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The spectrum of the algebra generated by two-sided convolutions on the Heisenberg group and by operators of multiplication by continuous functions. (English. Russian original) Zbl 0853.22005
Russ. Acad. Sci., Dokl., Math. 50, No. 1, 93-96 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 337, No. 4, 439-441 (1994).
Let $$\mathbb{H}^n$$ be the $$(2n + 1)$$-dimensional Heisenberg group. It has a group of dilations $$\{\delta_\tau; \tau \in \mathbb{R}^+\}$$. A function $$f$$ on $$\mathbb{H}^n$$ is homogeneous of degree $$k$$ if $$f(\delta_\tau (g)) = \tau^k f(g)$$. Let $${\mathcal S}$$ be the algebra generated by the right convolution operators on $$L^2 (\mathbb{H}^n)$$ with homogeneous functions of degree $$- 2n - 2$$ together with the multiplication operators. In this note, the author gives a complete description – without proofs – of the spectrum of $${\mathcal S}$$ as well as of its Jacobson topology.
Reviewer: M.B.Bekka (Metz)

##### MSC:
 22E25 Nilpotent and solvable Lie groups 43A80 Analysis on other specific Lie groups