcal{B}}^{n}_{1,1,\ldots,1}$, not just *-isomorphic to it [10, 2.2.P10]. Also, the subalgebras ${\mathcal{B}}$ and $\mathcal{E}$ of Example 3.8 are unitarily equivalent, not just *-isomorphic.

We use Theorem 3.9 to generalize our results to all unital C*-subalgebras once we verify our results for the standard subalgebras. One of the advantages of working with standard unital C*-subalgebras is that they have canonical bases which are orthogonal with respect to the inner products induced by faithful tracial states.

Any standard unital C*-subalgebra ${\mathcal{B}}\subseteq\bigoplus_{k=1}^{N}{\mathcal{M}}_{d_{k}}$ has a standard basis (up to ordering of terms) given by matrix units or sums of distinct matrix units, which we denote by $\beta_{\mathcal{B}}$.

All cases in the example above can be recovered as follows.

For these bases, although some of the elements are sums of distinct matrix units, we note that the summands are from distinct blocks. For example, for ${\mathcal{B}}^{4}_{2^{2}}$, no element of the form $E_{i,j}+E_{i,m}$ appears in the standard basis. This easily verified fact and more are summarized in the following.

We now provide an explicit way of calculating the conditional expectations associated with standard unital C*-subalgebras. This is a complete generalization of [1, Proposition 2.8].

We next show how the Frobenius–Rieffel norms recover the Frobenius norm.

The next two examples show that Frobenius–Rieffel norms are not generally sub-multiplicative or unitarily invariant.

As discussed in the introduction, it is important to be able to compare the Frobenius–Rieffel norms with the operator norm. Theorem 2.3 says that

for all $A\in\bigoplus_{k=1}^{N}{\mathcal{M}}_{d_{k}}$, any unital C*-subalgebra ${\mathcal{B}}\subseteq{\mathcal{A}}$, and any conditional expectation $P:{\mathcal{A}}\rightarrow{\mathcal{B}}$ onto ${\mathcal{B}}$. This equality is achieved by the identity matrix. Thus, the nontrivial task is to find a constant $\kappa^{+}_{P,{\mathcal{B}}}>0$ such that

for all $A\in\bigoplus_{k=1}^{N}{\mathcal{M}}_{d_{k}}.$

We begin with some general results and then focus on the case of ${\mathcal{M}}_{n}$. Then, we move to the general case, which is more involved since the Frobenius–Rieffel norms depend on the underlying subalgebra and faithful tracial state. We begin with an inequality that allows us to avoid dealing with $A^{*}A$.

The next lemma allows us to extend our results from standard unital C*-subalgebras to all unital C*-subalgebras. The following fact is surprising since, at the end of the last section, we showed that the Frobenius–Rieffel norms are not unitarily invariant in general. Also, it can be the case that $\|A\|_{\tau,{\mathcal{B}}}\neq\|A\|_{\tau,{\mathcal{C}}}$ for certain $A\in{\mathcal{A}}$, but the equivalence constants are the same for uniatrily equivalent subalgebras ${\mathcal{B}},{\mathcal{C}}\subseteq{\mathcal{A}}$.