Conceptual explanation for multiplicativity of theta generalization of extreme characters of U(infty)

Question

A character of $U(\infty)$ is a continuous, positive definite class function $\chi: U(\infty) \longrightarrow \mathbb{C}$, which is normalized by $\chi(e) = 1$. Observe that the set of characters of $U(\infty)$ is a convex set, so we can talk about its extreme points, called the extreme characters of $U(\infty)$.

It is a classical theorem of Edrei and Voiculescu that the extreme characters of $U(\infty)$ are parametrized by points $\omega = (\alpha^+, \alpha^-, \beta^+, \beta^-, \gamma^+, \gamma^-)$ in an infinite-dimensional simplex $\Omega \subset \mathbb{R}_+^{\infty}\times\mathbb{R}_+^{\infty}\times\mathbb{R}_+^{\infty}\times\mathbb{R}_+^{\infty}\times\mathbb{R}_+\times\mathbb{R}_+$ and they are given explicitly as follows. Let $\chi_{\omega}$ be the character associated to the point $\omega\in\Omega$. Since $\chi_{\omega}$ is a class function, it is determined completely by its restriction to the diagonal matrices in $U(\infty)$, that is, the subset $\mathbb{T}^{\infty} \subset U(\infty)$, where $\mathbb{T} = \{z \in \mathbb{C} : |z| = 1\}$. Then one has that $\chi_{\omega}(u_1, u_2, \ldots)$ has the form $$\prod_{i=1}^{\infty}{\phi_{\omega}(u_i)},$$ where $\phi_{\omega}(u_i)$ is given explicitly by an infinite product. It is, at least initially, surprising that the extreme characters are multiplicative. Roughly, this is explained (in some papers of Grigori Olshanski) by the fact that extreme characters $\chi_{\omega}$ are limits of "extreme normalized characters" of $U(n)$, and the functional equation for characters of $U(n)$ becomes the multiplicativity relation for extreme characters of $U(\infty)$.

In http://arxiv.org/abs/q-alg/9709011, a work of Okounkov and Olshanski, Theorem 1.3. proves a generalization of Edrei-Voiculescu's theorem that characterizes the extreme points of a subset of functions $C(\mathbb{T}^{\infty}) \longrightarrow \mathbb{C}$ that depends on a parameter $\theta > 0$. When $\theta = 1$, the result is Edrei-Voiculescu's theorem. The extreme points have also a multiplicative structure, i.e., they look like $\Psi(u_1, u_2, \ldots) = \prod_{i=1}^{\infty}{\phi_{\omega}^{\theta}(u_i)}$. The multiplicativity can be "conceptually explained" (in the same way as above) in the cases $\theta = 1/2, 2$, where some representation theory is available.

My question: is there any explanation for the multiplicativity of the "$\theta$-characters" in Theorem 1.3. of the paper above, for general $\theta$ (or at least $\theta\in\mathbb{N}$), that is independent of the proof of the Theorem in that paper?

In that paper, the proof finds all extreme characters and it happens that they have that multiplicative structure. I'd like to know if there are more conceptual explanations, either coming from representation theory or elsewhere.

Answer 1

The extreme characters of $U(\infty)$ correspond to pure traces on a C*-algebra associated to the group. Its (pre-ordered) $K_0$ group admits a ring structure (in this case, via $U(\infty) \times U(\infty) \to U(\infty)$, which induces a tensor product-like ring structure on the $K_0$-group). The pure traces are exactly the traces whose induced map on the $K_0$-group (which in this very special case is a commutative ring with $1$, the $1$ corresponding to the class of the free module on one generator) are multiplicative.

This is part of a more general phenomenon, that extremal harmonic functions for some random walks are suitably multiplicative (e.g., if we take $U(n)$, there is a natural RW---on the set of weights---associated to any character; the set of extremal harmonic functions can be identified with the Newton polyhedron of the restriction of the character to the maximal torus, modulo the action of the Weyl group). This type of observation goes back to Orey, and perhaps might be interpreted as originating in the Gelfand-Naimark theorem.

Edit (amplification): In its most general form, here is the relevant (easy) result: Let $R$ be a commutative partially ordered ring, having $1$ as an order unit. A normalized positive group homomorphism is pure iff it is a ring homomorphism (that is, is multiplicative).

Definitions

Here $G$ is a partially ordered abelian group with positive cone $G^+$.

order unit: element, $u$ of $G$, such that for all $g \in G$, there exists a positive integer $N$ with $-Nu \leq g \leq Nu$.

normalized positive group homomorphism: A group homomorphism $\phi: (G,G^+, u) \to ({\bf R},{\bf R}^+,1)$ (where $u$ is a specified order unit); known as a state or a trace. The set of these things is a compact convex set (wrt the weak topology), is nonempty if $G$ is not zero, and is the closed convex hull of its extreme (pure) points. When $G$ is a ring with $1$ as order unit, the set is a Bauer simplex.

See the AMS book, Partially ordered abelian groups, by Ken Goodearl, for more details on the aspects relating to po groups, their traces (there called states), and Choquet theory.

Edit (2) (references for RW connections with compact Lie group representations): The primary one is

D Handelman, Extending traces on fixed point C${}^*$ algebras under Xerox product type actions of compact Lie groups, Journal of functional analysis 72.1 (1987): 44-57.

others include

Positive Polynomials, Convex Integral Polytopes, and a Random Walk Problem, SLN (1987): 67-88.

Positive polynomials and product type actions of compact groups, Vol. 320. American Mathematical Soc., 1985.

Representation rings as invariants for compact groups and limit ratio theorems for them, International Journal of Mathematics 4.01 (1993): 59-88.

Iterated Multiplication of Characters of Compact Connected Lie Groups, Journal of Algebra Volume 173, Issue 1, 1 April 1995, Pages 67–96

Space-time boundaries for random walks obtained from diffuse measures, Israel journal of mathematics 86.1-3 (1994): 107-156.