Paschke's conjecture for the endpoint anisotropic series representations of the free group

J. Aust. Math. Soc. 74 (2003), 173-183

Paschke's conjecture for the endpoint anisotropic series representations of the free group

M. Gabriella Kuhn
Dipartimento di Matematica
Università di Milano ``Bicocca''
Viale Sarca 202
20126 Milano
Italia
kuhn@matapp.unimib.it

and

Tim Steger
Struttura di Matematica e Fisica
Università di Sassari
Via Vienna 2
07100 Sassari
Italia
steger@ssmain.uniss.it

Abstract

Let $\Gamma$ be a free noncommutative group with free generating set

. Let $\mu\in\ell^1(\Gamma)$ be real, symmetric, nonnegative and suppose that $\operatorname{supp}(\mu)=A_+\cup A_+^{-1}$ . Let $\lambda$ be an endpoint of the spectrum of $\mu$ considered as a convolver on $\ell^2(\Gamma)$ . Then $\lambda-\mu$ is in the left kernel of exactly one pure state of the reduced $C^*_{\text{reg}}(\Gamma)$ ; in particular, Paschke's conjecture holds for $\lambda-\mu$ .

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