[Paper Review] A tensor product of representations of UHF algebras arising from Kronecker products

Katsunori Kawamura|ArXiv.org|Oct 8, 2009

Algebraic structures and combinatorial models20 references4 citations

TL;DR

This paper introduces a non-symmetric tensor product of representations of UHF algebras using Kronecker products of matrices, establishing tensor product formulae for GNS representations induced by product states. The key contribution is proving that the resulting tensor product is non-commutative, even for irreducible representations, by constructing explicit unitary equivalences and demonstrating non-equivalence of swapped tensor products via state norms and disjointness of GNS representations.

ABSTRACT

We introduce a non-symmetric tensor product of representations of UHF algebras by using Kronecker products of matrices. We prove tensor product formulae of GNS representations by product states and show examples.

Motivation & Objective

• To develop a non-symmetric tensor product of representations for UHF algebras, which lack a known comultiplication.
• To extend the framework of Kronecker product-based tensor products—previously used for Cuntz and Cuntz-Krieger algebras—to UHF algebras.
• To derive explicit tensor product formulae for GNS representations associated with product states on UHF algebras.
• To demonstrate the non-commutativity of the tensor product by constructing examples where swapped tensor products yield unitarily inequivalent representations.

Proposed method

• The tensor product is defined via a family of isomorphisms between UHF algebras associated with sequences in $\mathbf{N}_{\geq 2}^\infty$, using Kronecker products of matrix algebras.
• A product state $\omega_{\mathbf{T}}$ on a UHF algebra $\mathcal{A}(\mathbf{a})$ is constructed as an infinite tensor product of states on finite-dimensional matrix algebras $M_{a_n}$.
• The GNS representation $\pi_{\mathbf{T}}$ is associated with each product state $\omega_{\mathbf{T}}$, and the tensor product $\pi_{\mathbf{T}} \otimes_\varphi \pi_{\mathbf{R}}$ is defined via a specific $\varphi$-isomorphism between $\mathcal{A}(\mathbf{a} \cdot \mathbf{b})$ and $\mathcal{A}(\mathbf{a}) \otimes \mathcal{A}(\mathbf{b})$.
• The construction relies on the Kronecker product of density matrices $T^{(n)} \boxtimes R^{(n)}$ to define the state on the tensor product algebra.
• Unitary equivalence between the constructed representation $\pi_{\mathbf{T} \boxtimes \mathbf{R}}$ and the tensor product $\pi_{\mathbf{T}} \otimes_\varphi \pi_{\mathbf{R}}$ is established via a unitary operator $U^{(\mathbf{T},\mathbf{R})}$.
• Non-commutativity is proven by showing $\|\omega \otimes_\varphi \omega' - \omega' \otimes_\varphi \omega\| = 2$, implying disjointness of GNS representations.

Experimental results

Research questions

• RQ1 Can a non-symmetric tensor product of representations be defined for UHF algebras despite the absence of a comultiplication?
• RQ2 How do GNS representations associated with product states behave under tensor product operations?
• RQ3 Is the tensor product of GNS representations of UHF algebras commutative?
• RQ4 Can explicit tensor product formulae be derived using Kronecker products of matrix states?
• RQ5 What is the structure of the semigroup of unitary equivalence classes of irreducible representations under this tensor product?

Key findings

• The tensor product of GNS representations $\pi_{\mathbf{T}}$ and $\pi_{\mathbf{R}}$ via the $\varphi$-isomorphism is unitarily equivalent to the representation $\pi_{\mathbf{T} \boxtimes \mathbf{R}}$ constructed via Kronecker product of states.
• The tensor product $\otimes_\varphi$ is non-commutative: $[\pi_{\mathbf{T}}] \otimes_\varphi [\pi_{\mathbf{R}}] \neq [\pi_{\mathbf{R}}] \otimes_\varphi [\pi_{\mathbf{T}}]$ for certain pure product states.
• For $\mathbf{a} = \mathbf{b} = (2,2,\ldots)$, with $T = \mathrm{diag}(1,0)$, $R = \mathrm{diag}(0,1)$, the norm difference $\|\omega \otimes_\varphi \omega' - \omega' \otimes_\varphi \omega\| = 2$, implying disjoint GNS representations.
• The GNS representations $\pi_{\mathbf{T}(J)}$ for $J \in \{1,\ldots,n\}^\infty$ are irreducible when $\mathbf{T}(J)$ corresponds to pure product states.
• The semigroup $\mathcal{P} = \bigcup_{n \geq 2} \{P_n[J] : J \in \{1,\ldots,n\}^\infty\}$ of unitary equivalence classes of irreducible representations is closed under $\otimes_\varphi$, with $P_n[J] \otimes_\varphi P_m[K] = P_{nm}[J \cdot K]$.
• The construction generalizes permutative representations of Cuntz algebras restricted to gauge-invariant UHF subalgebras, identifying them as 'atoms' in the representation theory of UHF algebras.