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Let $\pi$ be the regular representation of $G=\operatorname{SL}_2(R)$ on $L^2(G)$. Let $M$ be the (commutative) convolution algebra generated by measures of the form $m_K * \delta_g * m_K$ where $m_K$ is Haar measure on $K$ where $K=\operatorname{SO}(2)$. Define a homomorphism $\pi'$ from $M$ to the bounded operators on $L^2(G)$ by $\pi'(\mu)= \int \pi(g) d\mu(g)$. Then the norm closure of the image of $\pi'$ is a commutative C*-algebra of operators on $L^2(G)$. What is the spectrum of this algebra? In other words, what are the nontrivial multiplicative linear functionals?

Each character pulls back via $\pi$ to a character on $M$, and the characters of $M$ are well-known. They correspond to irreducible representations of $G$ with K-fixed vectors. The spherical principal series appear in the decomposition of $\pi$ into irreducibles, but the complementary series do not. Is it possible for a character associated to a complementary series repn to be the pull-back of a character on the closure of $\pi'(M)$?

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  • $\begingroup$ I do not have an answer to your question, let alone a reference, but while re-reading it I found myself a bit puzzled by / sceptical of your claim that the characters of $M$ are parametrized by a subset of the irreps of SL(2,R). The reason is that if I consider instead the measure algebra $M({\bf T})$, say, then there are many more characters on $M({\bf T})$ than the ones coming from the dual group ${\bf Z}$. Is there some phenomenon whereby SO(2)-bi-invariant measures on SL(2,R) are forced to be absolutely continuous? $\endgroup$
    – Yemon Choi
    Commented Nov 26 at 0:21
  • $\begingroup$ Also: you talk about the spectrum of $\pi'(M)$, but I usually only see that term reserved for commutative Banach algebras, and $\pi'(M)$ is probably not norm-closed in $B(L^2(G))$. One can have multiplicative linear functionals on dense star-subalgebras of commutative Cstar algebras, that do not extend continuously to the containing Cstar algebra: consider the algebra ${\mathbb C}[z,z^{-1}]$ viewed as a subalgebra of $C({\mathbb T})$. $\endgroup$
    – Yemon Choi
    Commented Nov 26 at 0:37
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    $\begingroup$ @YemonChoi Thanks for your comments! I have edited the question to address these comments. I have restricted $M$ to be the subalgebra generated $m_K*\delta_g* m_K$ which is all that I really want. A character $\chi$ on this subalgebra is determined by its values on the generators, and one can show that $g \to \chi(m_K *\delta_g*m_K)$ is a spherical function on $G$ and these are in one-to-one correspondence with irreps of $G$. (I wonder, however, whether this subalgebra is dense in the space of all b-$K$-invariant measures on $G$.) $\endgroup$
    – Chris Judge
    Commented Nov 26 at 13:54

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