On the operator-valued analogues of the semicircle, arcsine and bernoulli laws

S. T. Belinschi, M. Popa, V. Vinnikov

Research output: Contribution to journalArticlepeer-review


We study the connection between operator valued central limits for monotone, Boolean and free probability theory, which we shall call the arcsine, Bernoulli and semicircle distributions, respectively. In scalar-valued noncommutative probability these distributions are known to satisfy certain arithmetic relations with respect to Boolean and free convolutions. We show that, generally, the corresponding operator-valued distributions satisfy the same relations only when we consider them in the fully matricial sense introduced by Voiculescu. In addition, we provide a combinatorial description in terms of moments of the operator valued arcsine distribution and we show that its reciprocal Cauchy transform satisfies a version of the Abel equation similar to the one satisfied in the scalar-valued case.

Original languageAmerican English
Pages (from-to)239-258
Number of pages20
JournalJournal of Operator Theory
Issue number1
StatePublished - 20 Aug 2013


  • Free and Boolean convolutions
  • Generalized Cauchy transform
  • Operator-valued distributions

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


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